Locally convex spaces over non-archimedean valued fields

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 119 EDITORIAL BOARD ´ B. BOLLOBAS, W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK,...

Author: C. Perez-Garcia | W. H. Schikhof

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CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 119 EDITORIAL BOARD ´ B. BOLLOBAS, W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO

LOCALLY CONVEX SPACES OVER NON-ARCHIMEDEAN VALUED FIELDS Non-Archimedean functional analysis, where alternative but equally valid number systems such as p-adic numbers are fundamental, is a fast-growing discipline widely used not just within pure mathematics, but also applied in other sciences, including physics, biology and chemistry. This book is the first to provide a comprehensive treatment of non-Archimedean locally convex spaces. The authors provide a clear exposition of the basic theory, together with complete proofs and new results from the latest research. A guide to the many illustrative examples provided, end-of-chapter notes and glossary of terms all make this book easily accessible to beginners at the graduate level, as well as specialists from a variety of disciplines. C. P e r e z -Ga r c i a is a Professor in the Department of Mathematics, Statistics and Computation at the University of Cantabria, Spain. W. H. Sc h i k h o f worked as a Professor at Radboud University Nijmegen, the Netherlands for 40 years.

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Locally Convex Spaces over Non-Archimedean Valued Fields C. PEREZ-GARCIA W. H. SCHIKHOF

cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521192439 C

C. Perez-Garcia and W. H. Schikhof 2010

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2010 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library ISBN 978-0-521-19243-9 Hardback

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To my family and to my friend Nicole -C.P.G.

Contents

Preface

page xi

1 1.1 1.2 1.3

Ultrametrics and valuations Ultrametric spaces Ultrametric fields Notes

1 1 5 12

2 2.1 2.2 2.3 2.4 2.5 2.6

Normed spaces Basics Orthogonality Spaces of countable type The absence of Hilbert space Examples of Banach spaces Notes

14 15 22 28 37 41 78

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10

Locally convex spaces Seminorms and convexity Absolutely convex sets of countable type Definition of a locally convex space Basic facts and constructions Metrizable and Fr´echet spaces Bounded sets Examples of locally convex spaces Compactoids Compactoidity vs orthogonality Characterization of compactoids in normed spaces by means of t-frames 3.11 Notes

vii

81 83 90 91 97 108 112 114 143 158 165 168

viii

Contents

4 The Hahn–Banach Theorem 4.1 A first Hahn–Banach Theorem: spherically complete scalar fields 4.2 A second Hahn–Banach Theorem: spaces of countable type 4.3 Examples of spaces (strictly) of countable type 4.4 A third Hahn–Banach Theorem: polar spaces 4.5 Notes

170 171 175 182 192 198

5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

The weak topology Weak topologies and dual-separating spaces Weakly closed convex sets Weak topologies and spaces of finite type Weakly bounded sets Weakly convergent sequences Weakly (pre)compact sets and “orthogonality” Admissible topologies and the Mackey topology Notes

210 210 213 218 224 228 232 235 239

6 6.1 6.2 6.3

C-compactness Basics Permanence properties Notes

244 244 250 252

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7

Barrelledness and reflexivity Polar barrelledness, hereditary properties Examples of (polarly) barrelled spaces The weak star and the strong topology on the dual Reflexivity Examples of reflexive spaces Metrizability considerations in duality theory Notes

254 255 262 272 275 282 294 300

8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8

Montel and nuclear spaces Compactoid operators Intermezzo: a curious property of ∞ Compactifying operators (Semi-)Montel spaces Nuclear spaces Semi-Montelness, nuclearity and metrizability Examples of (semi-)Montel and nuclear spaces Notes

301 302 308 309 311 320 324 328 334

Contents

9 9.1 9.2 9.3 9.4 9.5 9.6

Spaces with an “orthogonal” base Bases in locally convex spaces Spaces with an “orthogonal” base Fr´echet spaces with an “orthogonal” base Perfect sequence spaces K¨othe sequence spaces Barrelledness, reflexivity, Montelness and nuclearity of sequence spaces 9.7 Spaces of analytic functions 9.8 Basic counterexamples 9.9 Notes

ix

337 338 340 347 350 355 360 363 366 369

10 10.1 10.2 10.3 10.4 10.5 10.6 10.7

Tensor products The algebraic tensor product Algebraic tensor products, where the scalar field is valued Tensor products of locally convex spaces Tensor products of nuclear and semi-Montel spaces Examples of tensor products Non-Archimedean complexifications Notes

373 374 377 383 388 396 404 405

11 11.1 11.2 11.3 11.4 11.5 11.6

Inductive limits Basic facts and examples Stability properties of inductive limits Compactoid inductive limits Inductive topologies on sequence spaces Compactoid sets in inductive limits Notes

409 410 417 424 428 435 437

Appendix A Glossary of terms A.1 Sets A.2 Real numbers A.3 Groups, rings and fields A.4 Vector spaces A.5 Topological spaces A.6 Metric spaces A.7 Topological vector spaces

442 442 443 444 445 446 448 449

Appendix B Guide to the examples B.1 Spaces of continuous functions B.2 Spaces of differentiable functions

451 451 452

x

Contents

B.3 Spaces of analytic functions B.4 Valued field extensions B.5 Sequence spaces Notation References Index

452 452 452 453 457 468

Preface

Aim This book presents the basics of locally convex theory over a field K with a nonArchimedean valuation | . | : K−→[0, ∞) (see 1.2.3).1 The most important example of such a K is the field of the p-adic numbers (1.2.7). The strong triangle inequality |λ + µ| ≤ max(|λ| , |µ|) is the major difference between | . | and the absolute value function on the field of real numbers R and the field of complex numbers C. Likewise, the defining seminorms of our locally convex spaces will satisfy the strong triangle inequality. The book is self-contained in the sense that it does not require knowledge of any deep theory; only basic knowledge of (linear) algebra, analysis and topology are needed. It is intended for both (graduate) students and interested researchers in other areas, but is also of relevance for specialists.

History The founding father of non-Archimedean Functional Analysis was Monna, who wrote a series of papers in 1943 (see [152]–[155]). Over the years a wellestablished discipline developed, reflected in the 2000 Mathematics Subject Classifications 46S10 and 47S10 of the Mathematical Reviews. A milestone was reached in 1978 at the publication of van Rooij’s book [193], the most extensive treatment on non-Archimedean Banach spaces existing in the literature. In the meantime van Tiel had published his thesis [227] on non-Archimedean locally convex spaces. Both fundamental works still form a basis for new developments, and have been cited by many authors. We should also mention 1

Note that in this book all proofs, definitions, theorems, etc. are numbered decimally by chapter. They will be referred to by number only.

xi

xii

Preface

the proceedings of conferences on non-Archimedean analysis which were held every two years from 1990 onwards, [21], [88], [20], [209], [107], [135], [208], [45], [2], containing several publications on locally convex theory (we have listed the references in chronological order). The time had come for a volume on locally convex theory to appear and the present book hopefully provides an answer to this need. Our aim is to cover the fundamentals by setting up a general theory which allows a wide spectrum of subjects and examples. Our proof techniques are analytically oriented. In this context we would like to point out Schneider’s book [210]. Whereas our book is directed towards a rather general readership, Schneider, as he explains himself, had a different motivation, i.e., to offer a quick grasp to a reader working in other areas (such as number theory). Because of this, he allows for restrictions, for example working mainly over spherically complete fields. Also, the treatment in [210] has a more algebraic flavour. Despite these differences, Schneider’s book and ours are compatible and one can be used to complement the reading of the other.

Foreign affairs Complex numbers provide an excellent domain for forcing quadratic equations to have solutions. Likewise, the p-adic number field acts as a natural home for solving certain infinite systems of congruences, so it takes a prominent place in number theory. But also in other branches (such as algebraic geometry; representation of (Lie) groups; (several) complex variables; real analysis; even theoretical physics, see e.g. [34], [47] and [140]), one is more interested in the role non-Archimedean fields could play as fundamental objects. Researchers in those disciplines sometimes need a solid background in non-Archimedean locally convex spaces; we have mentioned Schneider’s reason for writing his book. This has also influenced the set up of our book, as we will later explain. The presence of the vast area of Functional Analysis over R or C, henceforth called “classical analysis” is, of course, a fruitful source of inspiration for the non-Archimedean case. However, the reader will notice that, in the course of the development of the theory, the non-Archimedean world is equally fascinating, and asks for a mathematical intuition of its own.

Book organization In Chapter 1 we present the basics of ultrametric spaces and valued fields. In Section 1.2 we quote several theorems on valued fields without proof (but with

Preface

xiii

references) as we feel that otherwise it would lead us too far away from our main track. For that matter, the results are hardly needed, but may serve to offer the reader an impression. Starting from Chapter 2 our policy changes: with obvious exceptions we present full proofs, as they really belong to the subject of the book. According to our experiences, in such cases not only the mere statements, but also their proofs, are needed in order to obtain understanding and intuition. We have also applied this philosophy to those parts in which the classical and the nonArchimedean theories seem to be similar. Not only does it facilitate a better grasp of the subtle differences but also serves those readers who may not be familiar with classical Functional Analysis theory (e.g. students and workers in other fields). In general, we hope that any reader, including the expert, will appreciate having the basic theory together with the proofs collected into a single volume. In compiling the scattered material in the literature we were often able to simplify and tidy up the original proofs of the inventors; we hope that it will add to the value of the book. The same process revealed several natural questions that have not been touched upon before. We have tried to the best of our ability to fill in those gaps. Consequently this book contains quite a few new results. To illustrate the theory we have included examples, mainly playing around spaces of continuous, analytic, and differentiable (C n , C ∞ ) K-valued functions. The reader will notice that these themes return in every chapter, connecting them with the newly developed subject. For the reader’s convenience we include a guide to the examples at the end of the book. Formally, the examples are independent of the main theory, so the reader can choose to pick his or her favourites; only occasionally will we use some space to provide a counterexample to a conjecture. We would like to point out that our examples do not cover all the present-day knowledge of these spaces. For this and related theories one should consult the following references. Spaces of continuous functions were studied by e.g. Aguayo, De Grande-De Kimpe, Katsaras, Mart´ınez-Maurica, Navarro and Perez-Garcia ([3], [5], [86], [87], [113], [114], [115], [120], [122], [123], [124], [149], [176]). Background on general theory of analytic functions was given by Escassut, [53], Robert, [190] and Robba and Christol, [188], [189]. The last reference also contains some locally convex examples and some applications to p-adic differential equations, as do [35] and [36]. More on locally convex theory of analytic functions can be found in [68], [78] and [82]. For C n (C ∞ )-functions, see [195] for general theory, and the works of De Grande-De Kimpe, Khrennikov, Navarro, Schikhof and van Hamme ([70], [83], [165], [206], [207]) for locally convex aspects. Generalizations (several variables, more abstract settings) were

xiv

Preface

treated by De Smedt ([223], [224], [225]) and by Bertram, Gl¨ockner and Neeb ([28], [61]). Throughout the book we have provided, for convenience, many crossreferences. This way the reader can have easy access to related results that have been treated earlier, and a link to developments later on in the book. The notes at the end of each chapter contain some history, references and comments. When mentioning results that are not covered by the book no completeness is pretended. In the glossary of terms (Appendix A) we explain a few concepts, terminology, notations, etc., used freely in the book, but that may be not familiar to all readers. In the guide to the examples (Appendix B) we list the most important examples treated in the book and indicate where their properties can be found. Finally, we wish to thank the Ministerio de Educaci´on y Ciencia of Spain (MTM2006-14786), for partially supporting the research concerning new results that are presented in this book.

1 Ultrametrics and valuations

Ultrametric spaces will form the building blocks of the locally convex spaces to be treated in this book, whereas valued fields will act as their scalar fields. We assume that the reader has a basic knowledge of ultrametrics and valuations; in this short chapter we will recall briefly some fundamentals that will be needed later on.

1.1 Ultrametric spaces Let X be a set. A metric on X assigns to every ordered pair (x, y) ∈ X × X a nonnegative real number d(x, y) such that: (i) d(x, y) = 0 if and only if x = y, (ii) d(x, y) = d(y, x), (iii) d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality), for all x, y, z ∈ X. The pair (X, d) is called a metric space. We often write X instead of (X, d). Let (X, d) be a metric space. Let a ∈ X, r > 0. The set B(a, r) := {x ∈ X : d(x, a) ≤ r} is called the closed ball with radius r about a. (Indeed, B(a, r) is closed in the induced topology.) Similarly, B(a, r − ) := {x ∈ X : d(x, a) < r} is called the open ball with radius r about a. In cases where we want to specify X in the notation, we shall write BX (a, r) (BX (a, r − )) rather than B(a, r) (B(a, r − )). It is also customary to call the point a the centre of the balls B(a, r), B(a, r − ). This will cause no harm as long as 1

2

Ultrametrics and valuations

we keep in mind that a ball may have more than one centre (and more than one radius for that matter), see 1.1.2(b). For a non-empty set Y ⊂ X its diameter is diam Y := sup{d(x, y) : x, y ∈ Y } (possibly ∞). Analogously, the distance between two non-empty sets Y, Z ⊂ X is dist (Y, Z) := inf{d(y, z) : y ∈ Y, z ∈ Z}. For a ∈ X and Y ⊂ X, instead of dist ({a}, Y ) we write dist (a, Y ). Occasionally we shall deal with semi-metrics, i.e., cases where condition (i) above is weakened to “d(x, x) = 0”. Then the induced topology need not be Hausdorff. To avoid confusion we will use in this context sometimes expressions we left undefined (such as: Y is d-dense in X) that yet will be clear to the reader. The (semi-)metrics featuring in this book are of a special kind. Definition 1.1.1 Let X = (X, d) be a metric space. X is called an ultrametric space, and d is called an ultrametric, if d satisfies the strong triangle inequality: (iii) d(x, z) ≤ max(d(x, y), d(y, z)), for all x, y, z ∈ X. Clearly (iii) is stronger than (iii). At this point many examples of ultrametric spaces would fit here. However, most of them will be treated later on in their own right, so here we shall content ourselves with two examples. Examples 1.1.2 (a) (A typical one: the p-adic metric) Let p be a fixed prime. For m, n ∈ Z, put d(m, n) := 0 if m = n and, for m = n, d(m, n) := p−r if r is the largest nonnegative integer such that pr divides m − n. One checks easily that d is an ultrametric on Z. It may be worth mentioning that for a ∈ {0, 1, . . . , p − 1} we have B(a, 1− ) = B(a, p−1 ) = {n ∈ Z : n ≡ a (mod p)}. Similarly, for m ∈ N, a ∈ {0, 1, . . . , pm − 1} we have B(a, p −m ) = {n ∈ Z : n ≡ a (mod pm )}. Thus, balls have a numbertheoretic interpretation. Further, observe that the set d(Z × Z) of values of d equals {1, p −1 , p −2 , . . .} ∪ {0}. The induced topology is non-discrete. In fact, limn n! = 0. (b) (An extreme one: the trivial metric) Let X be any set and for x, y ∈ X put 0 if x = y d(x, y) := 1 if x = y.

1.1 Ultrametric spaces

3

Then d is an ultrametric, called the trivial metric. It clearly induces the discrete topology. Notice that, for any a ∈ X and for any r ≥ 1, X = B(a, r). So X is a ball with infinitely many radii, and every point of X serves as a centre. Below we will state several basic facts on ultrametric spaces. Though simple to derive they are fundamental for all that follows. In the remaining part of this section X = (X, d) is an ultrametric space. 1. Every point of a ball is a centre. Proof. Consider the closed ball B(a, r). Let b ∈ B(a, r) and choose x ∈ B(b, r). Then d(x, a) ≤ max(d(x, b), d(b, a)) ≤ r, so x ∈ B(a, r). By symmetry we have B(a, r) = B(b, r). A similar proof works for open balls. 2. Every ball is open and closed in the topological sense. Proof. That a closed ball is open follows from 1. Now let B be an open ball with radius r. The requirement “x ∼ y if and only if d(x, y) < r” defines an equivalence relation on X whose classes are the open balls with radius r. Since B is among them it follows that X \ B, being a union of open balls, is open, so B is closed. 3. (Mercury drop behaviour) Two balls are either disjoint, or one is contained in the other. Proof. Let B1 , B2 be two closed balls with radius r1 , r2 respectively and let a ∈ B1 ∩ B2 . If r1 ≤ r2 we have by 1, B1 = B(a, r1 ) ⊂ B(a, r2 ) = B2 . A similar proof works for two open balls and for the case of an open and a closed ball. 4. (Isosceles triangle principle) Let x, y, z ∈ X. Then among the numbers d(x, z), d(x, y), d(y, z) the largest and second largest are equal. In other words, if d(x, y) = d(y, z) then d(x, z) = max(d(x, y), d(y, z)). Proof. Suppose d(x, y) < d(y, z). Then d(x, z) ≤ max(d(x, y), d(y, z)) = d(y, z) but also, since d(y, z) is not ≤ d(x, y), d(y, z) ≤ max(d(x, y), d(x, z)) = d(x, z), and we are done.

4

Ultrametrics and valuations

5. (Equal distances between points of disjoint balls) Let B be a ball, a ∈ X, a ∈ B. Then d(a, x) = d(a, y) for each x, y ∈ B. More generally, let B1 , B2 be disjoint balls. Then, for x ∈ B1 and y ∈ B2 , the number d(x, y) is constant. We leave the proof to the reader. 6. (Some other immediate consequences) The topology of X has a base of sets that are clopen, i.e., it is zero-dimensional and, by consequence, totally disconnected. For a ∈ X, r > 0, the “sphere” {x ∈ X : d(x, a) = r} is clopen and, unless it is empty, not the boundary of B(a, r). Another implication of the strong triangle inequality is that, to find the diameter of a non-empty set Y ⊂ X, it suffices to take an arbitrary a ∈ Y and compute sup{d(a, y) : y ∈ Y }. Also, a useful feature is the fact that if B is a ball in X and Y is a subset of X that meets B, then B ∩ Y is a ball in the metric space Y . 7. (No new values of the metric after completion) Let x1 , x2 , . . . be a sequence in X converging to x ∈ X. Suppose a ∈ X, a = x. Then d(xn , a) = d(x, a) for large n. This follows easily from the isosceles triangle principle. The metric completion of X is again an ultrametric space, denoted X ∧ , with the metric on X ∧ also denoted by d. By the above remark we have d(X × X) = d(X ∧ × X∧ ). 8. Finally, a sequence x1 , x2 , . . . in X is a Cauchy sequence if and only if limn d(xn+1 , xn ) = 0. This follows from d(xm , xn ) ≤ max(d(xm , xm−1 ), . . . , d(xn+1 , xn ))

(m > n).

For later use we include the following result. Theorem 1.1.3 If X is locally compact then there is a partition of X consisting of compact balls. Proof. Choose r1 , r2 , . . . ∈ R with r1 > r2 > · · · , limn rn = 0. Let B be the collection of all closed balls in X with radius in {r1 , r2 , . . .} that are compact. Then B is a covering of X by compact balls. Every member of B is contained in a maximal member of B. These maximal balls form the desired partition. The following concept will play a key role in this book. Definition 1.1.4 X is called spherically complete if each nested sequence of balls B1 ⊃ B2 ⊃ · · · has a non-empty intersection. We obtain equivalent definitions if in the above we require all Bn to be closed balls, or all Bn to be open balls.

1.2 Ultrametric fields

5

It might be interesting to note that (ordinary) completeness of X amounts to “each nested sequence B1 ⊃ B2 ⊃ · · · of balls for which limn diam Bn = 0 has a non-empty intersection”. Thus, spherical completeness implies completeness. The converse is not true, which might seem surprising at first sight; for examples see 1.2.12 and 1.2.14. Spherical completeness has to do with best approximations. Definition 1.1.5 Let Y ⊂ X, a ∈ X. We say that a has a best approximation in Y if dist (a, Y ) is attained, i.e., min{d(a, y) : y ∈ Y } exists. Points in Y having this minimal distance to a are called best approximations (of a in Y ). Note that in general there is more than one best approximation. Theorem 1.1.6 Let X = ∅ be spherically complete. Let Z be an ultrametric space containing X as a subspace. Then each z ∈ Z has a best approximation in X. Proof. There are r1 > r2 > · · · with limn rn = dist (z, X). Each one of the balls Bn := {x ∈ Z : d(x, z) ≤ rn } has non-empty intersection with X, so B1 ∩ X ⊃ B2 ∩ X ⊃ · · · is a nested sequence of balls in X. Thus, there is an a ∈ n∈N (Bn ∩ X). Clearly d(z, a) = dist (z, X).

1.2 Ultrametric fields All fields appearing in this book are commutative. A valuation on a field K is a map | . | : K−→[0, ∞) such that: (i) |λ| = 0 if and only if λ = 0, (ii) |λ µ| = |λ| |µ| (multiplicativity), (iii) |λ + µ| ≤ |λ| + |µ| (triangle inequality), for all λ, µ ∈ K. The pair (K, | . |) is called a valued field. We often write K instead −1 of (K, | . |). Note that, for all λ, µ ∈ K, µ = 0, we have |−λ| = |λ|, λ µ = |λ| / |µ| and |1| = 1 (here we adapt the bad habit to use the same symbol 1 for the unit element of K and of R; in fact, in (i) we did the same thing for the symbol 0). Let K be a valued field. Then the map (λ, µ) → |λ − µ| is a metric on K which induces a topology for which K is a topological field (i.e., addition, subtraction, multiplication and division are continuous). The metric completion of K is in a natural way a valued field which we denote by K ∧ (the unique continuous extension of the valuation is again usually denoted by | . |).

6

Ultrametrics and valuations

The (closed) unit disk (or ball) is the set {λ ∈ K : |λ| ≤ 1} = BK (0, 1), often abbreviated by BK . Similarly, the open unit disk (or ball) is BK− := {λ ∈ K : |λ| < 1} = BK (0, 1− ). The set BK \BK− is called the unit sphere. Clearly R, C, with the absolute value function, henceforth denoted | . |∞ , are examples of complete valued fields. (Q, | . |∞ ) is a non-complete valued field with completion (R, | . |∞ ). Definition 1.2.1 Two valuations on a field are called equivalent if they induce the same topology. √ For example, the map z → |z|∞ is a valuation on C that is equivalent to | . |∞ . Theorem 1.2.2 Two valuations | . |1 , | . |2 on a field K are equivalent if and only if there is an s > 0 such that | . |2 = | . |s1 . Proof. One direction is obvious; we sketch the proof of the “only if”. For a valued field, the open unit disk is precisely the set {λ ∈ K : limn λn = 0}, so | . |1 and | . |2 have the same open unit disk. By looking at K \ BK we see that | . |1 and | . |2 also have the same closed unit disk. Then also their unit spheres are the same. From mutiplicativity one infers the existence of a function φ for which |λ|2 = φ(|λ|1 ) for all λ ∈ K. From here it is an easy exercise to show that φ is a power function. In this book we focus on valued fields that are ultrametric. Definition 1.2.3 Let K = (K, | . |) be a valued field. The valuation | . | is called non-Archimedean, and K is called a non-Archimedean valued field if | . | satisfies the strong triangle inequality: (iii) |λ + µ| ≤ max(|λ| , |µ|), for all λ, µ ∈ K. It is not our purpose to present here a complete theory of (non-Archimedean) valued fields. Instead we shall recall the most important facts and examples together to form either a refresher or a crash course, depending on the reader’s point of view. One of the facts that we will use frequently is the identity (1 − n λ)−1 = ∞ n=0 λ for λ ∈ K, |λ| < 1, where K is a non-Archimedean valued field. We leave the proof of this familiar formula to the reader. Let (K, | . |) be a non-Archimedean valued field. Then, for each n ∈ N, |n .1 | = |1 + 1 + · · · + 1| ≤ max(|1| , |1| , . . . , |1|) = 1, showing that the multiples of the unit element of K form a bounded set. This is in contrast with the “axiom of Archimedes” stating that N is unbounded in R and, at the same time, it explains the term “non-Archimedean”.

1.2 Ultrametric fields

7

The isosceles triangle principle in K, “if |λ| = |µ| then |λ + µ| = max(|λ| , |µ|)”, shows that if λ1 , λ2 , . . . is a sequence in K converging to λ = 0 then |λn | = |λ| for large n. So no new values are added after completion. We set |K| K × := K \ {0}, the multiplicative group := {|λ| : λ ∈ K} and × × of K. Also, K := {|λ| : λ ∈ K } is a multiplicative group of positive real numbers, the value group of K. There are two possibilities: (i) 1 is not an accumulation point of K × . Then K × is a discrete subset of (0, ∞) and the valuation is called discrete. It may happen that K × = {1}. Then we are dealing with the trivial valuation given by |λ| :=

0 if λ = 0 1 if λ ∈ K × ,

whose associated metric is the trivial one (see 1.1.2(b)) and whose induced topology is discrete. If the valuation is not trivial but discrete then max{|λ| : λ ∈ BK− } exists, − so there ×is a ρ ∈ nBK such that |K| ∩ (|ρ| , 1) = ∅. It is not difficult to see that K = {|ρ| : n ∈ Z}. We will call such a ρ a uniformizing element. (ii) 1 is an accumulation point of K × . Then K × is a dense subset of (0, ∞) and the valuation is called dense. A second natural object that can be assigned to every non-Archimedean valued field is defined as follows. The unit disk BK of K is not only multiplicatively, but, due to the strong triangle inequality, also additively closed. Thus, BK is a commutative ring with identity. The open unit disk BK− is easily seen to be an ideal in BK and, since each element of BK \BK− is invertible, even maximal. Thus, BK /BK− is a field, called the residue class field of K, customarily denoted by k. The canonical map BK −→k is mostly written λ → λ. Now let L be a valued field extension of K (i.e., a non-Archimedean valued field containing K as a valued subfield) with residue class fields l and k respectively. Consider the diagram BK

/ BL

k

l

where the vertical arrows represent the canonical maps and the horizontal one is the inclusion. It is easily seen that there exists precisely one map φ : k−→l

8

Ultrametrics and valuations

making the diagram BK

/ BL

k

/l

φ

commute. Clearly φ is an injective ring homomorphism with φ(1) = 1. We call this φ the natural map of k into l. Thus, we have that the value group of K is a subgroup of the value group of L and that the residue class field of K is naturally embedded into the residue class field of L. The next result is well known. Theorem 1.2.4 ([193], 1.5) Let K be a non-Archimedean valued field that is algebraically closed. Then its residue class field is again algebraically closed, its value group is divisible, and its valuation is either trivial or dense. Example 1.2.5 The p-adic valuation on Q Let p be a prime number. Define a real-valued function | . |p on Z by |0|p := 0 and, for n = 0, |n|p := p −r(n) , where r(n) is the largest nonnegative integer such that pr(n) divides n (so, in fact |n|p = d(n, 0) of 1.1.2(a)). For a rational number mn (m = 0), set n |n|p . := |m|p m p Direct verification shows that in this way a non-Archimedean valuation | . |p is defined on the field Q of rational numbers. | . |p is called the p-adic valuation on Q. We will show that – essentially – there are no other non-Archimedean valuations on Q. Theorem 1.2.6 Let | . | be a non-Archimedean valuation on Q. Then either | . | is trivial or | . | is equivalent to some p-adic valuation. Proof. Suppose | . | is not trivial. Then |n| = 1 for some n ∈ N. By the strong triangle inequality |n| ≤ 1. Hence {m ∈ N : |m| < 1} is non-empty, let p be its smallest element. Then p = 1. If p = m n for some m, n ∈ N, m < p, n < p then |m| = |n| = 1, so |p| = |m| |n| = 1, an impossibility. Thus, p is a prime number. If n ∈ N is not divisible by p there exist a ∈ {0, 1, 2, . . .} and

1.2 Ultrametric fields

9

r ∈ {1, . . . , p − 1} such that n = a p + r. Then |r| = 1 and, by the isosceles triangle principle, also |n| = 1. Hence, for each n ∈ N we have |n| = |p|r(n) , with r(n) as above. It follows easily that, for all λ ∈ Q, |λ| = |λ|sp , where s := −log |p| (log p)−1 . Q is not complete with respect to the metric induced by the p-adic valuation. (The shortest way to see it is by means of a Baire Category argument, using the countability of Q and the fact that (Q, | . |p ) has no isolated points.) Definition 1.2.7 The completion of (Q, | . |p ) is called Qp , the field of the p-adic numbers. The extended valuation on Qp is also denoted | . |p . Thus, in some natural sense, the fields Q2 , Q3 , . . . present themselves as alternatives to R: completions of Q, but with respect to the various p-adic valuations, which are obviously inequivalent. The value group of Qp is {pn : n ∈ Z}, so | . |p is discrete with uniformizing element p ∈ Qp . The unit disk is frequently denoted Zp instead of BQp , and it is called the ring of p-adic integers; it is the closure of Z in Qp . The residue class field of Qp is the field of p elements. n The fact that (1 − p)−1 = ∞ n=0 p shows that there is no linear ordering in Qp that behaves decently with respect to algebraic structure and topology. Qp is not algebraically closed. An easy way to see this is to consider the equation x 2 = p. Any solution x would have the p-adic valuation p −1/2 , but this number is not in the value group. So x 2 = p has no solutions in Qp . By considering the equation x n = p for any n > 2 we infer that the algebraic closure Qap must be of infinite degree over Qp . Qp is locally compact, i.e., each point has a compact neighbourhood. In fact each ball in Qp , in particular Zp , is compact. This is a special case of the following. Theorem 1.2.8 ([195], 12.2) Let K be a non-trivially non-Archimedean valued complete field. Then K is locally compact if and only if the valuation is discrete and the residue class field is finite. The following lemma will be needed in 2.5.22 and 2.5.34. Lemma 1.2.9 Let K be a non-trivially non-Archimedean valued field. Then K contains a closed subfield K1 that has a non-trivial discrete valuation. Proof. Let K0 be the prime field of K. If it is non-trivially valued then K0 = Q and the valuation on it is equivalent to a p-adic one (1.2.6), so is discrete and we can take K1 := K 0 .

10

Ultrametrics and valuations

If K0 is trivially valued, choose a ∈ K, 0 < |a| < 1. By using the isosceles triangle principle we see that, for λ0 , λ1 , . . . , λn ∈ K0 , λ0 + λ1 a + · · · + λn a n = max λi a i . 0≤i≤n

f , g

So the elements where f, g are polynomials in a with coefficients in K0 , g = 0 form a valued subfield K0 (a) of K with discrete value group {|a|n : n ∈ Z}, and we can take K1 := K0 (a). Before giving the next example we first quote an extension theorem. Theorem 1.2.10 ([195], 14.1, 14.2) Let (K, | . |) be a non-Archimedean valued field. Let L be a field extension of K, i.e., L is a field containing K as a subfield. Then | . | can be extended to a non-Archimedean valuation on L. If, moreover, K is complete and L is of finite degree over K (i.e., L is finite-dimensional as a K-vector space) then this extension is unique. Example 1.2.11 The field C p of the p-adic complex numbers Each point of the algebraic closure Qap of Qp lies in a field extension of Qp of finite degree over it, so by the previous theorem there is precisely one nonArchimedean valuation on Qap that extends | . |p . Unfortunately, Qap is no longer complete ([195], 16.6), so we define Cp := (Qap )∧ . Fortunately, the completion of an algebraically closed field is again algebraically closed ([195], 17.1). So Cp is in some sense the smallest complete algebraically closed extension of Qp , hence deserves the name field of the p-adic complex numbers. We will denote the extension of the p-adic valuation to Cp again by | . |p . We now quote some facts on Cp . Theorem 1.2.12 ([195], 17.2, 20.6) The value group of Cp is {p r : r ∈ Q}, its residue class field is the algebraic closure of the field of p elements. So the valuation is dense and the residue class field is infinite. Cp is not locally compact, but still separable, complete but not spherically complete. That the field of the p-adic numbers is spherically complete follows from the next theorem. Theorem 1.2.13 If K is a discretely valued complete non-Archimedean field then K is spherically complete. Proof. Let B1 ⊃ B2 ⊃ · · · be a nested sequence of balls in K, let dn := × diam Bn . Then d1 ≥ d2 ≥ · · · and dn ∈ K for each n. By discreteness either the sequence B1 , B2 , . . . becomes stationary or limn dn = 0. For both cases (for the last one use completeness) we find n∈N Bn = ∅.

1.2 Ultrametric fields

11

Remark 1.2.14 A similar proof works for any non-empty closed subset of K. However, if K is complete with respect to a dense valuation then K has closed subsets that are complete but not spherically complete. Indeed, take X := K \ BK . Choose λ1 , λ2 , . . . ∈ K such that |λ1 | > |λ2 | > . . . and limn |λn | = 1. Then n → BX (λn , |λn |) is a nested sequence of balls and n∈N BX (λn , |λn |) = ∅. Next we quote the existence of a “spherical completion” of K. Definition 1.2.15 Let (K, | . |) be a non-Archimedean valued field. A valued field extension (L, | . |) of (K, | . |) is called immediate if the value groups of K and L are the same and their residue class fields are naturally isomorphic. (K, | . |) is called maximal if it has no proper immediate extensions. It is easy to see that the extension is immediate if and only if for each non-zero a ∈ L there is a λ ∈ K with |a − λ| < |a|. Theorem 1.2.16 ([193], 4.47, 4.49) Let K = (K, | . |) be a non-Archimedean valued field. Then we have the following: (i) K is maximal if and only if K is spherically complete. (ii) K has an immediate extension that is maximal. Now let L1 and L2 be two maximal immediate extensions of K. One can prove that there is a bijective K-linear isometry L1 −→L2 that leaves K pointwise fixed, but we cannot always choose this map to be a field homomorphism ([193], 4.59). Despite this we shall denote any maximal immediate extension of K by K ∨ , and even call K ∨ the spherical completion of K! This will be harmless as long as the properties of K ∨ we will be interested in have to do with | . | and the structure of K ∨ as a K-vector space. Non-Archimedean valued fields admit a natural version of the well-known Weierstrass Theorem, which we will use in the sequel several times. Theorem 1.2.17 ([195], 43.3) Let K = (K, | . |) be a non-Archimedean valued field, let X be a compact subset of K, let f : X−→K be a continuous function, and let ε > 0. Then there exists a polynomial function P : K−→K such that |f (x) − P (x)| < ε for all x ∈ X. The remaining part of this section is not needed for the sequel but has been included to serve as some background account. We first present an example of a non-Archimedean valued field of different nature. Let L(X) be the field of rational functions in one indeterminate X over

12

Ultrametrics and valuations

a field L. Let h ∈ L(X), h = fg , where f, g are polynomials in X, g = 0. Put 0 if f = 0 |h| := 2degree f −degree g if f = 0. One can easily check that | . | defines a non-trivial non-Archimedean valuation on L(X). By choosing for L a finite field we obtain that L(X) has non-zero characteristic. Thus, with the extension theorem 1.2.10 it is possible to construct many examples of non-Archimedean valued fields of arbitrary characteristic. The following theorem says roughly that, except for the absolute value | . |∞ , all valuations are non-Archimedean. Theorem 1.2.18 ([18], p. 127) Let (K, | . |) be a valued field. Then either: (i) there is an equivalent valuation on K for which K is isometrically isomorphic to a valued subfield of (C, | . |∞ ), or (ii) the valuation | . | is non-Archimedean. The next two theorems also demonstrate the abundance of non-Archimedean valued fields. Theorem 1.2.19 ([193], 1.3, 1.J, 2.G) Let k be any field, let be any multiplicative subgroup of (0, ∞). Then there exists a non-Archimedean valued field K whose value group is and whose residue class field is isomorphic to k. This K can be chosen to be (spherically) complete. Theorem 1.2.20 ([194], Theorem 1) Every ultrametric space can isometrically be embedded into a suitable non-Archimedean valued field. Theorem 1.2.21 ([193], 2.2) Let K be a non-trivially non-Archimedean valued complete field. Then every compact ultrametric space can homeomorphically be embedded into K.

1.3 Notes We will briefly describe the so-called Krull valuations; these are generalizations of our non-Archimedean ones. Inspection of the axioms for valuations shows that, by passing from the ordinary to the strong triangle inequality, addition of real numbers is no longer involved, but only ordering. Of course multiplicativity is also required. This gives rise to the idea of allowing the range of the valuation to lie in an ordered group, rather than just (0, ∞). We give the formal definitions.

1.3 Notes

13

Let G be a multiplicatively written linearly ordered abelian group (i.e., there is a linear ordering < on G satisfying g1 < g2 =⇒ h g1 < h g2 for all g1 , g2 , h ∈ G). Augment G by adjoining an element 0 and define 0 < g, 0 . g := g . 0 := 0 . 0 := 0 for all g ∈ G. Let K be a field. A Krull valuation with value group G is a surjective map | . | : K−→G ∪ {0} such that: (i) |λ| = 0 if and only if λ = 0, (ii) |λ µ| = |λ| |µ|, (iii) |λ + µ| ≤ max(|λ| , |µ|), for all λ, µ ∈ K. For systematic theory, see [58] for ordered groups and [230] for Krull valued fields. We here only mention the following. The map (λ, µ) → |λ − µ| behaves like an ultrametric in the sense that it induces a field topology in the usual way. However, the situation may be different from what we are used to. For example, the topology on K may not be metrizable and even when it is, it still may happen that 1, λ, λ2 , . . . does not tend to 0 for some λ ∈ K, |λ| < 1.

2 Normed spaces

From now on in this book, unless explicitly stated otherwise, K = (K, | . |) is a complete non-Archimedean valued field. The valuation | . | is assumed to be non-trivial. By imposing the strong triangle inequality x + y ≤ max(x, y) we will develop in this chapter the theory of normed K-vector spaces, roughly only inasmuch it is needed for locally convex theory. See [193] for a treatment of normed spaces in its own right. More on normed spaces will also appear in Sections 3.10 and 8.2 and in Subsection 7.5.1. Fundamentals, like finite-dimensional spaces, completions, Banach spaces, quotient spaces, the Open Mapping Theorem, are considered in Section 2.1, parallel to the classical set up. After that the theory is going to diverge from the classical one. A curious difference is already that the range of the norm function may be different from |K|, so that non-zero vectors cannot always be normalized, leading to the notion of a solid norm (2.1.10). Attempts to define Hilbert spaces over K, starting from an inner product will not lead to satisfactory results; we provide the details in Section 2.4. On the other hand, there is a notion of orthogonality (x ⊥ y if dist (x, Ky) = x) that is symmetric and works for every normed space. We study this in Section 2.2 and prove, unlike in the Hilbert space case, that orthogonality is stable under small perturbations. The concept of orthogonality will be an important tool throughout this book. In Section 2.3 we “linearize” the notion of separability to obtain the concept “of countable type”, useful for spaces over not necessarily separable K. These spaces will turn out to be more important than their classical counterparts. We will prove the curious facts that all infinite-dimensional Banach spaces of

14

2.1 Basics

15

countable type over K are linearly homeomorphic, and that closed subspaces have a closed complement. As for the examples in Section 2.5, the study of a valued field L ⊃ K, considered as a normed space over K (Subsection 2.5.4) is a typical nonArchimedean feature. Notice that, as an application, in 2.5.48 we construct an operator on a Banach space E over a non-spherically complete K having no closed invariant subspaces other that {0} and E. The other examples look familiar at first sight, but one has to keep in mind that the definitions of “analytic function” and “C n -function” are different from the classical ones. For convenience we provide most details that are needed to present the examples. References for deeper study were given in the preface to this book (in the section “Book organization”). Finally, the reason for excluding trivial valuations will become clear as soon as we have defined locally convex spaces, see the end of the notes to Chapter 3.

2.1 Basics Definition 2.1.1 Let E be a K-vector space. A norm on E is a map . : E−→[0, ∞) such that: (i) x = 0 if and only if x = 0, (ii) λ x = |λ| x, (iii) x + y ≤ max(x, y), for all x, y ∈ E, λ ∈ K. We call (E, . ) a K-normed space or a normed space over K. We frequently write E instead of (E, . ). E is called a K-Banach space or a Banach space over K if it is complete with respect to the induced ultrametric (x, y) → x − y. The closed unit ball of a K-normed space E is the set BE (0, 1) usually abbreviated by BE . Similarly we define the open unit ball BE− . Also, E := {x : x ∈ E}. From now on we often drop the prefix “K -” and write vector space, normed space, Banach space instead of K -vector space, K -normed space, K -Banach space respectively. The induced topology makes a normed space into a Hausdorff topological vector space (addition and scalar multiplication are continuous).

16

Normed spaces

Remark 2.1.2 Maps . : E−→[0, ∞) satisfying (i), (ii) and the ordinary triangle inequality x + y ≤ x + y (x, y ∈ E), but not (iii) are called Anorms in the literature. For example, (λ1 , λ2 ) → |λ1 |2 + |λ2 |2 is an A-norm on K 2 , whereas (λ1 , λ2 ) → max(|λ1 | , |λ2 |) is a norm. Although A-norms can be a source for interesting topological problems, all spaces that appear naturally in the duality theory of this book carry norms (or seminorms, see Chapter 3) in the sense of 2.1.1. So A-norms will not be our main concern. A sequence x1 , x2 , . . . in a normed space E is summable with sum x ∈ E if limn x − ni=1 xi = 0. We sometimes express summability of x1 , x2 , . . . as “ i xi exists” and write x = ∞ i=1 xi . Now let x1 , x2 , . . . be summable with sum x. From the fact that x − n i=1 xi ≤ sup{xn+1 , xn+2 , . . .} for all n ∈ N, we easily conclude that the summation is unconditional i.e., if σ is a permutation of N then xσ (1) , xσ (2) , . . . is summable, again with sum x. So here the situation is simpler than in the classical case. In the same non-Archimedean spirit we have the following. Theorem 2.1.3 (Completeness Criterion) A normed space is complete if and only if each sequence converging to 0 is summable. Proof. Left to the reader.

The completion of a normed space E is in a natural way again a normed space. We denote it by E ∧ . Theorem 2.1.4 Let E, F be normed spaces, let T : E−→F be a continuous linear map. Then T is Lipschitz and T extends uniquely to a continuous linear T ∧ : E ∧ −→F ∧ . Proof. By continuity of T at 0 there is a δ > 0 such that z ∈ E, z ≤ δ implies T (z) ≤ 1. Choose any ρ ∈ K, 0 < |ρ| < 1; we prove that T (x) − T (y) ≤ |ρ|−1 δ −1 x − y for all x, y ∈ E. To this end we may assume y = 0. Let x ∈ E, x = 0. There is an n ∈ Z such that |ρ|n x ≤ δ ≤ |ρ|n−1 x. From the first inequality we obtain T (ρ n x) ≤ 1, so by the second inequality, T (x) ≤ |ρ|−n ≤ |ρ|−1 δ −1 x. The rest of the proof is straightforward. For normed spaces E, F we will denote the collection of all continuous linear maps E−→F by L(E, F ). Under pointwise operations L(E, F ) is again a vector space, normed by the Lipschitz norm T −→ T := inf{M ≥ 0 : T (x) ≤ M x for all x ∈ E}. This infimum is in fact a minimum as we have T (x) ≤ T x for all x ∈ E. If G is a normed space, T ∈ L(E, F ), S ∈ L(F, G) then S ◦ T ∈ L(E, G) and

2.1 Basics

17

S ◦ T ≤ S T . As customary we denote the identity E−→E by I . Also, we will write L(E) for L(E, E) and E for L(E, K). (We tacitly assume that K is normed by its valuation.) E is called the dual space of E. By E F we mean that there is a linear isometrical bijection E−→F and we say that E is isometrically isomorphic to F . An embedding T : E−→F is a linear homeomorphism of E onto the image. If, in addition, T is surjective we say that E is linearly homeomorphic to F . Theorem 2.1.5 Let E, F be normed spaces. If F is a Banach space then so is L(E, F ). In particular, E is a Banach space.

Proof. Classical.

Let D be a closed subspace of a normed space E, let π : E−→E/D be the canonical map. Then the formula π(x) := dist (x, D)

(x ∈ E)

defines a norm on E/D, the so-called quotient norm. Its induced topology is the quotient topology (i.e., the strongest topology on E/D for which π is continuous). Unless explicitly stated otherwise, we always assume E/D to be equipped with the quotient norm. Clearly π ∈ L(E, E/D) and π ≤ 1. Let F be a normed space and let T ∈ L(E, F ) be such that D ⊂ Ker T . Then there is a unique map T1 : E/D−→F making the diagram E? ?? ?? ? T ??

π

F

/ E/D zz zz z z T z }z 1

commute. T1 ∈ L(E/D, F ) and T1 = T . We will say that F is a quotient of E (by D) if T1 is an isometrical bijection. Theorem 2.1.6 Let D be a closed subspace of a normed space E. Then we have the following: (i) If E is a Banach space then so is E/D. (ii) If D and E/D are Banach spaces then so is E. Proof. Although the proof is classical, (ii) may be less known, so for convenience we provide its proof. Let x1 , x2 , . . . be a Cauchy sequence in E. Then, with π as above, π (x1 ), π (x2 ), . . . is Cauchy in E/D, hence there exists a y ∈ E such that limn π(xn ) = π (y). Thus, dist (y − xn , D) → 0, i.e., there are d1 , d2 , . . . ∈ D such that y − xn − dn → 0. Therefore, d1 , d2 , . . . is Cauchy and so converges to some d ∈ D. It follows that x1 , x2 , . . . converges to y − d.

18

Normed spaces

Two norms on a vector space are called equivalent if they induce the same topology. Theorem 2.1.7 Two norms . 1 and . 2 on a vector space E are equivalent if and only if there exist positive constants c and C such that c x1 ≤ x2 ≤ C x1 for all x ∈ E. Proof. Apply 2.1.4 to the identity (E, . 1 )−→(E, . 2 ) and its inverse. The following lemma we need in 2.2.12, but its logical place seems to be here. Lemma 2.1.8 Let (E, . ) be a normed space, let D be a subspace of E. Suppose . is a norm on D such that c x ≤ x ≤ C x

(x ∈ D)

for some constants 0 < c ≤ C. Then . can be extended to a norm . on E such that c x ≤ x ≤ C x

(x ∈ E).

Proof. For x ∈ E set x := inf max(C x − d, d ). d∈D

(2.1)

We obtain (by taking d = 0 in (2.1)) x ≤ C x and, since for every d ∈ D max(C x − d, d ) ≥ max(c x − d, c d) ≥ c x, also x ≥ c x, so the required inequalities are established. Straightforward verification shows that . is a norm. Finally, let x ∈ D. Then (by taking d = x in (2.1)) we obtain x ≤ x . On the other hand, for each d ∈ D we have max(C x − d, d ) ≥ max(x − d , d ) ≥ x , showing that x ≥ x . Thus, . is an extension of . , which finishes the proof. In classical analysis one can normalize a non-zero vector x by taking x−1 x. In our case this is not possible, but one can hope for a λ ∈ K such that |λ| = x; then λ−1 x would have length 1. Unfortunately if |K| is strictly contained in [0, ∞) we can give counterexamples. It suffices to choose s ∈ [0, ∞) \ |K| and put on K the unorthodox norm λ → |λ| s. (A more natural example is Cp as a Qp -vector space with norm | . |p .) We have, however, the following. Theorem 2.1.9 Let (E, . ) be a normed space over a discretely valued field K. Then there is an equivalent norm . on E for which E ⊂ |K|.

2.1 Basics

19

Proof. For x ∈ E take x := max{|λ| : λ ∈ K, |λ| ≤ x}. One checks easily that . is a norm on E with values in |K| and |ρ| x ≤ x ≤ x, where ρ is a uniformizing element. Curiously, the analogue for densely valued base fields is unknown. Renorming problem Let (E, . ) be a normed space over a densely valued field K . Does there exist an equivalent norm . on E for which E ⊂ |K |? In many circumstances, however, it is “approximate normalizing” (for every non-zero x ∈ E there is a λ ∈ K such that λ x is close to 1; more precisely: E ⊂ |K|) that is important, rather than the exact version. As a case in point, for normed spaces E, F the norm T −→ T 0 := sup{T (x) : x ∈ BE } on L(E, F ) is equivalent, but not always equal to, the usual Lipschitz norm . ! Nevertheless, if E ⊂ |K| and F ⊂ |K| we have equality. We leave the easy proofs to the reader. Definition 2.1.10 A norm . on a vector space E is called solid if E ⊂ |K|. Using this terminology we arrive at the next immediate conclusions. Theorem 2.1.11 If the valuation of K is dense all norms are solid. For discretely valued K a norm . on a vector space E is solid if and only if E ⊂ |K|. Every norm is equivalent to a solid one. Notice that the quotient norm of a solid norm is solid, and that the Lipschitz norm on L(E, F ) is solid if the norms on E, F are. We will return to the subject of solidity in 3.1.1. The following lemma on one-dimensional spaces is obvious; we state it for reference. Lemma 2.1.12 Any norm on a one-dimensional space E = Kx (x ∈ E, x = 0) has the form λ x → |λ| C for some C > 0. E is complete. E is spherically complete if and only if K is spherically complete. Any linear map between one-dimensional normed spaces is continuous. The next statements are well-known. The proofs are classical. Theorem 2.1.13 (see also 3.4.26) Let E be a normed space. An element f ∈ E ∗ is continuous if and only if Ker f is closed.

20

Normed spaces

Proof. Let f ∈ E ∗ , f = 0 such that Ker f is closed. Let π : E−→E/Ker f be the canonical map. Then π is continuous. Let f1 : E/Ker f −→K be such that the diagram E? ?? ?? ? f ??

/ E/Ker f vv vv v vv f1 zv v

π

K

commutes. Now f1 is continuous by the above lemma. Hence so is f1 ◦ π = f . Theorem 2.1.14 (i) Each finite-dimensional normed space is complete. Each subspace is closed. (ii) Each linear map between finite-dimensional normed spaces is continuous. (iii) All norms on a finite-dimensional normed space are equivalent. Proof. (i). Completeness is shown by induction with respect to the dimension. In fact, 2.1.12 takes care of the one-dimensional case and 2.1.6(ii) of the induction step. Closedness of subspaces is now obvious. (ii). By (i) and 2.1.13 linear functionals are continuous. For the general case let T : E−→F be linear, where E, F are finite-dimensional normed spaces. Let y1 , . . . , ym be an algebraic base of T (E). Then there are f1 , . . . , fm ∈ E ∗ such that T (x) = f1 (x) y1 + · · · + fm (x) ym

(x ∈ E),

and the continuity of T follows. (iii). This is a consequence of the fact that linear bijections are continuous. A standard example is the space K n , where n ∈ N, equipped with the canonical norm (λ1 , . . . , λn ) → maxi≤i≤n |λi |. From 2.1.14(iii) we obtain the following. Corollary 2.1.15 Every finite-dimensional normed space E is linearly homeomorphic to K n , with n := dim E. We now introduce direct sums and products of normed spaces. Definition 2.1.16 Let I be an index set. For every i ∈ I let Ei be a normed space.

2.1 Basics

21

The normed product of the Ei is the vector space Ei : {xi : i ∈ I } is bounded (xi )i∈I ∈ i∈I

equipped with the norm (xi )i∈I −→ sup xi . i∈I

b

b

We denote this space by i∈I Ei , or i Ei . The normed direct sum of the Ei is the (normed) subspace of all (xi )i ∈ b that, for every ε > 0, {i ∈ I : xi ≥ ε} is finite. This space is i Ei such

denoted by i∈I Ei , or i Ei .

Clearly if all Ei are Banach spaces then so are bi Ei and i Ei . b

If I = {1, . . . , n} for some n ∈ N then i Ei and i Ei are identical. So

instead of bi Ei or i Ei we often write E1 × · · · × En . Note that E1 × · · · × En = K n when Ei = K for all i ∈ {1, . . . , n}. The classical Banach Open Mapping Theorem is derived from the Baire Category Theorem and scalar multiplication considerations, so there will be no problem in carrying over the proof to the non-Archimedean case. However, we can obtain slightly more. Theorem 2.1.17 (Banach’s Open Mapping Theorem) Let E, F be Banach spaces, let T ∈ L(E, F ) be surjective. Then T is an open mapping; more generally, for every ball B in E the image T (B) is clopen. The proof follows easily from the lemma below. Lemma 2.1.18 Let E, F be normed spaces, let T ∈ L(E, F ). Then we have the following: (i) If T is surjective and F is complete then T (BE ) is open. (ii) If E is complete and T (BE ) is open then T (BE ) = T (BE ). Proof. (i). Let λ1 , λ2 , . . . ∈ K, 0 < |λ1 | ≤ |λ2 | ≤ · · · , limn |λn | = ∞. From E = n∈N λn BE and surjectivity we obtain F = n∈N λn T (BE ) = n∈N λn T (BE ). By completeness of F and the Baire Category Theorem there is an n such that A := λn T (BE ) has a non-empty interior, so there is an x ∈ A and an r > 0 such that x + BF (0, r) ⊂ A. But, by the strong triangle inequality, BE is an additive group, hence so is A. Therefore, for each y ∈ A we have y + BF (0, r) = y − x + x + BF (0, r) ⊂ A + A + A ⊂ A; it follows that A is open. Then so is T (BE ) = λ−1 n A.

22

Normed spaces

(ii). As T (BE ) is an additive group, it suffices to prove the following. If BF (0, r) ⊂ T (BE ) for some r > 0 then BF (0, r) ⊂ T (BE ). To this end, let y ∈ BF (0, r), let ρ ∈ K, 0 < |ρ| < 1. Since y ∈ T (BE ) there is an x0 ∈ BE such that y − T (x0 ) < |ρ| r. Then ρ −1 (y − T (x0 )) < r and again there is an x1 ∈ BE such that ρ −1 (y − T (x0 )) − T (x1 ) < |ρ| r, i.e., y − T (x0 ) − T (ρ x1 ) < |ρ|2 r. Carrying on like this we inductively find x0 , x1 , . . . ∈ BE such that, for every n, (2.2) y − T (x0 + ρ x1 + ρ 2 x2 + · · · + ρ n xn ) < |ρ|n+1 r. n Now limn ρ n xn = 0, so by assumption x := ∞ n=0 ρ xn exists and x ∈ BE . From (2.2) and continuity of T we obtain T (x) = y, so y ∈ T (BE ) and we are done. We have the following three traditional results; we state them for reference, the proofs run as in the Archimedean case. Corollary 2.1.19 (i) A continuous linear bijection between Banach spaces is a homeomorphism. (ii) (Closed Graph Theorem) Let E, F be Banach spaces, let T : E−→F be a linear map whose graph is closed in E × F . Then T is continuous. Theorem 2.1.20 (Uniform Boundedness Principle) Let E be a Banach space, let F be a normed space. Let {Ti : i ∈ I } be a collection in L(E, F ) that is pointwise bounded, i.e., for every x ∈ E the set {Ti (x) : i ∈ I } is bounded. Then {Ti : i ∈ I } is uniformly bounded, i.e., supi Ti < ∞. Corollary 2.1.21 (Banach–Steinhaus Theorem) Let E, F be as in 2.1.20, let T1 , T2 , . . . be a sequence in L(E, F ) such that T (x) := limn Tn (x) exists for every x ∈ E. Then T ∈ L(E, F ).

2.2 Orthogonality We introduce a concept of orthogonality, valid in every normed space, and therefore sometimes called norm orthogonality. In Section 2.4 we shall study the possibility of form orthogonality, i.e., a notion derived from a bilinear or Hermitian form. Recall the isosceles triangle principle (Section 1.1) that in the normed case can be written as if x = y then x + y = max(x, y). The following simple consequence will be used frequently.

2.2 Orthogonality

23

Theorem 2.2.1 (Principle of van Rooij) Let t ∈ (0, 1], let x, y be elements of a normed space such that x + y ≥ t x. Then x + y ≥ t y. Proof. The statement is clear if x ≥ y. Now suppose x < y. Then, by the isosceles triangle principle, x + y = y ≥ t y. Definition 2.2.2 Let x, y ∈ E, where E is a normed space. We say that x is . -orthogonal to y, notation x ⊥ y, if dist (x, Ky) = x. We often omit the prefix “ . -”. So x ⊥ y means that the function λ → x − λ y (λ ∈ K) attains a minimum at 0. However, apart from trivial cases, this minimum is attained also at other points! Notice that dist (x, Ky) ≤ x holds for all vectors x, y, so the crucial point of 2.2.2 is that dist (x, Ky) ≥ x. Finally the natural interpretation of 2.2.2 in classical Hilbert space leads to the usual “form orthogonality”, as is easily seen. Theorem 2.2.3 (Symmetry) Let x, y be vectors in a normed space. Then x ⊥ y if and only if, for all λ, µ ∈ K, λ x + µ y = max(λ x, µ y).

(2.3)

Proof. Trivially λ x + µ y ≤ max(λ x, µ y), and also (2.3) holds for λ = 0. Now let x ⊥ y and λ = 0. Then λ x + µ y = |λ| x + λ−1 µ y ≥ |λ| x = λ x. By the Principle of van Rooij, λ x + µ y ≥ λ y and (2.3) is proved. Conversely, if (2.3) holds then, by taking λ := 1, x + µ y = max(x, µ y) ≥ x, so inf{x + µ y : µ ∈ K} ≥ x, whence x ⊥ y. So we have x ⊥ y if and only if y ⊥ x and we can use the expression “x and y are ( . -)orthogonal (to one another)”. Also, we have 0 ⊥ x for all x, x ⊥ x implies x = 0, and x ⊥ y implies λ x ⊥ µ y for all λ, µ ∈ K. Notice, however, that there are crucial differences with orthogonality in Hilbert space. For example, in K 2 with the canonical norm (λ1 , λ2 ) → max(|λ1 | , |λ2 |), choose the vectors e1 := (1, 0), e2 := (0, 1). Then e1 ⊥ e2 , e1 ⊥ e1 + e2 , e2 ⊥ e1 + e2 , but not e1 + e2 ⊥ e1 + e2 . Thus, {x ∈ K 2 : x ⊥ e1 + e2 } is not a vector space! The following definitions will not come as a surprise to the reader. Definition 2.2.4 Let E be a normed space. For two sets X1 , X2 ⊂ E we say that X1 and X2 are ( . -)orthogonal, notation X1 ⊥ X2 , if x1 ⊥ x2 for all x1 ∈ X1 , x2 ∈ X2 . For x ∈ E we write x ⊥ X2 instead of {x} ⊥ X2 .

24

Normed spaces

Now let D1 , D2 be subspaces of E. We say that D2 is an ( . -)orthogonal complement of D1 if D1 ⊥ D2 and E = D1 + D2 . A subspace D is called ( . -)orthocomplemented if it has an orthogonal complement. Clearly D1 ⊥ D2 implies D1 ∩ D2 = {0}, so the above sum D1 + D2 is direct. Notice that a subspace may have more than one orthogonal complement. In fact, in the above example K 2 the subspaces Ke2 and K(e1 + e2 ) are both orthogonal complements of Ke1 . If D1 , D2 are subspaces of E with E = D1 + D2 , then D1 and D2 are orthogonal complements of each other if and only if d1 + d2 = max(d1 , d2 ) for each d1 ∈ D1 , d2 ∈ D2 . Also, for x ∈ E we have x ⊥ D2 if and only if dist (x, D2 ) = x. Orthocomplemented subspaces are closed. (The converse is false, as we will see in 2.3.23.) Definition 2.2.5 Let E be a normed space. A linear map P : E−→E is called a projection if P 2 = P . It is an orthogonal projection, or ( . -) orthoprojection if, in addition, Ker P ⊥ Im P . Observe that every orthoprojection is automatically continuous and its norm is ≤ 1. More precisely, if P = 0 is a continuous projection then P is an orthoprojection if and only if P = 1. Now we relax the above notions. A subspace D of E is called complemented if there exists a continuous projection P ∈ L(E) (not necessarily an orthoprojection) with P (E) = D. Then D = Im P and D1 := Ker P are closed subspaces (such a D1 is called a complement of D) and E is the algebraic direct sum of D and D1 . Conversely, if E is a Banach space and D, D1 are closed subspaces such that the algebraic direct sum of them is E, then D is complemented and D1 is a complement of D. In fact, the norm d + d1 → max(d, d1 ) (d ∈ D, d1 ∈ D1 ) makes E into a Banach space as well as is greater than or equal to the original one on E. Now use 2.1.19(i) to see that the projection d + d1 → d is continuous. We will return to complementation in 3.4.27 and 3.4.28. Next, we extend the notion of orthogonality to more than two points. The vectors e1 , e2 , e1 + e2 from our previous example in K 2 are mutually orthogonal but the system {e1 , e2 , e1 + e2 } is linearly dependent and it would be silly to call this system “orthogonal”. Therefore, it seems natural to call a system of vectors orthogonal if each element is orthogonal to the linear span of the remaining vectors. To avoid annoying trivialities we shall exclude 0 from our orthogonal systems. Definition 2.2.6 A subset X of a normed space E, 0 ∈ X, is called an ( . -) orthogonal system (set) if for each x ∈ X we have x ⊥ [X \ {x}]. If, in addition, x = 1 for all x ∈ X, X is called an orthonormal system.

2.2 Orthogonality

25

Clearly X is an orthogonal system if and only if each finite subset of X is an orthogonal system; hence any orthogonal system is linearly independent. The following gives a practical test for orthogonality and ties in with formula (2.3) of 2.2.3. Theorem 2.2.7 Let e1 , . . . , en be distinct non-zero vectors of a normed space E. Then the following are equivalent: (α) {e1 , . . . , en } is an orthogonal system. (β) For all λ1 , . . . , λn ∈ K, λ1 e1 + · · · + λn en = max λi ei . 1≤i≤n

(γ ) For each j ∈ {1, . . . , n − 1}, ej +1 ⊥ [e1 , . . . , ej ]. Proof. Clearly (α) and (β) are equivalent and (α) =⇒ (γ ). To prove (γ ) =⇒ (β), notice that λ1 e1 + · · · + λn en ≥ λn en . This is a consequence of (γ ) if λn = 0, and trivial if λn = 0. By the Principle of van Rooij we have λ1 e1 + · · · + λn en ≥ λ1 e1 + · · · + λn−1 en−1 . Now again (γ ) can be used to arrive at λ1 e1 + · · · + λn−1 en−1 ≥ λn−1 en−1 . Continuing in this way we arrive after n steps at (β). Note When we are considering orthogonality of a set X ⊂ E \ {0} it is sometimes useful to “index” X by taking a set I with the same cardinality as X, a bijection i → ei of I onto X, and write X = (ei )i∈I . If I ⊂ N we sometimes are sloppy by talking about “orthogonal sequences”. This will be harmless as long as we define orthogonality of a sequence e1 , e2 , . . . of non-zero vectors as en ⊥ [em : m = n] for all n ∈ N: it then follows that e1 , e2 , . . . are mutually distinct. Remark 2.2.8 It is easily seen that if e1 , e2 , . . . is a sequence in a normed space such that, for n = m, en and em are in different cosets of K × then e1 , e2 , . . . is orthogonal. Contrary to the Hilbert space case, perturbing the vectors of an orthogonal system a little does not disturb orthogonality. Theorem 2.2.9 (Perturbation Lemma) Let (ei )i∈I be an orthogonal system in a normed space E. For each i ∈ I let ei ∈ E be such that ei − ei < ei . Then (ei )i∈I is also an orthogonal system.

26

Normed spaces

Proof. We may assume I = {1, . . . , n} for some n ∈ N. By the isosceles triangle principle we have ei = ei for each i. Now let λ1 , . . . , λn ∈ K, not all λi = 0. Write n n n λi ei = λi (ei − ei ) + λi ei . i=1

i=1

i=1

By assumption and 2.2.7

n

λi (ei − ei ) ≤ max |λi | ei − ei < max |λi | ei = 1≤i≤n

i=1

1≤i≤n

n

λi ei .

i=1

Again by the isosceles triangle principle

n

λi ei =

i=1

n

λi ei = max λi ei = max λi ei ,

i=1

1≤i≤n

1≤i≤n

and by 2.2.7 we obtain the orthogonality of (ei )i∈I .

Remark 2.2.10 An easy application of Zorn’s Lemma shows that every orthogonal system is contained in a maximal one. In [193], 5.4 it is proved that two maximal orthogonal systems in a normed space have the same cardinality, but we will not need it here. However, see 2.3.18. In several circumstances a less restrictive form of orthogonality is appropriate. Definition 2.2.11 A subset X of a normed space is called (an) “orthogonal” (system) if there is an equivalent norm for which X is orthogonal. We now prove that “orthogonality” is equivalent to a weak form of (β) of 2.2.7. Theorem 2.2.12 Let E be a normed space and let 0 ∈ X ⊂ E. Then the following are equivalent: (α) X is “orthogonal”. (β) There is a t ∈ (0, 1] such that, for all distinct e1 , . . . , en ∈ X and λ1 , . . . , λn ∈ K, n λi ei ≥ t max λi ei . i=1

1≤i≤n

Proof. (α) =⇒ (β). Let . be an equivalent norm for which X is orthogonal; let c, C be positive constants such that c . ≤ . ≤ C . . Then

n i=1

λi ei ≥ C −1

n

λi ei = C −1 max λi ei ≥ c C −1 max λi ei .

i=1

So (β) is established with t := c C −1 .

1≤i≤n

1≤i≤n

2.2 Orthogonality

27

(β) =⇒ (α). By linear independence every x ∈ [X] has a unique represen tation x = i∈J λi ei , where J ⊂ I is finite. Set x := maxi∈J λi ei . Then . is a norm on [X], x ≥ x ≥ t maxi∈J λi ei = t x and, since ei = ei for each i, X is orthogonal with respect to . . It suffices now to extend . to an equivalent norm on E, see 2.1.8. This leads to the following more quantitative notion. See 3.9.9, 3.9.10 for an extension to the locally convex case. Definition 2.2.13 Let E be a normed space, let t ∈ (0, 1]. A vector x ∈ E is t-orthogonal to y ∈ E (with respect to . ) if dist (x, Ky) ≥ t x. By the Principle of van Rooij, if x is t-orthogonal to y then λ x + µ y ≥ t max(λ x, µ y) for all λ, µ ∈ K, so t-orthogonality is symmetric. 1-orthogonality is what we called earlier orthogonality. The same goes for the next concept. Definition 2.2.14 Let t ∈ (0, 1], let X be a subset of a normed space E, 0 ∈ X. X is called a t-orthogonal system (set) (with respect to . ) if for each x ∈ X, y ∈ [X \ {x}], x is t-orthogonal to y. Clearly X is t-orthogonal if and only if each finite subset of X is t-orthogonal, if and only if for each n ∈ N, each distinct e1 , . . . , en ∈ X, and each λ1 , . . . , λn ∈ K, λ1 e1 + · · · + λn en ≥ t max λi ei . 1≤i≤n

Concluding we can say that X is “orthogonal” if and only if there is a t ∈ (0, 1] such that X is t-orthogonal. Remark 2.2.15 In contrast, the natural interpretation of (γ ) of 2.2.7, dist (ej +1 , [e1 , . . . , ej ]) ≥ t ej +1 for all j ∈ {1, . . . , n − 1} is not enough to conclude t-orthogonality. In fact, take E := K 3 with the canonical maximum norm and let ρ ∈ K, 0 < |ρ| < 1. Choose e1 := (1, 0, 0), e2 := (1, ρ, 0) and e3 := (0, 1, ρ). Then dist (e3 , [e1 , e2 ]) = dist (e2 , [e1 ]) = |ρ|. But e1 − e2 + ρ e3 = |ρ|2 < |ρ| = |ρ| max(e1 , e2 , ρ e3 ). We can easily prove the following.

28

Normed spaces

Theorem 2.2.16 Let e1 , . . . , en be distinct non-zero vectors in a normed space E. Let t2 , . . . , tn ∈ (0, 1] be such that dist (e2 , [e1 ]) ≥ t2 e2 , dist (e3 , [e1 , e2 ]) ≥ t3 e3 , .. . dist (en , [e1 , . . . , en−1 ]) ≥ tn en . Then {e1 , . . . , en } is t2 t3 · · · tn -orthogonal. Remark 2.2.17 Let {ei : i ∈ I } be an orthogonal system and let {λi : i ∈ I } ⊂ K × . Then {λi ei : i ∈ I } is also an orthogonal system. In particular, let ρ ∈ K, 0 < |ρ| < 1. Then we can choose λi ∈ K × such that |ρ| ≤ λi ei ≤ 1, for each i. We will often use this fact. Similar procedures can be carried out for t-orthogonal systems, where t ∈ (0, 1).

2.3 Spaces of countable type Recall that a topological space is called separable if it has a countable dense subset. Now let E be a normed space over K, E = {0} and suppose E is separable. Then its one-dimensional subspaces are homeomorphic to K, so K must be separable as well. Thus, for normed spaces the concept of separability is of no use if K is not separable. (Such K exist, see 4.2.7.) However, by “linearizing” the notion of separability we obtain a generalization, useful for every scalar field K. Definition 2.3.1 A normed space E is of countable type if it contains a countable set whose linear hull is dense in E. Clearly every separable normed space is of countable type. So, for example, (Cp , | . |p ) is of countable type, viewed as a normed space over any closed subfield of Cp . It is not hard to see that for separable K a normed space is of countable type if and only if it is separable. But also for non-separable K we have non-trivial examples such as finite-dimensional normed spaces. Now we give an infinite-dimensional example. Example 2.3.2 Standard space of countable type Let c0 be the vector space of all sequences (λ1 , λ2 , . . .) ∈ K N for which limn λn = 0, with coordinatewise operations and norm . ∞ : (λ1 , λ2 , . . .) −→ max |λn | . n∈N

2.3 Spaces of countable type

29

Clearly the linear span of the unit vectors (1, 0, 0, . . .), (0, 1, 0, 0, . . .), . . . is dense in c0 . Also, c0 is easily seen to be a Banach space. As for hereditary properties, it is straightforward to prove that finite products (compare with 2.5.15) and countable normed direct sums of spaces of countable type are of countable type. In this context, observe that the space c0 equals

n∈N Kn , where Kn = K for each n. Also, continuous linear images, in particular quotients by closed subspaces, of spaces of countable type are of countable type. The problem for subspaces is more subtle and will be solved in 2.3.14. The completion of a normed space of countable type is obviously of countable type. Somewhat less trivial is the following. Theorem 2.3.3 Let E be a normed space with a dense subspace D. Then E is of countable type if and only if D is of countable type. Proof. Suppose E is of countable type. Let X ⊂ E be countable, [X] dense in E. For each x ∈ X choose a countable subset Yx of D such that x ∈ Yx . Then Y := x∈X Yx is countable, lies in D and X ⊂ Y , so [Y ] = D, from which we deduce that D is of countable type. Remark 2.3.4 The reader will notice that the concept “of countable type” will be studied thoroughly in this book; we devote more attention to it than “separability” usually gets in classical text books. The reason is that in nonArchimedean theories spaces of countable type have nice properties concerning duality (Hahn–Banach Theorem, see Section 4.2) that other spaces may not have. Therefore, it is often important to know whether a given space is of countable type or not. The system {e1 , e2 , . . .} of unit vectors in c0 is clearly orthogonal. Further, for each (λ1 , λ2 , . . .) ∈ c0 the sequence λ1 e1 , λ2 e2 , . . . is summable and (λ1 , λ2 , . . .) = ∞ n=1 λn en . Hence c0 is not only of countable type, but even {e1 , e2 , . . .} is acting as a base in a “topological” sense. We now will study this feature in more detail. Definition 2.3.5 Let E be a normed space, let t ∈ (0, 1]. A sequence e1 , e2 , . . . in E is called a t-orthogonal base (of E) if {e1 , e2 , . . .} is t-orthogonal and every x ∈ E has an expansion x = ∞ n=1 λn en , where λn ∈ K. The base e1 , e2 , . . . is orthogonal if t = 1, orthonormal if, in addition, en = 1 for all n ∈ N.

30

Normed spaces

By continuity of the norm, for x ∈ E as above we have limn λn en = 0 and x = lim m

m

λn en ≥ t lim max λn en = t max λn en , m 1≤n≤m

n=1

n∈N

so the expansion of x is unique: if also x = ∞ n=1 µn en for certain µn ∈ K then ∞ 0 = n=1 (λn − µn ) en , so that t maxn∈N (λn − µn ) en = 0, i.e., λn = µn for all n. Note In the spirit of the Note following 2.2.7 we will speak about torthogonality of indexed systems (ei )i∈I and t-orthogonal sequences. Clearly the unit vectors form an orthonormal base of c0 . Any space with some t-orthogonal base is of countable type. For the converse we first prove a useful test to decide whether a t-orthogonal sequence is a base. Theorem 2.3.6 Let t ∈ (0, 1], let e1 , e2 , . . . be a t-orthogonal sequence in a normed space E. If [e1 , e2 , . . .] is dense in E then e1 , e2 , . . . is a t-orthogonal base of E. Proof. We may assume that E is a Banach space. Let ρ ∈ K, 0 < |ρ| < 1. By suitable scalar multiplication we may assume that |ρ| ≤ en ≤ 1 for all n. Now let (λ1 , λ2 , . . .) ∈ c0 . Then limn λn en = 0, so by completeness ∞ n=1 λn en exists in E. The formula T : (λ1 , λ2 , . . .) −→

∞

λn en

n=1

defines a linear map T : c0 −→E for which T (c0 ) is dense in E. But also, for x = (λ1 , λ2 , . . .) ∈ c0 , T (x) =

∞

λn en ≤ max λn en ≤ max |λn | = x n∈N

n=1

n∈N

and, by t-orthogonality, T (x) =

∞ n=1

λn en ≥ t max λn en ≥ t |ρ| max |λn | = t |ρ| x, n∈N

n∈N

so that T is a homeomorphism. Thus, T (c0 ) is complete, hence closed in E, and dense, so T is surjective, which proves the theorem. Now we are ready to prove the main theorem of this section. Theorem 2.3.7 Let E be a normed space of countable type. Then, for each t ∈ (0, 1), E has a t-orthogonal base.

2.3 Spaces of countable type

31

Proof. For convenience we assume that E is infinite-dimensional. (For finitedimensional spaces the inductive construction below breaks off.) There are linearly independent x1 , x2 , . . . ∈ E such that [x1 , x2 , . . .] is dense in E. Set Dn := [x1 , . . . , xn ] for n ∈ N. There are t2 , t3 , . . . ∈ (0, 1) such that ∞ n=2 tn exists and is ≥ t. For each n ≥ 2 the distance of xn to Dn−1 is positive since Dn is closed (2.1.14(i)) and xn ∈ Dn−1 , therefore there is a dn−1 ∈ Dn−1 such that tn xn − dn−1 ≤ dist (xn , Dn−1 ). Now choose e1 := x1 , en := xn − dn−1 for each n ≥ 2. We claim that e1 , e2 , . . . is a t-orthogonal base of E. In fact, for each n we have Dn = [e1 , . . . , en ], so [e1 , e2 , . . .] is dense in E, and for n ≥ 2 dist(en , Dn−1 ) = dist(xn , Dn−1 ) ≥ tn xn − dn−1 = tn en . So by 2.2.16 {e1 , · · · , en } is t2 · · · tn -orthogonal, hence t-orthogonal. From this we get t-orthogonality of e1 , e2 , . . . and the theorem is proved after applying 2.3.6. Remark 2.3.8 For a discussion on what happens for the case t = 1, see 2.3.25 and 2.3.26. Corollary 2.3.9 Each normed space of countable type is linearly homeomorphic to a subspace of c0 . Each infinite-dimensional Banach space of countable type is linearly homeomorphic to c0 . Proof. We only prove the second assertion. Let t ∈ (0, 1), let e1 , e2 , . . . be a t-orthogonal base of a Banach space E of countable type, and let ρ ∈ K, 0 < |ρ| < 1. Without loss, assume |ρ| ≤ en ≤ 1 for all n. Now take the map T : c0 −→E of the proof of 2.3.6. This result shows that, up to linear homeomorphisms, there exists, for given K, only one infinite-dimensional Banach space of countable type, viz. c0 . This is in contrast to the Archimedean theory, where there exist separable Banach spaces without a Schauder base, [51], and where Banach spaces having Schauder bases (see below for this concept) may not be linearly homeomorphic (e.g. 1 and 2 ). Because we will need them later on, we now introduce two other concepts of “base”. Definition 2.3.10 Let E be a normed space, let e1 , e2 , . . . be a sequence in E of non-zero vectors. It is called a (topological) base if for every x ∈ E there exist unique λ1 , λ2 , . . . ∈ K such that x = ∞ n=1 λn en , a Schauder base if, in addition, the coefficient functionals x → λn (n ∈ N) are continuous. Clearly every space with a base is of countable type and every t-orthogonal base is Schauder.

32

Normed spaces

Theorem 2.3.11 In a Banach space, each base is Schauder, each Schauder base is t-orthogonal for some t ∈ (0, 1]. Proof. Let e1 , e2 , . . . be a base of a Banach space (E, . ). It is enough to prove it to be t-orthogonal for some t ∈ (0, 1]. For every x = ∞ n=1 λn en ∈ E we have limn λn en = 0, so x1 := maxn∈N λn en exists. . 1 is a norm ≥ . , and by using completeness of E it is not difficult to see that (E, . 1 ) is also complete. But then . 1 and . are equivalent by 2.1.19(i), so there is a C ≥ 1 with . 1 ≤ C . . Hence, for x = ∞ n=1 λn en ∈ E we have x ≥ C −1 x1 = C −1 max λn en , n∈N

which proves C

−1

-orthogonality.

Remarks 2.3.12 (a) There are non-complete normed spaces with bases that are not Schauder. In fact, let K ⊃ Qp for some prime number p, let E be the vector space of all polynomial functions of Zp to K equipped with the canonical supremum norm. Consider the sequence e1 , e2 , . . . in E given by en (x) := x n−1 (x ∈ Zp ). Clearly e1 , e2 , . . . is an algebraic base of E. Also, for each f ∈ E its expansion f = ∞ n=1 λn en is unique. So e1 , e2 , . . . is a base of E. But the associated second coefficient functional f → f (0) is not continuous (where f (0) is the derivative of f at 0, defined in the usual way, see (x ∈ Subsection 2.5.7). In fact, f1 , f2 , . . . given by fn (x) := x(x−1)···(x−n+1) n! Zp ) is a bounded sequence in E for which f1 (0), f2 (0), . . . is unbounded in K. (b) There are non-complete normed spaces with Schauder bases that are torthogonal for no t ∈ (0, 1]. In fact, let ρ ∈ K, 0 < |ρ| < 1 and let E be the linear hull of the unit vectors of c0 , with the norm induced by the supremum one of c0 . Set en := (1, ρ, ρ 2 , . . . , ρ n−1 , 0, 0, . . .) (n ∈ N). It is easily seen that this sequence is a Schauder (algebraic) base of E that satisfies the required conditions. The next result on spaces of countable type may also be surprising. Theorem 2.3.13 Let E be a Banach space of countable type. Then each closed subspace D of E is complemented. In fact, for every ε > 0 there exists a continuous projection P ∈ L(E) onto D with P ≤ 1 + ε. Proof. Let π : E−→E/D be the canonical map; it suffices to construct a T ∈ L(E/D, E) with T ≤ 1 + ε and for which π ◦ T is the identity on E/D. (Then take P := I − T ◦ π.) To this end, let t := (1 + ε)−1/2 and let z1 , z2 , . . . be a t-orthogonal base of E/D (2.3.7). There are e1 , e2 , . . . ∈ E

2.3 Spaces of countable type

33

with π (en ) = zn , en ≤ t −1 zn for each n. For z = ∞ n=1 λn zn ∈ E/D set ∞ T (z) := n=1 λn en . (Since limn λn en = 0 and E is complete the latter sum exists.) Clearly T is linear, π (T (z)) = z for all z ∈ E/D, and T (z) = =t

∞

λn en ≤ max λn en ≤ t −1 max λn zn ≤ t −2

n=1 −2

n∈N

n∈N

∞

λn zn

n=1

z = (1 + ε) z,

hence T ≤ 1 + ε.

This leads to a result for which we do not know a more direct proof. Corollary 2.3.14 If a normed space is of countable type then so are its subspaces. Proof. Let D be a subspace of a normed space E of countable type. Then E ∧ is of countable type, so we may assume that E is Banach. By 2.3.13 there is a continuous projection P ∈ L(E) onto D, hence D = P (E) is of countable type, then so is D (2.3.3). Corollary 2.3.15 If D is a closed subspace of a normed space E, then E is of countable type if and only if D and E/D are of countable type. Proof. Left to the reader.

Problem Let E be a non-complete normed space of countable type. Is every closed subspace D of E complemented? For countable systems we formulate an extension of the Perturbation Lemma 2.2.9. Theorem 2.3.16 (Full Perturbation Lemma) Let e1 , e2 , . . . be non-zero vectors in a normed space E, t-orthogonal for some t ∈ (0, 1]. Let e1 , e2 , . . . ∈ E. Then we have the following: (i) If en − en < t en for each n ∈ N then e1 , e2 , . . . is also t-orthogonal. (ii) If e1 , e2 , . . . form a t-orthogonal base of E and, for all n ∈ N, en − en < s t en for some s, 0 < s < 1 then e1 , e2 , . . . is again a t-orthogonal base of E. Proof. (i). The proof of the Perturbation Lemma 2.2.9 applies with obvious modifications. (ii). By 2.3.6, e1 , e2 , . . . is also a t-orthogonal base of E ∧ , so it suffices to show that [e1 , e2 , . . .] is dense in E ∧ . (Then it is dense in E and we apply (i) and again 2.3.6.) Thus, it is enough to prove (ii) for Banach spaces E.

34

Normed spaces

Let x ∈ E have expansion ∞ n=1 λn en . Then λn en = λn en → 0, so y := ∞ n=1 λn en exists and x − y ≤ s t maxn∈N λn en ≤ s x. So, with D := [e1 , e2 , . . .], we have dist (x, D) ≤ s x for all x ∈ E. It follows easily that x ∈ D, i.e., E = D. Next we prove that t-orthogonal systems in spaces of countable type are at most countable. To this end, a lemma. Lemma 2.3.17 Let E be a normed space of countable type. Let X ⊂ E be such that [X] = E. Then there is a countable Y ⊂ X with [Y ] = E. Proof. We first prove that for every x ∈ E there is a countable Yx ⊂ X with x ∈ [Yx ]. In fact, there are x1 , x2 , . . . ∈ [X] with x − xn < n1 for each n. There are also finite sets F1 , F2 , . . . ⊂ X such that xn ∈ [Fn ] for each n. Set Yx := n∈N Fn . Then Yx is countable and x1 , x2 , . . . ∈ [Yx ]. Hence x ∈ [Yx ]. Now let e1 , e2 , . . . ∈ E be such that [e1 , e2 , . . .] = E. By the first part of the proof there are countable sets Y1 , Y2 , . . . ⊂ X with en ∈ [Yn ] for each n. Set Y := n∈N Yn . Then Y is countable, [Y ] contains e1 , e2 , . . . , so [Y ] = E. Theorem 2.3.18 Every t-orthogonal system in a normed space of countable type is countable. Proof. Let (ei )i∈I be a t-orthogonal system. Then D := [ei : i ∈ I ] is by 2.3.14 of countable type. By the lemma above there is a countable J ⊂ I such that D = [ej : j ∈ J ]. If I were uncountable there would be an i ∈ I , i ∈ J . By t-orthogonality we would have ei ∈ [ej : j ∈ J ] = D, a contradiction. Now we shall study spaces with an orthogonal base. Lemma 2.3.19 Let E be a normed space with an orthogonal base e1 , e2 , . . .. Then each one-dimensional subspace has an orthogonal complement. Proof. Let x ∈ E, x = 0 has expansion x = ∞ n=1 λn en . Then there is an m ∈ N with x = λm em . Let D be the closed linear span of {en : n = m}. Then each d ∈ D has an expansion ∞ n=1 µn en , where µm = 0. We see that x − d ≥ x, so x ⊥ D. Also, Kx + D = Kem + D = E, so D is an orthogonal complement of Kx. Lemma 2.3.20 Let E be a normed space with the property that every onedimensional subspace has an orthogonal complement. Then every finitedimensional subspace has an orthogonal complement. Proof. We prove the following. Let D be an orthocomplemented subspace, let D1 := D + Kx for some x ∈ E, x ∈ D. Then D1 is orthocomplemented. In fact, since D is orthocomplemented there is an orthogonal projection P of E

2.3 Spaces of countable type

35

onto D. Choose a non-zero d ∈ D1 with P (d) = 0. (Then D1 = D + Kd.) By assumption there is an orthogonal projection Q of E onto Kd. One verifies directly that P + Q − Q ◦ P is an orthogonal projection of E onto D1 . Corollary 2.3.21 Let E be a finite-dimensional normed space with an orthogonal base. Then every subspace has an orthogonal base and is orthocomplemented. Theorem 2.3.22 Let E be a normed space with an orthogonal base e1 , e2 , . . . . Then every subspace has an orthogonal base. Proof. Let D be a subspace of E. For convenience we assume that D is infinite-dimensional (if not the process below stops after finitely many steps). D is of countable type (2.3.14), so there are subspaces D1 ⊂ D2 ⊂ · · · with dim Dn = n for each n and n∈N Dn dense in D. Choose d1 ∈ D1 , d1 = 0. By the above lemmas, for each n > 1, Dn−1 is orthocomplemented in Dn , so we can find a dn ∈ Dn , dn = 0 with dn ⊥ Dn−1 . Then by 2.2.7, d1 , d2 , . . . is orthogonal and, since Dn = [d1 , . . . , dn ], their linear span is dense in D, so it is an orthogonal base of D, by 2.3.6. Remark 2.3.23 In the above theorem we cannot conclude, as we did in 2.3.21, that every closed subspace of E is orthocomplemented. As an example, let K have a dense valuation, choose ρ1 , ρ2 , . . . ∈ BK− with limn |ρn | = 1. Let E := c0 and let f ∈ E be defined by f (x) :=

∞

λn ρ n

(x = (λ1 , λ2 , . . .) ∈ E).

n=1

One can easily check that f = 1 and |f (x)| < x for all x ∈ E, x = 0. Take D := Ker f . Suppose there is an x ⊥ D, x = 0. Then, for each d ∈ D, |f (x + d)| |f (x + d)| |f (x)| ≥ = , x max(x, d) x + d so

|f (x)| x

≥ f = 1, a contradiction.

Spherical completeness enters the picture when we ask the question whether every space of countable type has an orthogonal base. First a lemma. Lemma 2.3.24 Let E1 , E2 be spherically complete Banach spaces. Then so is E1 × E2 . Proof. Straightforward (each nested sequence of balls in E1 × E2 induces naturally nested sequences of balls in E1 and E2 ).

36

Normed spaces

Theorem 2.3.25 Let K be spherically complete. Then each normed space of countable type has an orthogonal base. Proof. Again assume that the space E is infinite-dimensional. Then there are subspaces D1 ⊂ D2 ⊂ · · · of E with dim Dn = n for each n and n∈N Dn dense in E. Choose e1 ∈ D1 , e1 = 0. Then Ke1 is spherically complete (2.1.12), so by 1.1.6 there is an e2 ∈ D2 , e2 ∈ D1 , e2 ⊥ Ke1 . Then D2 is isometrically isomorphic to Ke1 × Ke2 , hence spherically complete by the previous lemma, so there is an e3 ∈ D3 , e3 ∈ D2 , e3 ⊥ D2 , etc. The sequence e1 , e2 , . . . is orthogonal by 2.2.7. Further, [e1 , e2 , . . .] is dense, so by 2.3.6 it is an orthogonal base. For non-spherically complete K the above conclusion is not true. In fact, we proceed to construct a two-dimensional space over K without orthogonal base! Example 2.3.26 Let K be not spherically complete. Then there is a norm ν on K 2 such that ν(x) ∈ |K| for all x ∈ K 2 and such that there is no ν-orthogonal base. Proof. Let B1 ⊃ B2 ⊃ · · · be balls in K with empty intersection. Choose λn ∈ Bn for each n ∈ N and define ν(α1 , α2 ) := lim |α1 − α2 λn | . n

To show that this limit exists, assume α2 = 0. Then there is an m ∈ N such that α1 α2−1 ∈ Bn for n ≥ m, so α1 α2−1 − λn = dist (α1 α2−1 , Bn ) is a non-zero constant for n ≥ m (see Section 1.1, property 5 of ultrametrics). So ν is welldefined and ν(α1 , α2 ) = 0 if and only if α1 = α2 = 0. The other requirements for a norm are clear. To see that there is no ν-orthogonal base it suffices by 2.3.21 to show that the linear hull of e1 := (1, 0) has no orthogonal complement, i.e., that x is ν-orthogonal to e1 for no x except 0. Without loss, take x = (α, 1), α ∈ K. There is an m ∈ N with α ∈ Bm , so ν(x) = lim |α − λn | = |α − λm | > |λn − λm | n

for all n > m (since λn ∈ Bm , α ∈ Bm ), which again is ≥ limn dist (λm , Bn ) = ν(λm , 1) = ν(x − (α − λm ) e1 ). Thus, there is a β ∈ K (viz. β = α − λm ) such that ν(x − β e1 ) < ν(x), so x is not ν-orthogonal to e1 . Note The above space – which we call Kν2 from now on – will be used in the sequel to obtain (counter)examples. Remark 2.3.27 Let K be not spherically complete. A much shorter way (but using the spherical completion K ∨ of Section 1.2) to obtain a two-dimensional space without orthogonal base is the following.

2.4 The absence of Hilbert space

37

∨ Take × λ ∈ K , λ = 0. Let us prove that λ is not orthogonal to K. Since K = |K| we may assume |λ| = 1. As the residue class fields of K and K ∨ are isomorphic there is a µ ∈ K such that |λ − µ.1| < 1, and we are done. Hence no two-dimensional subspace of K ∨ containing K has an orthogonal base.

One could ask if perhaps quotients of spaces with an orthogonal base have an orthogonal base. Thanks to 2.3.25 the answer is positive if K is spherically complete. However, for any non-spherically complete scalar field K we have a counterexample. It suffices to take E := Kν2 in the following. Theorem 2.3.28 Every Banach space E of countable type with solid norm is a quotient of c0 . Proof. Let 0 < t1 < t2 < · · · , limm tm = 1. Let m ∈ N. By 2.3.7 E has a tm orthogonal base {amn : n ∈ M}, where M = N if E is infinite-dimensional, otherwise M = {1, . . . , dim E}. By solidity of the norm we may assume tm ≤ amn ≤ 1 for all n ∈ M. Now let σ be a bijection N−→N × M, and define T : c0 −→E by (λ1 , λ2 , . . .) −→

∞

λn aσ (n) .

n=1

Clearly T is well-defined, linear and T ≤ 1. To see that T makes E into a quotient of c0 (by Ker T ), let z ∈ E, ε > 0; we produce an x ∈ c0 with T (x) = z and x∞ ≤ (1 + ε) z. Choose m ∈ N be such that tm ≥ (1 + ε)−1/2 . Then z= ∞ λ a , λ ∈ K. Denoting the unit vectors of c0 by e1 , e2 , . . ., set n=1 n mn n x := ∞ n=1 λn eσ −1 ((m,n)) . Then clearly T (x) = z and z ≥ tm max λn amn ≥ tm2 max |λn | = tm2 x∞ . n∈N

n∈N

Hence, x∞ ≤ tm−2 z ≤ (1 + ε) z.

2.4 The absence of Hilbert space In Section 2.2 we have studied “norm” orthogonality. In this section we follow the more classical path of deriving orthogonality from an inner product. Unfortunately this approach will not lead to spaces over K deserving the name “Hilbert space” (2.4.5). Because Hilbert space plays a dominant role in classical Functional Analysis, it is worth the trouble to check in detail what goes wrong in the non-Archimedean setting.

38

Normed spaces

In this section we assume that we have an isometrical map λ → λ∗ in K that is either the identity or a field automorphism of order 2. Then K H := {λ ∈ K : λ = λ∗ } is a closed subfield. Definition 2.4.1 Let E be a vector space. An inner product is a map assigning to every ordered pair (x, y) ∈ E × E an element < x, y > of K such that: (i) < x, x > = 0 whenever x = 0 (definiteness), (ii) < x, y > = < y, x >∗ for all x, y ∈ E (antisymmetry), (iii) x → < x, y > is linear for each y ∈ E. Note “Positivity” has no meaning in K, so the classical condition of positive definiteness has been replaced by (i). This may seem rather arbitrary, but it is not. In fact, if we assume the axioms (i), (ii), (iii) in the complex case (reading ∗ as complex conjugation) then an easy exercise shows that either < , > or − < , > is positive definite. With this in mind the appearance of the absolute value signs in the definition of the associated norm x −→ x := |< x, x >| (x ∈ E) (see 2.4.2) becomes quite natural. For a subset X of E we set X⊥ := {y ∈ E : < y, x > = 0 for all x ∈ X}. Clearly for subspaces D of E we have D ∩ D⊥ = {0}. If x1 , x2 , . . . is a linearly independent sequence of vectors then the usual Gram-Schmidt procedure < xn , ei > < x2 , e1 > e1 , . . . , en := xn − ei , . . . < e1 , e1 > < e i , ei > i=1 n−1

e1 := x1 , e2 := x2 −

leads to a sequence e1 , e2 , . . . that is “form orthogonal”, i.e., < en , em > = 0 whenever n = m, and for which [e1 , . . . , en ] = [x1 , . . . , xn ] for each n. The next theorem looks promising. Theorem 2.4.2 Let E be a vector space with inner product < , >. Suppose that |2| = 1 (i.e., the characteristic of the residue class field is = 2). Then for all x, y ∈ E we have the following: (i) |< x, y >|2 ≤ |< x, x > < y, y >| (Cauchy–Schwarz inequality). (ii) . is a norm on E. (iii) If < x, y > = 0 then x + y = max(x, y) (form orthogonality implies norm orthogonality).

2.4 The absence of Hilbert space

39

Proof. We first prove < x, y > = 0 =⇒ |< x + y, x + y >| ≥ |< y, y >| .

(2.4)

To this end we may suppose x = 0, y = 0. Put µ := < x, x > < y, y >−1 . Then µ ∈ K H . We have to show that |1 + µ| ≥ 1. Suppose not. Then |1 + µ| < 1 and |µ| = 1. Define a sequence λ1 , λ2 , . . . ∈ K H by λ1 := µ, λn+1 := λn + 12 (λ2n + µ) for n ≥ 1. Inductively one proves |λn | ≤ 1, |1 + λn | ≤ |1 + µ| for all n, |λn+1 − λn | ≤ |1 + µ| |λn − λn−1 | for all n ≥ 2. It follows that λn+1 − λn → 0, so λ1 , λ2 , . . . converges to a λ ∈ K H . Then λ = limn λn+1 = limn (λn + 12 (λ2n + µ)) = λ + 12 (λ2 + µ), so λ2 + µ = 0. But on the other hand 0 = < x + λ y, x + λ y > = < x, x > +λ2 < y, y > = < y, y > (µ + λ2 ) = 0, a contradiction, proving (2.4). To prove (i) we may assume x = 0, y = 0. Write y = z + u, where z = y− < y, x > < x, x >−1 x, and u = < y, x > < x, x >−1 x. Then < z, u > = 0, so by (2.4) we have |< z + u, z + u >| ≥ |< u, u >| yielding (i) after substitution. For (ii) we only need to check the strong triangle inequality. Now x + y2 = |< x + y, x + y >| ≤ max(|< x, x >| , |< x, y >| , |< y, x >| , |< y, y >|). By (i) this is ≤ max(x2 , x y, y2 ) = max(x, y)2 , and we have (ii). Finally, to prove (iii), let < x, y > = 0. From (2.4) we obtain x + y2 ≥ y2 . By symmetry x + y ≥ max(x, y). Then use (ii) to arrive at (iii). Corollary 2.4.3 < , > is continuous with respect to the associated norm. Proof. Just observe that by 2.4.2(i) we have |< x, y >| ≤ x y for all x, y ∈ E. The next two “experiments” may have already crossed the reader’s mind: 1. Let K := Qp (p = 2). For x := (λ1 , λ2 , . . .) ∈ c0 and y := (µ1 , µ2 , . . .) ∈ c0 , set < x, y > :=

∞

λn µn .

n=1

of Qp is the (As Q is dense in Qp the only continuous field automorphism 1/2 n is summable (−X) identity.) The formal series (1 − X)1/2 = ∞ n=0 n 1/2 n 2 ∈ Q and α = 1 − p (for details (−p) for X = p, so α := ∞ p n=0 n see [195], Section 47). Now set y := (λ1 , . . . , λp , 0, 0, . . .), where λi = 1

40

Normed spaces

if 1 ≤ i ≤ p − 1, and λp = α. Then < y, y > = 0, so < , > has isotrope vectors and therefore is not an inner product. 2 2 2. Let 2 := {(λ1 , λ2 , . . .) ∈ K N : ∞ n=1 |λn | < ∞}. Then is a vector space but (λ1 , λ2 , . . .) −→ (

∞

|λn |2 )1/2

n=1

is an A-norm (see 2.1.2) that does not satisfy the strong triangle inequality and therefore is not associated to any inner product. Now we consider a true inner product in infinite-dimensional space. Example 2.4.4 Let K be such that (∗) λ21 + · · · + λ2n = max(|λ1 |2 , . . . , |λn |2 ) for all λ1 , . . . , λn ∈ K, n ∈ N. Then obviously the formula < x, y > :=

∞

λn µ n

(x = (λ1 , λ2 , . . .), y = (µ1 , µ2 , . . .))

n=1

an inner product on c0 (where λ∗ = λ for all λ ∈ K) with |< x, x >| = defines ∞ λ2 = maxn∈N |λn |2 = x2 . n=1

n

∞

To see that there exists a K with property (*), let K := R((X)) be the field of formal Laurent series in X over R (see Chapter 1 of [193] for details) with the valuation | . | given by |0| := 0 and for f = n∈Z an Xn = 0, an ∈ R, |f | := 2−j ,

where j := min{n : an = 0}. Then K is a complete non-Archimedean valued field and since a sum of squares in R is 0 only if all summands are, K has property (*). Now let us call a vector space E with an inner product Hilbert-like if E is a Banach space for the associated norm and such that for each closed subspace D of E we have D + D ⊥ = E. Here is the announced negative result. Theorem 2.4.5 Let |2| = 1. Then there do not exist infinite-dimensional Hilbert-like spaces over K. Proof. Let E be an infinite-dimensional Hilbert-like space; we derive a contradiction. The Gram–Schmidt process described earlier yields the existence of a sequence e1 , e2 , . . . of non-zero vectors that are form orthogonal. By suitable scalar multiplication we may assume that |ρ| ≤ en ≤ 1 for all n, where

2.5 Examples of Banach spaces

41

ρ ∈ K, 0 < |ρ| < 1. We may, after the observation that closed subspaces of Hilbert-like spaces are Hilbert-like, therefore assume that [e1 , e2 , . . .] is dense in E. Now by 2.4.2(iii) e1 , e2 , . . . is an orthogonal sequence in the sense of the norm (2.2) and by 2.3.6 is an orthogonal base of E. Hence every x ∈ E has a unique expansion x = ∞ n=1 λn en , where λn ∈ K, λn en → 0, i.e., λn → 0. Let ∞ ∞ D := λn en ∈ E : λn = 0 . n=1

n=1

Then D is a closed subspace, D = E. Let y ∈ D ⊥ ; we prove that y = 0 arriving at the desired contradiction. In fact, let y have expansion ∞ n=1 µn en . For each m, n ∈ N we have em − en ∈ D, so that µm < em , em > = < y, em > = < y, en >= µn < en , en >. But, since limn µn = 0 and n → < en , en > is bounded, we have limn µn < en , en > = 0, implying µn = 0 for all n. Hence y = 0. Remarks 2.4.6 (a) The restriction that |2| = 1 made above is not necessary. See [97], Remark 35. (b) Finite-dimensional Hilbert-like spaces are less interesting from an analytic point of view as any finite-dimensional space with an inner product is Hilbert-like. It depends on algebraic properties of K whether or not such inner products on K n exist.

2.5 Examples of Banach spaces Before starting we need one more technicality on summation of more than countably many vectors. Let E be a normed space, let I be a set, and let for each i ∈ I an element xi ∈ E be given. We say that the family (xi )i∈I is summable with sum x ∈ E if for every ε > 0 there is a finite set J0 ⊂ I such that, for all finite J with J0 ⊂ J ⊂ I , we have x − i∈J xi < ε. We then write x = i∈I xi . The proofs of the following statements are straightforward and left to reader. Theorem 2.5.1 Let E be a normed space. Let (xi )i∈I be a family of elements of E. Then we have the following: (i) For I = N summability as defined above coincides with the one of Section 2.1.

42

Normed spaces

(ii) (xi )i∈I has at most one sum. If it has one then limi xi = 0 i.e., for every ε > 0 the set {i ∈ I : xi ≥ ε} is finite. Conversely, if E is a Banach space and limi xi = 0 then (xi )i∈I is summable. (iii) If (xi )i∈I is summable then there is a countable set J := {i1 , i2 , . . .} ⊂ I such that xi = 0 if i ∈ I \ J , and i∈I xi = ∞ n=1 xin . With the above machinery it is not hard to extend the notion of t-orthogonal base to this more general situation and extend the Perturbation Lemma 2.3.16 accordingly. We will not bore the reader with the details.

2.5.1 The space c0 (I)

For a set I let c0 (I ) := i∈I Ki , where Ki = K for all i ∈ I . In other words, c0 (I ) is the set of all (λi )i∈I ∈ K I for which limi λi = 0. It is a Banach space under pointwise operations and (solid) norm . ∞ given by (λi )i∈I −→ max |λi | . i∈I

For I = {1, . . . , n}, where n ∈ N, c0 (I ) is nothing but K n introduced before 2.1.15; the space c0 (N) has been called c0 earlier. For each i ∈ I we have the “ith unit vector” ei ∈ c0 (I ), given by ei := ξ{i} . The linear hull of {ei : i ∈ I } is denoted by c00 (I ); instead of c00 (N) we write c00 . Clearly the (ei )i∈I form an orthonormal base of c0 (I ) in the sense just suggested. Obviously every Banach space with an orthonormal base is isometrically isomorphic to c0 (I ) for some I . Also, every Banach space with a t-orthogonal base for certain t ∈ (0, 1], is linearly homeomorphic to some c0 (I ). We have the following generalizations of 2.3.6 and 2.3.19 respectively. Theorem 2.5.2 Let t ∈ (0, 1], let (ei )i∈I be a t-orthogonal system in a normed space E. If [ei : i ∈ I ] is dense in E then (ei )i∈I is a t-orthogonal base of E. Proof. In the proof of 2.3.6, replace c0 by c0 (I ).

Theorem 2.5.3 Let E be a normed space with an orthogonal base. Then each finite-dimensional subspace has an orthogonal complement. Proof. For one-dimensional subspaces the proof is a simple adaptation of the one given for 2.3.19. Now use 2.3.20. As an application of 2.5.2 we have the following. Theorem 2.5.4 Let E be a Banach space with a solid norm over a discretely valued field K. Then E is isometrically isomorphic to c0 (I ) for some I . Each

2.5 Examples of Banach spaces

43

maximal orthogonal system in E is an orthogonal base. Every closed subspace of E has an orthogonal complement. Proof. We first prove the second assertion. Let {ei : i ∈ I } be a maximal orthogonal system in E, let D := [ei : i ∈ I ]. By 2.5.2 it suffices to show that D = E. Suppose x ∈ E, x ∈ D; we derive a contradiction. Let d1 , d2 , . . . ∈ D be such that x − d1 ≥ x − d2 ≥ · · · , limn x − dn = dist (x, D) > 0. By solidity, n → x − dn is a decreasing sequence in |K|, not converging to 0. Hence the sequence becomes stationary, i.e., there is a d0 ∈ D with x − d0 = dist (x, D). But then x − d0 ⊥ D, conflicting maximality. Now the first assertion follows easily by using scalar multiplications and solidity. To finish the proof, let D be a closed subspace of E. By the first part D has an orthogonal base (ei )i∈I , which can be extended to a maximal orthogonal system in E, (ej )j ∈J with J ⊃ I . Then clearly [ej : j ∈ J \ I ] is an orthogonal complement of D. Remark 2.5.5 The conclusion of the theorem does not hold in case K has a dense valuation, as we will see in 8.2.2. The spaces c0 (I ) are “generic” in the following sense. Compare with 2.3.28. Theorem 2.5.6 Every Banach space with a solid norm is a quotient of c0 (I ) for some I . Proof. Let E be a Banach space with solid norm. Let I be a set with the same cardinality as the unit ball BE , let σ : I −→BE be a bijection. Define a map T : c0 (I )−→E by T λi ei := λi σ (i). i∈I

i∈I

(Here the ei are the unit vectors of c0 (I ).) Clearly T is well-defined (by completeness of E), linear, surjective, and T ≤ 1. To show the crucial property, let y ∈ E, y = 0, let ε > 0; we produce an x ∈ c0 (I ) with T (x) = y and x∞ ≤ (1 + ε) y. By solidity of the norm there is a λ ∈ K such that (1 +ε)−1 ≤ λ y ≤ 1. Let x := λ−1 eσ −1 (λ y) . Then T (x) = y and x∞ = −1 λ ≤ (1 + ε) y. Remark 2.5.7 In a similar way we obtain that every normed space with a solid norm is a quotient of some c00 (I ). Theorem 2.5.8 c0 (I ) is of countable type if and only if I is countable.

44

Normed spaces

Proof. Direct consequence of 2.3.18.

Theorem 2.5.9 Let D ⊂ c0 (I ) be a non-zero subspace of countable type. Then we have the following: (i) D has an orthonormal base. (ii) If D is closed then it is complemented. In fact, for every ε > 0 there is a continuous projection P of E onto D with P ≤ 1 + ε. If D is finitedimensional it is orthocomplemented. Proof. First observe that for J ⊂ I we have a natural embedding c0 (J ) ⊂ c0 (I ) and a natural projection PJ of c0 (I ) onto c0 (J ). We first prove that there exists a countable J ⊂ I such that D ⊂ c0 (J ). In fact, let d1 , d2 , . . . ∈ D be such that [d1 , d2 , . . .] is dense in D. Each dn has an expansion i∈Jn λi ei , where Jn ⊂ I is countable. Let J := n∈N Jn . Then J is countable and all dn are in c0 (J ), hence D ⊂ c0 (J ). Now (i) follows from 2.3.22. For (ii), by 2.3.13 there is a continuous surjective projection Q : c0 (J )−→D with Q ≤ 1 + ε. Now take P := Q ◦ PJ . If D is finite-dimensional we can take ε = 0 in the above by 2.3.19 and 2.3.20. Remark 2.5.10 For a discussion on the conclusions of 2.5.9 for arbitrary D, see the chapter notes.

2.5.2 The space ∞ (I)

For a set I let ∞ (I ) := bi∈I Ki , where Ki = K for all i ∈ I . In other words, ∞ (I ) is the set of all bounded maps I −→K. It is a Banach space under pointwise operations and (solid) norm . ∞ given by (λi )i∈I −→ sup |λi | , i∈I

containing c0 (I ) as a closed subspace. For I = {1, . . . , n}, where n ∈ N, ∞ (I ) = c0 (I ) = K n . For ∞ (N) we usually write ∞ . In the Archimedean world the dual space of c0 is isometrically isomorphic to 1 . In our case, however, c0 is isometrically isomorphic to ∞ , as we will see below. For each x = (λi )i∈I ∈ c0 (I ) and y = (µi )i∈I ∈ ∞ (I ), the formula λi µi B(x, y) := i∈I ∞

defines a bilinear form B : c0 (I ) × (I )−→K.

2.5 Examples of Banach spaces

45

Theorem 2.5.11 The map y → B(., y) is an isometrical isomorphism ∞ (I )−→c0 (I ) . Proof. Writing fy = B(., y) we see that for each y = (µi )i∈I ∈ ∞ (I ) the map fy is in c0 (I ) and since fy (x) = |B(x, y)| ≤ x∞ y∞ (x ∈ c0 (I )) we have fy ≤ y∞ . Applying fy to the unit vectors ei of c0 (I ) we see fy (ei ) = |µi | = |µi | ei ∞ , so that fy ≥ |µi | for each i, i.e., fy ≥ y∞ . Thus, y → fy is isometrical. To show surjectivity, let g ∈ c0 (I ) . Then z := (g(ei ))i∈I ∈ ∞ (I ) since |g(ei )| ≤ g ei ∞ = g for all i. To see that fz = g, let x := (λi )i∈I = i∈I λi ei ∈ c0 (I ). Then, by linearity and continu ity, fz (x) = i∈I λi g(ei ) = g( i∈I λi ei ) = g(x). In the rest of the book we will identify ∞ (I ) and c0 (I ) by using the above isometrical isomorphism. Remark 2.5.12 For the dual of ∞ see 5.5.5 and 5.5.6. The following theorem is a “dual” to 2.3.28. Theorem 2.5.13 Any normed space E of countable type with a solid norm can linearly and isometrically be embedded into ∞ . Proof. We may assume that E is infinite-dimensional. Let 0 < t1 < t2 < · · · , limm tm = 1. By 2.3.7, for each m ∈ N, E has a tm -orthogonal base {emn : n ∈ N} such that (solidity) tm−1 ≤ emn ≤ tm−2 for each n. For x ∈ E, ∞ let n=1 λmn emn be its expansion with respect to {emn : n ∈ N}. Define T : E−→ ∞ (N × N) by T (x) := (λmn )m,n∈N . Then, for each m, tm2 x ≤ tm2 max |λmn | emn ≤ max |λmn | ≤ tm max |λmn | emn ≤ x, n∈N

n∈N

n∈N

so that, by taking the supremum over m, T (x) = x, i.e., T is an isometry. Now the conclusion follows after the observation that ∞ (N × N) is isometri cally isomorphic to ∞ . Remark 2.5.14 The dual version of 2.5.6 “every normed space with a solid norm can linearly and isometrically be embedded into ∞ (I ) for some I ” is not true. We will clarify the situation in 4.4.9. Next we prove the following theorem. Theorem 2.5.15 ∞ is not of countable type.

46

Normed spaces

Proof. As Q is countable it suffices to show that ∞ (Q) is not of countable type. By 2.3.18 it is enough to construct an uncountable orthonormal system in ∞ (Q). For each α ∈ R, let fα ∈ ∞ (Q) be defined via the formula 0 if x ∈ Q, x ≤ α fα (x) := 1 if x ∈ Q, x > α. Clearly fα ∞ = 1 for all α. To prove that {fα : α ∈ R} is orthonormal, choose α1 < · · · < αn in R and λ1 , . . . , λn in K. By selecting n x1 ∈ Q ∩ (α1 , α2), x2 ∈ Q ∩ (α 2 , α3 ), . . ., xn ∈ Q ∩ (αn , ∞) we obtain i=1 λi fαi (xj ) = λ1 + · · · + λj for each j , from which

n i=1

λi fαi ∞ ≥ max λ1 + · · · + λj = max |λi | = max λi fαi ∞ . 1≤j ≤n

1≤i≤n

1≤i≤n

Remark 2.5.16 From 2.5.4 it follows that if the valuation of K is discrete then c0 is orthocomplemented in ∞ (8.1.11). On the other hand, we will show in 8.2.1 that when K is densely valued there is not even a continuous linear surjection ∞ −→c0 . We will need the following generalizations in the sequel. Let I be a set, let s : I −→(0, ∞). Let ∞ (I, s) be the subspace of all (λi )i∈I ∈ K I for which i → |λi | s(i) is bounded, with the norm (λi )i∈I −→ sup |λi | s(i). i∈I

Then ∞ (I, s) is a Banach space. Its norm is solid if and only if the range of s lies in |K|. If s(i) = 1 for all i then ∞ (I, s) = ∞ (I ). In the same spirit we can define c0 (I, s) as the subspace of ∞ (I, s) consisting of those (λi )i∈I for which limi |λi | s(i) = 0. ∞ (I, s) is linearly homeomorphic to ∞ (I ). In fact, let ρ ∈ K, 0 < |ρ| < 1. For each i ∈ I choose µi ∈ K with |ρ| s(i) ≤ |µi | ≤ s(i). Then ∞ (I, s)−→ ∞ (I ) given by (λi )i∈I −→ (λi µi )i∈I does the job. In the same vein, c0 (I, s) is linearly homeomorphic to c0 (I ).

2.5.3 Banach spaces of continuous functions For a topological space X, let C(X−→K) be the vector space of all continuous functions X−→K. If no confusion is to be expected we write C(X) for

2.5 Examples of Banach spaces

47

C(X−→K). In the sequel we shall study connections between properties of C(X−→K) and certain subspaces on one hand, and topological properties of X on the other. In this context it is not satisfactory to have two different topologies on one set X leading to the same space of continuous functions. To overcome we can require that the topology on X is the weakest one making all f ∈ C(X−→K) continuous. This condition can be viewed as the non-Archimedean counterpart of complete regularity (see e.g. [52], p. 39). Theorem 2.5.17 The following statements on a topological space X are equivalent: (α) X has the weakest topology making all f ∈ C(X−→K) continuous. (β) X is zero-dimensional. Proof. Since K is zero-dimensional it is clear that (α) implies (β). Now suppose (β) holds. Let τ be the original topology on X, let τ be a topology on X making all f ∈ C(X−→K) continuous. Then, for any τ -clopen subset A of X, ξA is τ -continuous, so A is τ -clopen. Hence, τ ≤ τ , and we have (α). It is also reasonable to ask that the continuous functions on X separate the points of X. Theorem 2.5.18 The following statements on a zero-dimensional topological space X are equivalent: (α) The continuous functions X−→K separate the points of X. (β) X is Hausdorff. Proof. Only (α) =⇒ (β) needs a proof. Let x, y ∈ X, x = y. By (α) there is an f ∈ C(X−→K) with f (x) = f (y). Then U := {z ∈ X : |f (z) − f (x)| < |f (x) − f (y)|} is clopen, contains x, not y, and X \ U is a neighbourhood of y. So X is Hausdorff. Remark 2.5.19 Observe that both properties (β) above do not depend on K. From now on in Subsection 2.5.3, X is a non-empty zero-dimensional Hausdorff topological space. Definition 2.5.20 Let (X) be the collection of all clopen subsets of X. A (continuous) function f : X−→K is locally constant if for each x ∈ X there exists a neighbourhood of x on which f is constant. A step function is a continuous function X−→K with only finitely many values.

48

Normed spaces

Let BC(X−→K) (BC(X)) be the subspace of C(X−→K) consisting of all bounded continuous functions. For f ∈ BC(X), set f ∞ := sup{|f (x)| : x ∈ X}. Let P C(X−→K) (P C(X)) be the space of all f ∈ C(X−→K) for which f (X) is compact (i.e., f (X) is precompact). Finally, let C0 (X−→K)) (C0 (X)) be the space of all continuous functions f : X−→K “vanishing at infinity”, i.e., such that for each ε > 0 the set {x ∈ X : |f (x)| ≥ ε} is compact. From now on in Subsection 2.5.3 we will equip BC(X) and its subspaces with the (solid) norm . ∞ . Theorem 2.5.21 (X) is a ring of subsets covering X. A function f : X−→K is locally constant if and only if there exists a partition of X into clopen sets on each of which f is constant. The locally constant functions form a subspace of C(X). For each f ∈ C(X) and each ε > 0 there is a locally constant g : X−→K with |f (x) − g(x)| < ε for all x ∈ X. BC(X) is a closed subspace of ∞ (X), hence is a Banach space. The bounded locally constant functions form a dense subspace of BC(X). Each step function is locally constant. P C(X) is a closed subspace of BC(X). The step functions form a dense subspace of P C(X). C0 (X) is a closed subspace of P C(X). The functions in C(X) having compact support form a dense subspace of C0 (X). If X has the discrete topology then C0 (X) = c0 (X), BC(X) = ∞ (X). Proof. Everything is straightforward except maybe for the density statements. For f ∈ C(X) and ε > 0 the assignment “x ∼ y if |f (x) − f (y)| < ε” is an equivalence relation and the classes form a partition {Ui : i ∈ I } of X into clopen sets, which is finite if f ∈ P C(X) since by precompactness, f (X) is covered by finitely many open disks in K of radius ε. Choose ui ∈ Ui for each i and set g(x) := f (ui ) if x ∈ Ui . Obviously g is locally constant (a step function if f ∈ P C(X)) and |f (x) − g(x)| < ε for each x ∈ X. Now assume that f ∈ C0 (X). Let ε > 0. Then U := {x ∈ X : |f (x)| ≥ ε} is clopen and compact. Hence f ξU has compact support and f − f ξU ∞ ≤ ε. C(X) for compact X Clearly if X is compact then C(X) = BC(X) = P C(X) = C0 (X). We now prove that C(X) has an orthonormal base. Theorem 2.5.22 Let X be compact. Then C(X) has an orthonormal base. More precisely, let U be a maximal collection of clopen sets for which {ξU : U ∈ U} is orthonormal. Then {ξU : U ∈ U} is an orthonormal base of C(X).

2.5 Examples of Banach spaces

49

Proof. Let K1 be a closed subfield of K with non-trivial discrete valuation (1.2.9). Then {ξU : U ∈ U} is also orthonormal in C(X−→K1 ); we first prove it to be a maximal orthonormal system in C(X−→K1 ). Let D be the K1 linear span of {ξU : U ∈ U}. We must prove for f ∈ C(X−→K1 ), f = 0 that dist (f, D) < f ∞ . First, let f := ξV , where V ∈ (X). By assumption there are U1 , . . . , Un ∈ U and λ1 , . . . , λn ∈ K such that, for all x ∈ X, n λi ξUi (x) < 1. (2.5) ξV (x) − i=1

Now consider the K1 -linear span S of 1, λ1 , . . . , λn . By 2.3.21 and 2.3.25 there is a surjective K1 -linear projection P : S−→K1 with |P (α)| ≤ |α| for all α ∈ S. Applying P to (2.5) results into n P (λi ) ξUi (x) < 1 (x ∈ X), ξV (x) − i=1

so that dist (ξV , D) < 1 = ξV ∞ . Now let f ∈ C(X−→K1 ) be arbitrary, f = 0. There is a step function g ∈ C(X−→K1 ) with f − g∞ < f ∞ (2.5.21). Write g := ni=1 µi ξVi , where V1 , . . . , Vn is a finite clopen partition of X and µ1 , . . . , µn ∈ K1 , not all 0. By the above there are di ∈ D such that ξVi − di ∞ < 1 for i ∈ {1, . . . , n}. Then g − ni=1 µi di ∞ < g∞ = f ∞ and f − ni=1 µi di ∞ ≤ max(f − n g∞ , g − i=1 µi di ∞ ) < f ∞ . Thus, {ξU : U ∈ U} is a maximal orthonormal set in C(X−→K1 ). From 2.5.4 it follows that {ξU : U ∈ U} is an orthonormal base of C(X−→K1 ). be the K-linear span of {ξU : U ∈ U}. Then ξV ∈ D ⊂ D Finally, let D It follows for each V ∈ (X). Hence all K-valued step functions are in D. Now 2.5.2 shows that {ξU : U ∈ U} is an from 2.5.21 that C(X−→K) = D. orthonormal base of C(X−→K). As an application we have the zero-dimensional version of the Tietze–Urysohn Extension Lemma. Corollary 2.5.23 Let Y be a compact subset of X. Then every f ∈ C(Y ) can be extended to an f ∈ P C(X) such that f ∞ = f ∞ . Even more, there is a linear isometry T : C(Y )−→P C(X) such that T (f )|Y = f for all f ∈ C(Y ). Proof. First we prove that for each U ⊂ Y , U clopen in Y , there exists a clopen V ⊂ X with V ∩ Y = U . In fact, there is an open W ⊂ X with W ∩ Y = U . By zero-dimensionality, for each x ∈ U we can find a clopen Wx ⊂ X with x ⊂ Wx ⊂ W . These Wx cover U and by compactness there is a finite subcover Wx1 , . . . , Wxn . Set V := 1≤i≤n Wxi . Then V is clopen in X and

50

Normed spaces

V ∩ Y = U . Now, to prove the corollary, let (Ui )i∈I be a collection of clopens in Y such that (ξUi )i∈I is an orthonormal base of C(Y ). For each i choose a clopen Vi ⊂ X with Vi ∩ Y = Ui . Then the map ξUi → ξVi can uniquely be extended to a continuous linear map T : C(Y )−→P C(X), which satisfies the requirements. Next we determine for which compact X the space C(X) is of countable type. Theorem 2.5.24 Let X be compact. Then C(X) is of countable type if and only if X is ultrametrizable. Proof. Let C(X) be of countable type. Then there is a sequence f1 , f2 , . . . , with fn ∞ = 1 for all n, whose linear hull is dense in C(X). The formula d(x, y) := max |fn (x) − fn (y)| 2−n n∈N

(x, y ∈ X)

defines an ultrametric on X (since the fn separate the points we have d(x, y) = 0 =⇒x = y), which induces a Hausdorff topology ς on X. Calling the initial topology on X τ , we prove that the identity (X, τ )−→(X, ς) is continuous (then by compactness we have τ = ς , hence X is ultrametrizable). Let (xi )i τ be a net in X, xi −→ x, for some x ∈ X. Let ε > 0. Then 2−n < ε for n > N. Using the fact that fn (xi ) → fn (x) for n ∈ {1, . . . , N } we see that for large i, ς |fn (xi ) − fn (x)| < ε for n ∈ {1, . . . , N }. But then d(xi , x) < ε. So xi −→ x, and we are done. Conversely, let X be ultrametrizable. Let d be an ultrametric defining the topology. Choose r1 > r2 > · · · , limn rn = 0. Let B be the collection of all balls of the form {x ∈ X : d(x, a) ≤ rn }, where a ∈ X, n ∈ N. Then by compactness and the mercury drop behaviour (see Section 1.1) B is countable. Let D := [ξB : B ∈ B]. We prove that D is dense in C(X). To this end it suffices to show that ξU ∈ D for each clopen U ⊂ X (then all step functions are in D and, by 2.5.21, D = C(X)). In fact, the members of B that lie in U cover U . By compactness there is a finite subcover. Again by the mercury drop behaviour there is a further subcover that is a partition of U . It follows that ξU is a finite sum of ξB , B ∈ B, and we are done. For the next result we need the following lemma. Lemma 2.5.25 Let A1 , A2 , . . . , An be subsets of some set I . Suppose A1 ⊂ A2 ∪ · · · ∪ An , A2 ⊂ A3 ∪ · · · ∪ An , . . . , An−1 ⊂ An . Then ξA1 , . . . , ξAn are orthonormal in ∞ (I ). Proof. Let i ∈ {1, . . . , n − 1}. According to 2.2.7 it suffices to show that dist (ξAi , [ξAi+1 , . . . , ξAn ]) = 1. By assumption there is an a ∈ Ai a ∈

2.5 Examples of Banach spaces

51

Ai+1 ∪ · · · ∪ An . Then for every f ∈ [ξAi+1 , . . . , ξAn ] we have f (a) = 0, so that ξAi − f ∞ ≥ ξAi (a) − f (a) = 1. Remark 2.5.26 The natural converse of the above lemma is not true. Let I = N. Suppose the characteristic of k is not 2. Choose A1 := {2, 3}, A2 := {1, 3}, A3 := {1, 2}. Then A1 , A2 , A3 do not satisfy the conditions about inclusions, even after changing the order. To show orthonormality, let λ1 ξA1 + λ2 ξA2 + λ3 ξA3 ∞ < 1, where max1≤i≤3 |λi | = 1. Applying the canonical map λ → λ of BK onto k (see Section 1.2) we find λ1 ξ A1 (n) + λ2 ξ A2 (n) + λ3 ξ A3 (n) = 0 for all n, where now ξ indicates k-valued characteristic functions. Substituting n = 1, 2, 3 respectively we find λ2 + λ3 = λ1 + λ3 = λ1 + λ2 = 0 leading to λ1 = λ2 = λ3 := ν and 2 ν = 0. As the characteristic of k is not 2 we have ν = 0, so |λi | < 1 for each i ∈ {1, 2, 3}, a contradiction, showing orthonormality. If the characteristic of k equals 2 then, with the same methods, A1 := {2, 3, 4}, A2 := {1, 3, 4}, A3 := {1, 2, 4}, A4 := {1, 2, 3} will do the job. This example shows that orthogonality of characteristic functions of sets is not independent of K. Theorem 2.5.27 Let X be a compact ultrametric space. Then C(X) has an orthonormal base consisting of characteristic functions of balls. Even more, each maximal system of balls whose characteristic functions are linearly independent is an orthonormal base of C(X). Proof. Let S be a maximal collection of balls such that {ξB : B ∈ S} is linearly independent. Then the characteristic function of each ball is in [ξB : B ∈ S], hence every step function is in [ξB : B ∈ S]. So [ξB : B ∈ S] is dense in C(X) (2.5.21). By 2.5.2 it remains to be shown that S is an orthonormal set. Take distinct B1 , . . . , Bk ∈ S such that diam B1 ≥ diam B2 ≥ · · · ≥ diam Bk . If B1 ⊂ 1
52

Normed spaces

With the natural operations and the norm . defined by (ii) the measures on X form a Banach space, called M(X). Next we are going to integrate continuous functions with respect to a measure µ as follows. First, let f : X−→K be a step function. Let U1 , . . . , Un be a finite clopen partition of X such that f is constant, say λi ∈ K, on each Ui . Then f = ni=1 λi ξUi and we set n f dµ := λi µ(Ui ). X

i=1

By considering refinements one shows that the right-hand side does not depend on the clopen partition chosen such that f is constant on its members. A similar procedure yields linearity of f → X f dµ. Since n f dµ = λ µ(U ) i i ≤ max |λi | µ = f ∞ µ, 1≤i≤n X

i=1

we have that f → X f dµ is a continuous linear map, with norm ≤ µ, on the space of the step functions. By 2.5.21 it has a unique continuous linear extension to C(X), again denoted f → X f dµ. µ ∈ M(X) the map ϕµ : f → Theorem 2.5.29 Let X be compact. For each f dµ defined above is an element of C(X) whose norm equals µ. X Proof. It remains to be shown that µ ≤ ϕµ . But this follows from |µ(U )| = ϕµ (ξU ) ≤ ϕµ for each U ∈ (X). We now prove that for compact X every element of C(X) can be represented as an integral with respect to a measure. This is the counterpart of the classical Riesz Representation Theorem, but less deep and much easier to prove. Theorem 2.5.30 (Riesz Representation Theorem) Let X be compact. Then for every ϕ ∈ C(X) there is exactly one measure µ on X such that ϕ(f ) = X f dµ for all f ∈ C(X). More precisely, the map µ → ϕµ (µ ∈ M(X)) is an isometrical isomorphism M(X)−→C(X) . Proof. Direct verification yields linearity. So, by 2.5.29, only surjectivity remains to be proved. Let ϕ ∈ C(X) . Then µ(U ) := ϕ(ξU ) (U ∈ (X)) defines a measure µ for which ϕµ = ϕ. Remark 2.5.31 In Subsection 7.2.2 we will continue studying C(X) in the locally convex setting.

2.5 Examples of Banach spaces

53

The space C0 (X) of continuous functions vanishing at infinity Firstly, we find out when C0 (X) has “enough” elements. Theorem 2.5.32 The following are equivalent: (α) For each a ∈ X there is an f ∈ C0 (X) with f (a) = 0. (β) (β) X is locally compact. Proof. (α) =⇒ (β). Let a ∈ X, let f ∈ C0 (X), f (a) = 0. Then Ua := {x ∈ X : |f (x) − f (a)| < |f (a)|} is a clopen neighbourhood of a, contained in {x ∈ X : |f (x)| = |f (a)|} which set lies in a compact one. Hence, Ua is also compact. So each point of X has a compact neighbourhood, i.e., X is locally compact. (β) =⇒ (α). Let a ∈ X. By assumption a has a compact neighbourhood V . By zero-dimensionality there is a clopen U with a ∈ U ⊂ V . Then U is compact, open and ξU ∈ C0 (X); also ξU (a) = 1 = 0. Remark 2.5.33 The condition that C0 (X) separates the points of X is not enough to guarantee local compactness. In fact, define on X := R a topology as follows. A set U ⊂ X is called open if either 0 ∈ U or X \ U is countable. This makes X into a Hausdorff zero-dimensional space such that X \ {0} is discrete, and compact subsets of X are finite. For each a, b ∈ X, a = b, a = 0 the function ξ{a} is in C0 (X) and separates a and b. But neighbourhoods of 0 are infinite, hence not compact, i.e., X is not locally compact. (This example was kindly pointed out to us by J. Araujo.) Theorem 2.5.34 Let X be locally compact. Then C0 (X) has an orthonormal base consisting of characteristic functions of open compact sets. Proof. Let us prove that the linear hull of {ξU : U ⊂ X open compact} is dense in C0 (X). For that, let ε > 0 and let f ∈ C0 (X). As in the proof of 2.5.21 define an equivalence relation on X by “x ∼ y if |f (x) − f (y)| < ε”. Then the classes form a finite partition of X consisting of clopen sets U1 , . . . , Un . Take ui ∈ Ui and let J := {i ∈ {1, . . . , n} : |f (ui )| ≥ ε}. We may assume J = ∅. For each i ∈ J , the set Ui is contained in {x ∈ X : |f (x)| = |f (ui )|} which lies in a compact set, so Ui is compact. Also, for g := i∈J f (ui ) ξUi we have f − g∞ ≤ ε. From this point the proof of 2.5.22 applies with only obvious modifications. Before characterizing when C0 (X) is of countable type it is useful to have some topological considerations (that we also will apply later on).

54

Normed spaces

Lemma 2.5.35 Let X be locally compact. Then the following are equivalent: (α) X is σ -compact and ultrametrizable. (β) X is σ -compact and every compact subset of X is ultrametrizable. (γ ) X is ultrametrizable and separable. Proof. (α) =⇒ (β) is trivial. Next we prove (β) =⇒ (γ ). Let Y1 , Y2 , . . . be compact sets covering X. By local compactness we may assume that the Yn are open. Then U1 := Y1 , Un := Yn \ i 1) form a partition of X into open compact sets. Choose, for each n, an ultrametric dn on Un , dn < 1, yielding the topology on Un and define, for x, y ∈ X, dn (x, y) if x, y ∈ Un for some n d(x, y) := 1 otherwise. One checks immediately that d is an ultrametric on X inducing the topology. Hence X is ultrametrizable. To show separability observe that each Un is separable ([52], 4.1.18), so we can choose a countable set Sn ⊂ Un that is dense in Un . Then n∈N Sn is countable and dense in n∈N Un = X. Finally, we treat (γ ) =⇒ (α). By 1.1.3 X has a partition consisting of compact balls {Bi : i ∈ I }. Let S be a countable dense subset of X. Then S must meet every Bi , so that I is countable and we obtain σ -compactness. Theorem 2.5.36 Let X be locally compact. Then the following are equivalent: (α) C0 (X) is of countable type. (β) X is σ -compact and ultrametrizable. (γ ) X is ultrametrizable and separable. Proof. It suffices to prove that (α) is equivalent to (β) of 2.5.35. First assume that C0 (X) is of countable type. Let f1 , f2 , . . . ∈ C0 (X) be such that [f1 , f2 , . . .] is dense in C0 (X). For each m, n ∈ N, 1 Un,m := x ∈ X : |fn (x)| ≥ m is open and compact. For each a ∈ X there is an f ∈ C0 (X) with f (a) = 0, so fn (a) = 0 for some n. This shows that n,m∈N Un,m = X and we have σ compactness. Now let Y ⊂ X be compact. To show ultrametrizability of Y we may assume by local compactness that Y is open. Then we have the standard linear isometrical embedding C(Y )−→C0 (X), so C(Y ) is of countable type (2.3.14), hence Y is ultrametrizable by 2.5.24. Conversely, assume (β) of 2.5.35. There are compact sets U1 , U2 , . . . covering X; without loss suppose that the Un are open and U1 ⊂ U2 ⊂ · · · . By assumption and 2.5.24 each C(Un ) is of countable type, hence so is

2.5 Examples of Banach spaces

55

{f ∈ C0 (X) : supp f ⊂ Un for some n}, and this last space is dense in C0 (X) (2.5.21). Then (α) follows. P C(X) and BC(X) We first construct a compactification of X as follows. Recall that X is zerodimensional and Hausdorff. For each x ∈ X the formula ζx (U ) := ξU (x), U ∈ (X) defines an element ζx ∈ {0, 1}(X) . The latter space, with the product topology, is compact by the Tychonov Theorem ([52], 3.2.4), and the map ζX : X−→{0, 1}(X) , x → ζx is a homeomorphism of X onto ζX (X). The closure Xζ of ζX (X) in the compact space {0, 1}(X) , more precisely, the pair (ζX , X ζ ), is called the Banaschewski compactification of X. Xζ is compact, zero-dimensional, Hausdorff and ζX (X) is dense in X ζ . In the sequel we identify X with its image ζX (X); we use this in 2.5.41. If Y is a second zero-dimensional Hausdorff topological space with Banaschewski compactification (ζY , Y ζ ) and σ : X−→Y is a continuous map, then there is a unique continuous map σ ζ : Xζ −→Y ζ making the diagram X

σ

ζX

Xζ

/Y

σζ

ζY

/ Yζ

commute. Thus, the Banaschewski compactification is the zero-dimensional ˘ analogue of the Stone–Cech compactification (see e.g. [52], Section 3.6). Now, in particular, let f ∈ P C(X). Then f (X) is compact and zeroζ dimensional, so f (X) can be identified with f (X) . So there is a unique continuous f ζ : Xζ −→K making the diagram XA AA AA A ζX AA

f

Xζ

/K }> } }} }} f ζ } }

commutative. The map f → f ζ is easily seen to be linear and isometrical; we also have (f g)ζ = f ζ g ζ for all f, g ∈ P C(X). Thus, we have proved: Theorem 2.5.37 The map f → f ζ of above is an isometrical ring isomorphism of P C(X) onto C(Xζ ).

56

Normed spaces

As a direct consequence of this theorem and 2.5.22 we obtain: Corollary 2.5.38 P C(X) has an orthonormal base consisting of characteristic functions of clopen sets. Remark 2.5.39 For the question whether BC(X) has an orthogonal base, see [193], 5.25. It is easy to see when BC(X) and P C(X) are of countable type. Theorem 2.5.40 The following are equivalent: (α) BC(X) is of countable type. (β) P C(X) is of countable type. (γ ) X is compact and ultrametrizable. For the proof we use a lemma on the Banaschewski compactification. Lemma 2.5.41 If X ζ is ultrametrizable then X is compact. Proof. We show that X = Xζ . Suppose there exists an a ∈ X ζ , a ∈ X. There is a sequence x1 , x2 , . . . in X such that d(x1 , a) > d(x2 , a) > · · · , limn d(xn , a) = 0, where d is an ultrametric on Xζ yielding the topology. Let U0 := {x ∈ X : d(x, a) > d(x1 , a)} and for n ≥ 1 set Un := {x ∈ X : d(xn+1 , a) < d(x, a) ≤ d(xn , a)}. Then U0 , U1 , . . . form a clopen partition of X. Define f : X−→K by 0 if x ∈ Un , n is odd f (x) := 1 if x ∈ Un , n is even. Then f ∈ P C(X), but limx→a f (x) does not exist and f cannot be extended to a continuous function on X ζ , a contradiction. Proof of 2.5.40. The implication (α) =⇒ (β) is a consequence of 2.3.14, and (γ ) =⇒ (α) follows from 2.5.24. We prove (β) =⇒ (γ ). From (β) and 2.5.37 we have that C(Xζ ) is of countable type. Again by 2.5.24, X ζ is ultrametrizable so, by the above lemma, X = X ζ is compact and ultrametrizable.

2.5.4 Valued field extensions If L is a complete valued field extension of K, i.e., L is a complete valued field containing K as a subfield and the valuation of K is the restriction of the valuation of L, then L is in a natural way a Banach space over K. This observation generates a new class of examples of Banach spaces.

2.5 Examples of Banach spaces

57

For a ∈ L, let K(a) be the smallest subfield of L containing K and a. Below we shall study, under the assumption that K is algebraically closed, the K-Banach space K(a) and answer the natural question whether K(a) is of countable type (2.5.44). As a by-product we will construct a continuous linear map with only trivial closed invariant subspaces (2.5.48). Our main goal is to prove the following. Theorem 2.5.42 Let K be algebraically closed with residue class field k. Let L be a complete valued field extension of K, and let a ∈ L \ K. Then the K-Banach space K(a) has, for each t ∈ (0, 1), a t-orthogonal base. More specifically, we have the following: (i) If dist (a, K) is not attained then K(a) is of countable type but has no orthogonal base. (ii) If dist (a, K) is attained, but not in |K| then K(a) is of countable type, has an orthogonal, but no orthonormal base. (iii) If dist (a, K) is attained and in |K| then K(a) has an orthonormal base of cardinality k. Remark 2.5.43 Note that, by 2.3.25, case (i) only can happen if K is not spherically complete. Corollary 2.5.44 K(a) is not of countable type if and only if k is uncountable and dist (a, K) is attained and in |K|. Remark 2.5.45 To see that this case really occurs, choose K such that k is uncountable, let K(X) be the field of rational functions with coefficients in K with the valuation defined by λ0 + λ1 X + · · · + λn X n max0≤i≤n |λi | , µ + µ X + · · · + µ Xm := max 0 1 m 0≤j ≤m µj where λi , µj are in K, not all µj equal to 0, and let L be the completion of (K(X), | . |), and choose a := X. Problem Study K (a) in the spirit of 2.5.42 for non-algebraically closed K . The proof of 2.5.42 runs in several steps. Notice that by 2.3.7 we only need to prove (i), (ii) and (iii). Until 2.5.48 we assume that K , k, L, a are as in 2.5.42.

58

Normed spaces

Lemma 2.5.46 Let λ1 , . . . , λn and µ1 , . . . , µn be elements of L such that |λi − µi | < |µi | for each i ∈ {1, . . . , n}. Then n n n µi < µi . λi − i=1

i=1

i=1

Proof. Follows directly by observing that µλii − 1 < 1 for each i ∈ {1, . . . , n} and that {λ ∈ L : |λ − 1| < 1} is a multiplicative group. Proof of 2.5.42(i). First we show that K(a) is of countable type. For that we prove that the ring K[a] := [1, a, a 2 , . . .] is dense in K(a). (a) For every b ∈ K[a], b = 0 there is a µ ∈ K such that |b − µ| < |µ|. To see this, let b = λ0 + λ1 a + · · · + λn a n ; we may suppose n ≥ 1, λ0 , λ1 , . . . , λn ∈ K, λn = 0. By algebraic closedness there exist ω1 , . . . , ωn ∈ K such that b = λn (a − ω1 ) · · · (a − ωn ). Since dist (a, K) = dist (a − ωi , K) is not attained there are µ1 , . . . , µn ∈ K such that |a − ωi − µi | < |µi | for each i ∈ {1, . . . , n}. By 2.5.46 we have |b − µ| < |µ|, where µ := λn µ1 · · · µn . (b) For each b ∈ K[a], b = 0 we have b−1 ∈ K[a]. In fact, by (a) there is a µ ∈ K such that |b − µ| < |µ|. Then µ−1 b − 1 < 1, so that µ b−1 = −1 b)n ∈ K[a]. (1 − (1 − µ−1 b))−1 = ∞ n=0 (1 − µ (c) Conclusion. From (b) it follows that K(a) ⊂ K[a]. This, together with the obvious inclusion K[a] ⊂ K(a) leads to K[a] = K(a). Finally we prove that K(a) has no orthogonal base. From the assumption we obtain, via 2.3.21, that [1, a] has no orthogonal base and by 2.3.22 neither has K(a) (note that by 2.3.18 any orthogonal base of K(a) must be countable). Proof of 2.5.42(ii). Let λ0 ∈ K be such that |a − λ0 | = dist (a, K). Then K(a − λ0 ) = K(a) and dist (a − λ0 , K) = dist (a, K), so we may replace a by a − λ0 ; in other words, we may assume that |a| ∈ |K|. It suffices to show that {a n : n ∈ Z} is an orthogonal base of K(a). First observe that |a|n ∈ |K| for n ∈ N. Indeed, if |a|n = |λ| for some n ∈ N and λ ∈ K then, by algebraic closedness, there is a µ ∈ K with µn = λ, so that |a| = |µ| ∈ |K|, a contradiction. Next we prove orthogonality of {a n : n ∈ Z}. Let x :=

m i=s

λi a i ,

2.5 Examples of Banach spaces

59

wherei λs ,. . . , λj m ∈ K, not all 0. From what we have just proved it follows that λi a = λj a for all i, j ∈ {s, . . . , m} unless i = j or λi = λj = 0. Then |x| = maxs≤i≤m λi a i and orthogonality follows. We proceed to show that x −1 ∈ [a n : n ∈ Z], where x is as above. There is a unique j ∈ {s, . . . , m} with |x| = λj a j . Then λi a i = 1 + v, (λj a j )−1 x = 1 + (λj a j )−1 i=j

where v ∈ [a n : n ∈ Z], |v| < 1. Thus (λj a j ) x −1 =

∞

(−v)n ∈ [a n : n ∈ Z],

n=0

implying x −1 ∈ [a n : n ∈ Z]. Now continuity of the inverse map shows that [a n : n ∈ Z] is a field, hence must be equal to K(a). By 2.3.6 we obtain that {a n : n ∈ Z} is an orthogonal base of K(a). The proof of 2.5.42(iii) is somewhat more involved. Lemma 2.5.47 Let a ⊥ K, |a| = 1. Then for each polynomial P = λ0 + λ1 X + · · · + λn Xn ∈ K[X] and µ ∈ BK we have |P (µ)| ≤ max |λi | ≤ |P (a)| . 0≤i≤n

Proof. Only the second inequality needs a proof. We may assume λn = 0. By algebraic closedness there are ω1 , . . . , ωn ∈ K such that P = λn (X − ω1 ) · · · (X − ωn ) and by assumption we have |a − ωi | ≥ |a| = 1 for all i, so that |P (a)| ≥ |λn |. By the same token we obtain λ0 + λ1 a + · · · + λn−1 a n−1 ≥ |λn−1 |, i.e., |λn−1 | ≤ |P (a) − λn a n | ≤ max(|P (a)| , |λn |) = |P (a)|, and we can proceed inductively. Proof of 2.5.42(iii). Let λ0 ∈ K with |a − λ0 | = dist (a, K) ∈ |K|. Then K(a − λ0 ) = K(a) and dist (a − λ0 , K) = dist (a, K), so we may assume that a ⊥ K and |a| = 1. Let σ : k−→BK be such that σ (u) = u for all u ∈ k, where the bar indicates the canonical map BK −→k. Now let S := {a s : s ∈ N ∪ {0}} ∪ {(a − µ)−m : µ ∈ σ (k), m ∈ N}. S is a subset of K(a) and, since k is infinite, we have S = k. We now establish (a)–(d) below which, together with 2.5.2, show that S is an orthonormal base of K(a): (a) S is an orthonormal system. (b) [S] is a subring of K(a).

60

Normed spaces

(c) For each β ∈ K, (a − β)−1 ∈ [S]. (d) K(a) ⊂ [S]. Proof of (a). Clearly each member of S has length 1. Take a linear combination :=

s

ξr a r +

n m

λij (a − µi )−j

i=1 j =1

r=0

(s ∈ N ∪ {0}, m, n ∈ N, ξr , λij ∈ K, µi ∈ σ (k)). For orthonormality of S it suffices to show, assuming || < 1, that all |ξr | and λij are < 1. Via (downward) induction on n we only prove that all |ξr | and all |λin | are < 1 (r ∈ {0, . . . , s}, i ∈ {1, . . . , m}). To obtain polynomials we multiply by L(a) := (a − µ1 )n · · · (a − µm )n , which does not change the absolute value as |L(a)| = 1 and the assumption || < 1 turns into |V1 (a) + V2 (a)| < 1,

(2.6)

where V1 , V2 ∈ K[X]. In fact, for x ∈ L, s ξr x r ) L(x), V1 (x) := ( r=0

V2 (x) :=

n m

λij Lij (x),

i=1 j =1

where Lij (x) := (x − µi )n−j

(x − µl )n . l=i

Let r ∈ {0, . . . , s}. If ξr = 0 the degree of V1 is ≥ m n, whereas V2 has degree < m n. Thus, ξr is a coefficient of the polynomial V1 + V2 , so that by 2.5.47 we have |ξr | ≤ |V1 (a) + V2 (a)| < 1. So all |ξr | are < 1 and |V1 (a)| < 1, hence (2.6) reduces to |V2 (a)| < 1. Choose q ∈ {1, . . . , m}. Then since Lij (µq ) = 1 if q = i, j = n 0 otherwise we find by 2.5.47 1 > |V2 (a)| ≥ V2 (µq ) = λqn , so that |λ1n |, . . . , |λmn | are all < 1.

2.5 Examples of Banach spaces

61

Proof of (b). It suffices to show that S .S ⊂ [S]. For µ ∈ σ (k) and m ∈ N the identity a (a − µ)−m = (a − µ)1−m + µ (a − µ)−m shows that a S ⊂ [S]. Then a 2 S = a a S ⊂ a [S] = [a S] ⊂ [S], and so on proving that a r S ⊂ [S] for each r ∈ N ∪ {0}. It remains to be shown that (a − µ1 )−m1 (a − µ2 )−m2 ∈ [S] for m1 , m2 ≥ 1, µ1 , µ2 ∈ σ (k). If µ1 = µ2 this is clear, so suppose µ1 = µ2 . We use induction with respect to n := m1 + m2 . If n = 2 (then m1 = m2 = 1) the formula (a − µ1 )−1 (a − µ2 )−1 =

1 ((a − µ1 )−1 − (a − µ2 )−1 ) µ1 − µ2

(2.7)

does the job. For the step n − 1 → n observe that we have, using (2.7), (a − µ1 )−m1 (a − µ2 )−m2 1 = ((a − µ1 )−1 − (a − µ2 )−1 ) (a − µ1 )−m1 +1 (a − µ2 )−m2 +1 , µ1 − µ2 which is a linear combination of the elements (a − µ1 )−m1 (a − µ2 )−m2 +1 and (a − µ1 )−m1 +1 (a − µ2 )−m2 , which are by the induction hypothesis in [S]. −1 Proof of (c). If |β| > 1 we have (a − β)−1 = −β −1 ∞ a)n ∈ [S], so n=0 (β let |β| ≤ 1. Then there is a µ ∈ σ (k) with |β − µ| < 1 and β − µ −1 −1 −1 −2 1− (a − β) − (a − µ) = (β − µ) (a − µ) a−µ ∞ = (β − µ) (a − µ)−2 (β − µ)n (a − µ)−n ∈ [S]. n=0

Proof of (d). Every element of K(a) can be written as P (a) Q(a)−1 for some polynomials P , Q ∈ K[X], Q(a) = 0. By algebraic closedness we can decompose Q into linear factors, whose inverses are in [S] by (c). Then, since [S] is a ring by (b), P (a) Q(a)−1 ∈ [S], which finishes the proof. As an application we will construct an example of a continuous linear map on c0 without non-trivial closed invariant subspaces. Theorem 2.5.48 Let K be algebraically closed and not spherically complete (e.g. K = Cp ). Then there is a T ∈ L(c0 ) such that for each closed subspace D with T (D) ⊂ D it follows that D = {0} or D = c0 . Proof. Let L ⊃ K be the spherical completion of K, let a ∈ L \ K. Then dist (a, K) is not attained, since L is an immediate extension of K. So we

62

Normed spaces

are in case (i) of 2.5.42 showing that K(a) is of countable type (and infinitedimensional as K is algebraically closed), hence linearly homeomorphic to c0 (2.3.9), so it suffices to define a continuous linear map T : K(a)−→K(a) with the required properties. In fact, let T (x) := a x (x ∈ K(a)), let D = {0} be a closed invariant subspace. For d ∈ D, d = 0 we have d, a d, a 2 d, . . . ∈ D, so K[a] d = [1, a, a 2 , . . .] d ⊂ D. Now in the proof of 2.5.42(i) we found that K[a] is dense in K(a). Hence K(a) d ⊂ D. But K(a) is a field. So K(a) d = K(a), and we are done. The above proof reveals the following interesting fact. Theorem 2.5.49 Let K be algebraically closed and not spherically complete. Then on c0 there exists an equivalent norm . with c0 = |K| such that: (i) no two non-zero vectors are . -orthogonal, (ii) there is a multiplication on c0 making (c0 , . ) into a valued field. Proof. From the proof of 2.5.48 it follows that K(a) is linearly homeomorphic to c0 and that K(a) is a subfield of the spherical completion of K. Let . be the valuation on K(a). Now, if λ, µ ∈ K(a), λ ⊥ µ with respect to . and λ, µ = 0 then λ µ−1 = 0, λ µ−1 ⊥ K, which is not possible since K(a) is an immediate extension of K. Remark 2.5.50 The conclusion of 2.5.48 is true for any field K, as was recently ´ proved by Sliwa in [222].

2.5.5 Spaces of power series Throughout Subsections 2.5.5 and 2.5.6 K is algebraically closed (so its valuation is dense). By a disk we mean either K or a ball in K . Let B be a disk, f : B−→K, a ∈ B, and suppose f has a power series expan sion about a, i.e., there exist λ0 , λ1 , . . . ∈ K such that f (x) = ∞ n=0 λn (x − n a)n for all x ∈ B. Now let b ∈ B. By writing (x − a) = (x − b + b − a)n = n n i n−i and some elementary calculus one concludes that i=0 i (x − b) (b − a) n there are µ0 , µ1 , . . . ∈ K such that f (x) = ∞ n=0 µn (x − b) for all x ∈ B (for the full proof see [195], 25.1). Thus, if f has a power series expansion about some point of B then it has one about any point of B. Definition 2.5.51 For a disk B let P S(B) be the set of all f : B−→K that can be developed into a power series about some (any) point of B. The elements of P S(K) are called entire functions. Direct verification shows that P S(B) is a vector space, even a K-algebra with respect to pointwise multiplication.

2.5 Examples of Banach spaces

63

For the properties of P S(B) it makes a lot of difference whether B is “fenced” or not. Definition 2.5.52 A disk B is called fenced if it has the form B(a, r), where a ∈ K, r ∈ K × . Let us introduce some notations. Let ∅ = Y ⊂ X ⊂ K, let f : X−→K. We write f Y := sup{ |f (y)| : y ∈ Y } (admitting the value ∞). Sometimes we write f r (f − r ) instead of f B(0,r) ( f B(0,r − ) ). Finally, we shall denote the function K−→K, x → x, and its restrictions to non-empty subsets, by X. Theorem 2.5.53 Let B be the fenced disk B(a, r), where a ∈ K, r ∈ K × . Then every f ∈ P S(B) is bounded and f B(a,r) = max{|f (x)| : x ∈ B(a, r)} = f B(a,r − ) . ∞ If f (x) = n=0 λn (x − a)n is an expansion of f then limn |λn | r n = 0 and f B(a,r) = max |λn | r n . n

Proof. Choose µ ∈ K with |µ| = r. The affine transformation σ : x → a + µ x sends BK onto B, hence f → f ◦ σ sends P S(B) onto P S(BK ). Using this map the proof is reduced to the case where a = 0, r = 1. n By taking x = 1 in the expansion f (x) = ∞ n=0 λn x we see that limn λn = n 0. Since also |f (x)| ≤ maxn |λn x | ≤ maxn |λn | for all x ∈ BK we have that f is bounded and f 1 ≤ maxn |λn |. We prove that there is an x ∈ BK with |f (x)| = maxn |λn |. To this end we may assume maxn |λn | = 1. Let |λn | < 1 for n > N . From the assumption |f (x)| < 1 for all x ∈ BK we obtain N n n=0 λn x = 0 for all x ∈ k (for the meaning of the bar and k, see Section 1.2). Then a non-trivial polynomial has all the elements of k as zeros. As k is infinite this is impossible. × , r < 1}. By what we just have Finally, f − 1 = sup{f r : r ∈ K proved this equals supr<1 maxn |λn | r n = supn supr<1 |λn | r n = supn |λn | = f 1 . This finishes the proof. As a consequence we obtain that for any disk B and f ∈ P S(B) its coefficients in the expansion about any point of B are uniquely determined. Also, we derive the following. Corollary 2.5.54 Let B = B(a, r), where a ∈ K, r ∈ K × . For f ∈ P S(B), n let f (x) = ∞ n=0 λn (x − a) be its expansion. Then the map f → (λ0 , λ1 , . . .)

64

Normed spaces

is an isometrical isomorphism of P S(B) (equipped with the norm . B ) onto c0 (N ∪ {0}, s), where s(n) = r n , n ∈ N ∪ {0}. Corollary 2.5.55 For each fenced disk B the space P S(B) is isometrically isomorphic to c0 , hence of countable type. Now we study the case of non-fenced disks. Theorem 2.5.56 For every non-fenced disk B there exist unbounded functions in P S(B). Proof. Without loss, assume 0 ∈ B. The function X is in P S(K) and unbounded, so we restrict ourselves to the case B = B(0, r − ) for some r > 0. By density of the valuation we can find λn ∈ K such that 2n n ≤ |λn | ≤ n rn r

(n ∈ {0, 1, . . .}).

Because |λn x n | ≤ 2n (|x| /r)n we have limn λn x n = 0 for all x ∈ B, so n f (x) := ∞ n=0 λn x is defined on B and f ∈ P S(B). To show that f is unbounded, consider f B = sup{f v : v ∈ K × , v < r} = sup max |λn | v n v
(for the second equality apply 2.5.53).

Definition 2.5.57 For a disk B let BP S(B) be the space of all bounded functions in P S(B). Theorem 2.5.58 Let B = B(a, r − ), where a ∈ K, r > 0. For f ∈ BP S(B), n let f (x) = ∞ n=0 λn (x − a) be its expansion. Then the map f → (λ0 , λ1 , . . .) is an isometrical isomorphism of BP S(B) (equipped with the norm . B ) onto ∞ (N ∪ {0}, s), where s(n) = r n , n ∈ N ∪ {0}. Proof. We may assume a = 0. Then, by 2.5.53, f B = sup{f v : v < r, v ∈ K × } = sup max |λn | v n = sup |λn | r n , v
n

n

and the statement follows.

Corollary 2.5.59 For every non-fenced bounded disk B the space BP S(B) is linearly homeomorphic to ∞ and hence not of countable type. It remains to determine BP S(K). This is in fact the Liouville Theorem.

2.5 Examples of Banach spaces

65

Theorem 2.5.60 BP S(K) contains only the constants, i.e., a bounded entire function is constant. n Proof. Let f be entire with expansion f (x) = ∞ n=0 λn x . Let |f (x)| ≤ M for all x ∈ K (M > 0). Then, for each r > 0 and n ≥ 1, M ≥ f r ≥ |λn | r n . By letting r → ∞ we find λn = 0 for each n ≥ 1. Remark 2.5.61 In 3.7.25 we will meet another way how to deal with unboundedness of functions in P S(B(0, r − )), namely by equipping the space with the family of norms { . s : s < r}. In a sort of dual way we can also consider for B(0, r), functions that are in P S(B(0, s)) for some s > r and take s>r {f |B(0, r) : f ∈ P S(B(0, s))}. This is a subspace of P S(B(0, r)), the space of the “overconvergent power series”. See Subsection 3.7.2.

2.5.6 Analytic elements To arrive at a useful definition of analytic function on sets that are not disks we cannot simply take over the principle of analytic continuation from complex theory, because of the mercury drop behaviour of disks. Also, local analyticity is not an option since, for example, a non-empty open subset of K decomposes into disjoint disks, on each of which we can define an arbitrary power series function, so that there is no relation between the behaviour of the function in one neighbourhood and its behaviour elsewhere. To overcome this difficulty Krasner, [145], has proposed the following, see also [53]. Definition 2.5.62 Let A be a non-empty subset of K. Let R(A) be the space of all rational functions that can be defined on A (i.e., that have no poles in A). A function f : A−→K is called an analytic element on A if there exist f1 , f2 , . . . ∈ R(A) such that limn fn = f uniformly. The analytic elements on A form a vector space called H (A). If f, g ∈ H (A) are bounded then f g ∈ H (A) and f gA ≤ f A gA . Theorem 2.5.63 Let A ⊂ K be closed and bounded. Then each f ∈ R(A), hence each f ∈ H (A), is bounded. Proof. By algebraic closedness we may assume that f has the form γ

(X − λ1 )(X − λ2 ) · · · (X − λn ) (X − µ1 ) · · · (X − µm )

66

Normed spaces

for some γ , λ1 , . . . , λn ∈ K, µ1 , . . . , µm ∈ K \ A, n, m ∈ N. For each i ∈ {1, . . . , m} we have 1 1 = sup : x ∈ A = inf{|x − µi | : x ∈ A}−1 X − µ |x − µi | i A = dist (µi , A)−1 . Further, X − λj A ≤ max(M, λj ) for each j ∈ {1, . . . , n}, where M := sup{|x| : x ∈ A}. Thus, we get dist (µi , A)−1 . f A ≤ |γ | max(M, |λ1 | , . . . , |λn |) 1≤i≤m

Corollary 2.5.64 For closed bounded A ⊂ K the space H (A), equipped with the norm . A , is a Banach space, and also a K-algebra with respect to pointwise multiplication. Remark 2.5.65 It is easy to see that H (K) is the set of all polynomials in K and therefore H (K) = P S(K). The uniform topology is the discrete one and so H (K) is not a normed space. At this point it is natural to ask whether H (A) = BP S(A), where A is a bounded disk. Again, the answers are different according to whether A is fenced or not. Theorem 2.5.66 For any bounded disk A we have H (A) ⊂ BP S(A). Proof. We may assume 0 ∈ A. We show that rational functions are in P S(A) (then they are in BP S(A), by 2.5.63). Since R(A) and P S(A) are K-algebras it suffices to show that f ∈ P S(A) for all f ∈ R(A) from a set that generates R(A) as K-algebra viz. {1, X} ∪ {(X − µ)−1 : µ ∈ K \ A}. This is done by showing that (X − µ)−1 ∈ P S(A), where µ ∈ K \ A. In fact, |µ| > |x| for all x ∈ A, so ∞ 1 Xn −1 −1 = −µ ∈ P S(A). = X−µ µn µ(1 − X ) µ n=0

Next we look into the question of equality. Theorem 2.5.67 Let A = B(a, r), where a ∈ K, r > 0. Then f : A−→K n is in H (A) if and only if f has an expansion x → ∞ n=0 λn (x − a) with n limn |λn | r = 0.

2.5 Examples of Banach spaces

67

Proof. We may assume a = 0. Clearly if f has an expansion as required then f is the uniform limit on A of polynomials, hence f ∈ H (A). To prove the converse it suffices to show the statement for an f of the form (X − µ)−1 , where µ ∈ K \ A, i.e., |µ| > r. For x ∈ A we have |x/µ| < 1, so ∞ 1 −1 −1 µ−n x n , = x = −µ x−µ µ(1 − µ ) n=0

and clearly µ−n r n tends to 0, which completes the proof.

Corollary 2.5.68 Let A = B(a, r), where a ∈ K, r > 0. Then H (A) is of countable type. In fact, H (A) c0 (N ∪ {0}, s), where s(n) = r n for all n ∈ N ∪ {0}. Also, we have the following: (i) If A is fenced then H (A) = BP S(A) = P S(A). (ii) If A is not fenced then H (A) is strictly contained in BP S(A). n Proof. Let f = ∞ n=0 λn (X − a) ∈ H (A). By either 2.5.53 or 2.5.58 we have n f A = supn |λn | r , which together with 2.5.67 show that H (A) c0 (N ∪ {0}, s). Next we prove (i). By 2.5.66, it suffices to see that P S(A) ⊂ H (A). Let f ∈ P S(A). Applying 2.5.53, f is an uniform limit of polynomials, so f ∈ R(A) = H (A). Finally (ii) follows from 2.5.59. We now consider the case A = B(a, r − ) with r ∈ K × . Theorem 2.5.69 Let A = B(a, r − ), where a ∈ K, r ∈ K × . Then H (A) has an orthonormal base of cardinality k, where k is the residue class field of K. In particular, H (A) is of countable type if and only if the residue class field of K is countable. Proof. With the same reasoning as in 2.5.53 the proof is reduced to the case A = BK− . Let L be the completion of (K(X), | . |) as in 2.5.45. The assignment X → X defines a unique linear ring homomorphism : R(BK− )−→K(X). To prove that is also an isometry, first let f be a polynomial function N − n − n=0 λn X ∈ R(BK ). Then by 2.5.58 we have f BK = maxn |λn | = |(f )|. f − Now consider an element g ∈ R(BK ), where f, g are polynomial functions and g has the form (X − µ1 ) · · · (X − µm ) with |µi | ≥ 1. Then |g| is the constant |µ1 · · · µm | on BK− , hence f f BK− f . = = g − gBK− g BK

68

Normed spaces

By continuity extends to a linear isometry : H (BK− )−→L. Like in the proof of 2.5.42(iii), let σ : k−→BK be such that σ (u) = u for all u ∈ k. By using one obtains easily, following the same arguments as in 2.5.42(iii), that {Xs : s ∈ N ∪ {0}} ∪ {(X − µ)−m : µ ∈ σ (k × ), m ∈ N} is an orthonormal base of H (BK− ) (notice that here µ ∈ σ (k × ) instead of µ ∈ σ (k)). Corollary 2.5.70 Let the residue class field of K be countable. Let A ⊂ K be a bounded disk. Then, A is fenced if and only if H (A) = BP S(A). Proof. If A = B(a, r) with a ∈ K, r > 0, the statement follows from 2.5.68. If A = B(a, r − ) with r ∈ |K|, then by 2.5.69 we have H (A) = c0 (I ) where I = k, so H (A) c0 and BP S(A) ∞ (2.5.58). We see that H (A) = BP S(A). Remark 2.5.71 In 8.2.2 we will prove that ∞ does not have a base. By using this fact in the above proof we see that in 2.5.70 we may drop the countability condition. If K happens to be separable (which includes the important case K = Cp ) we can say more. Theorem 2.5.72 Let K be separable. Then, for each closed bounded set A, H (A) is separable (hence of countable type). Proof. Let S be a countable dense subset of K, S × a countable dense subset of K × , and T a countable dense subset of K\A. For each n, m ∈ {0, 1, . . .} consider the set λ0 (X − λ1 ) · · · (X − λn ) : λ0 , λ1 , . . . , λn ∈ S, Wn,m := µ0 (X − µ1 ) · · · (X − µm ) µ0 ∈ S × , µ1 , . . . , µm ∈ T . Wn,m is countable, hence so is W := n.m Wn,m . The map K−→R(A), λ → (X − λ) is continuous. To show that also K\A−→R(A), µ → (X − µ)−1 is continuous, let µ, β ∈ K \A be such that |µ − β| < dist (µ, A) = dist (β, A). Then, (X − µ)−1 − (X − β)−1 A = |µ − β| (X − µ)−1 (X − β)−1 A ≤ |µ − β| (X − µ)−1 A (X − β)−1 A = |µ − β| dist(µ, A)−1 dist(β, A)−1 = |µ − β| dist (µ, A)−2 , and continuity follows.

2.5 Examples of Banach spaces

69

Thus, (λ0 , λ1 , . . . , λn ) × (µ0 ) × (µ1 , . . . , µm ) −→

λ0 (X − λ1 ) · · · (X − λn ) µ0 (X − µ1 ) · · · (X − µm )

is a continuous surjection K n+1 × K × × (K \ A)m −→ Rn,m (A) f := ∈ R(A) : degree f = n, degree g = m . g Now S n+1 × S × × T m is dense in K n+1 × K × × (K \ A)m , so by continuity Wn,m is dense in Rn,m (A). Hence, W is dense in n,m Rn,m (A) = R(A), and we are done. Problem For non-separable K , characterize those closed and bounded A for which H( A) is of countable type. Remark 2.5.73 We leave it as an exercise for the reader to show that for the annulus A := {λ ∈ K : 0 < r1 ≤ |λ| ≤ r2 }, H (A) consists of all functions of the form x → n∈Z λn x n , where limn |λn | r2n = limn |λ−n | r1−n = 0. One can show that, indeed, if f ∈ H (A), f = 0 in some non-empty open subset of A then f = 0. However, this fails if we take for A the “less connected looking” set {λ ∈ K : |λ| ≤

1 } ∪ {λ ∈ K : 2 ≤ |λ| ≤ 3}. 2

In fact, the functions fn := (1 − Xn )−1 are in R(A), limn fn = 1 uniformly on the disk and limn fn = 0 uniformly on the annulus. So H (A) contains a nontrivial characteristic function! An even more dramatic example is furnished by compact A (e.g. A := Zp , K := Cp ). By the Weierstrass Theorem (1.2.17) every f ∈ C(A) can be uniformly approximated on A by polynomials, so we have H (A) = C(A): elements of H (A) may not even be differentiable! In order to avoid such pathologies one has restricted the theory to a class of subsets of K having properties resembling connectedness. We refer the interested reader to [53].

2.5.7 C n -functions Throughout this subsection X is a non-empty subset of K , without isolated points.

70

Normed spaces

We start with a brief introduction. Like in the classical case we say that a function f : X−→K is differentiable at a ∈ X if f (a) := limx→a (f (x) − f (a))/(x − a) exists. As usual, f (a) is called the derivative of f at a. If f is differentiable at all points of X it is called differentiable (on X); the function f : X−→K is called the derivative of f . For n ≥ 2 we define inductively the n-th derivative f (n) of f as the derivative of f (n−1) , when this derivative exists. Also, f (0) := f . The well-known rules for differentiation of sums, products, quotients and compositions of differentiable functions apply. Rational functions without poles in X are differentiable. Locally constant functions are differentiable with derivative 0. Differentiable functions are continuous. The classical approach to define C 1 -functions has disadvantages in the ultrametric context (See [195] for more objections.) Example 2.5.74 Let X be compact. Then the space {f : X−→K : f is differentiable and f is continuous} is not complete with respect to the norm f → max(f ∞ , f ∞ ). Proof. There exist (2.5.21) locally constant functions f1 , f2 , . . . on X such that limn fn (x) = x uniformly in x ∈ X. The sequence f1 , f2 , . . . is Cauchy but not convergent for the given norm as 0 = limn fn (x) = 1 for all x ∈ X. Thus, we make the requirements for a function to be C 1 somewhat stronger. Definition 2.5.75 A function f : X−→K is called continuously differentiable (sometimes called strictly differentiable), or a C 1 -function, if the difference quotient 1 f : (x, y) −→

f (x) − f (y) x−y

(x, y ∈ X, x = y)

can be extended to a continuous function 1 f : X2 −→K. The collection of all C 1 -functions on X is denoted by C 1 (X−→K). (If no confusion is to be expected we write C 1 (X) for C 1 (X−→K).) Remarks 2.5.76 (a) Clearly a C 1 -function f is continuous. Since X has no isolated points the continuous extension 1 f is unique. We have f (a) = 1 f (a, a) (a ∈ X), so f ∈ C 1 (X) implies continuity of f . The converse is not true (see [195], Remark 1 following 27.1). (b) We will see in 2.5.84 that for compact X the space C 1 (X) is a Banach space with the norm f −→ max(f ∞ , 1 f ∞ ).

2.5 Examples of Banach spaces

71

By considering second-order difference quotients f (x)−f (y) x−y

−

f (x)−f (z) x−z

y−z we can define C 2 -functions, etc. We proceed formally as follows. Definition 2.5.77 For n ∈ N set n X := {(x1 , . . . , xn ) ∈ Xn : if i = j then xi = xj }. For f : X−→K define its n-th difference quotient n f : n+1 X−→K inductively by 0 f := f and, for (x1 , . . . , xn+1 ) ∈ n+1 X, n f (x1 , . . . , xn+1 ) :=

n−1 f (x1 , x3 , . . . , xn+1 ) − n−1 f (x2 , x3 , . . . , xn+1 ) . x1 − x2

Of course some computational machinery is needed. Theorem 2.5.78 (Rules for n ) Let f, g : X−→K and let λ, µ ∈ K, n ∈ {0, 1, . . .}. Then we have the following rules: (I) For (x, y, z, x1 , . . . , xn−1 ) ∈ n+2 X, (x − y) n f (x, y, x1 , . . . , xn−1 ) + (y − z) n f (y, z, x1 , . . . , xn−1 ) = (x − z) n f (x, z, x1 , . . . , xn−1 ). (II) n f is a symmetric function of n + 1 variables. (III) For (x1 , . . . , xn+1 , a1 , . . . , an+1 ) ∈ 2n+2 X, n f (x1 , . . . , xn+1 ) − n f (a1 , . . . , an+1 ) =

n+1 (xi − ai ) n+1 f (a1 , . . . , ai , xi , . . . , xn+1 ). i=1

(IV) n (λ f + µ g) = λ n f + µ n g. (V) For (x1 , . . . , xn+1 ) ∈ n+1 X, n (f g)(x1 , . . . , xn+1 ) n k f (x1 , . . . , xk+1 ) n−k g(xk+1 , . . . , xn+1 ). = k=0

(VI) For (x1 , . . . , xn+1 ) ∈ n+1 X, n f (x1 , . . . , xn+1 ) =

n+1 (xi − xj )−1 f (xi ). i=1 j =i

72

Normed spaces

(VII) If f (x) = 0 for all x ∈ X then for g =

1 , f

(x1 , . . . , xn+1 ) ∈ n+1 X,

n g(x1 , . . . , xn+1 ) n k f (x1 , . . . , xk+1 ) n−k g(xk+1 , · · · , xn+1 ). = −f (x1 )−1 k=1

(VIII) If f is locally constant then n f = 0 in a neighbourhood of the set {(x1 , . . . , xn+1 ) ∈ X n+1 : x1 = · · · = xn+1 }. (IX) n+1 f = 0 if and only if f is a polynomial function of degree ≤ n. (X) For each r > 0 and k ∈ N ∪ {0}, k−n if k ≥ n n (Xk )(x1 , . . . , xn+1 ) ≤ r 0 if k < n, for all (x1 , . . . , xn+1 ) ∈ n+1 X for which |xi | ≤ r, i ∈ {1, . . . , n + 1}. (XI) If 0 ∈ X and f (x) = x1 for all x ∈ X then, for (x1 , . . . , xn+1 ) ∈ n+1 X, n f (x1 , . . . , xn+1 ) = (−1)n

n+1

xi−1 .

i=1

Proof. Straightforward.

Definition 2.5.79 Let f : X−→K, n ∈ {0, 1, . . .}. We say that f ∈ C n (X−→K) (or f ∈ C n (X)) if n f can (uniquely) be extended to a continuous function n f : Xn+1 −→K. The map x → Dn f (x) := n f (x, x, . . . , x) is called the n-th Hasse derivative of f . Further, let C ∞ (X−→K) (C ∞ (X)) := n C n (X−→K). From rule III we conclude that f ∈ C n+1 (X) implies f ∈ C n (X). Hence C 0 (X) = C(X) ⊃ C 1 (X) ⊃ C 2 (X) ⊃ · · · ⊃ C ∞ (X). Simple continuity arguments show that rules I, II, IV, V, VII, VIII, IX, X remain valid for functions f, g ∈ C n (X), where k is replaced by k and k X by Xk for all occurring k. It follows that C n (X) is a vector space (IV), closed under products (V) and that locally constant and polynomial functions are in C ∞ (X) (VIII and IV). By VII, if f ∈ C n (X) and f is nowhere 0 then f1 ∈ C n (X). Thus, rational functions without poles in X are in C ∞ (X). We also have a Taylor formula. Theorem 2.5.80 Let f ∈ C n (X). Then for all x, y ∈ X, f (x) = f (y) + (x − y) D1 f (y) + · · · + (x − y)n−1 Dn−1 f (y) + (x − y)n n f (x, y, y, . . . , y).

2.5 Examples of Banach spaces

73

Proof. Directly by induction; for the step n − 1 → n use the identity n−1 f (x, y, y, . . . , y) − Dn−1 f (y) = (x − y) n f (x, y, y, . . . , y). We also have the following, see [195], 29.5 for its proof.

Corollary 2.5.81 Let f ∈ C n (X). Then f is n times differentiable and f (m) = m! Dm f for 0 ≤ m ≤ n. Next we define the Banach spaces of C n -functions. Definition 2.5.82 For n ∈ {0, 1, . . .}, let BC n (X−→K) (BC n (X)) be the space of all f ∈ C n (X−→K) such that f, 1 f, . . . , n f are bounded, with the norm f −→ f n := max(f ∞ , 1 f ∞ , . . . , n f ∞ ). Remark 2.5.83 Observe that boundedness of n+1 f implies, by rule III of 2.5.78, that n f is uniformly continuous, and so f is a C n -function. Thus, if all n f are bounded then f is a C ∞ -function. Clearly if X is compact then BC n (X) = C n (X) for each n. Theorem 2.5.84 For every n ∈ {0, 1, . . .}, (BC n (X), . n ) is a Banach space. If X is compact then C n (X) is isometrically isomorphic to c0 , hence of countable type. Proof. The map T : f → (f, 1 f, . . . , n f ) is a linear isometry of BC n (X) into the product 1≤k≤n+1 BC(X k ), and a straightforward check shows that the image of T is closed in the (complete) product space. If X is compact T maps into 1≤k≤n+1 C(Xk ). Now each Xk is ultrametrizable, so by 2.5.24 C(X k ) is of countable type, and by 2.3.18 and 2.5.22, C(X k ) has a countable orthonormal base. Hence so has the product space and its closed subspace T (C n (X)) (2.3.22). Therefore, C n (X) has a countable orthonormal base. Since X is infinite C n (X) cannot be finite-dimensional, and the theorem follows. The following provides a natural converse of 2.5.84. Theorem 2.5.85 Let n ∈ {0, 1, . . .}. Suppose BC n (X) is of countable type. Then X is compact. For the proof it is convenient to extend the norm . n of 2.5.82 to any f : X−→K, as follows.

74

Normed spaces For n ∈ {0, 1, . . .} set, allowing the value ∞, n f ∞ := sup{|n f (v)| : v ∈ n+1 X} f n := max{i f ∞ : 0 ≤ i ≤ n}.

(For n = 0 we will write f ∞ rather than 0 f ∞ .) In addition, we introduce another quantity by |f (x) − f (y)| : x, y ∈ X, x = y . nf := sup |x − y|n Let Xn be the collection of all subsets of X containing precisely n + 1 elements. ˜ nf : By symmetry of n f (rule II of 2.5.78) it induces naturally a function Xn −→K via the formula ˜ n f (S) = n f (x1 , . . . , xn+1 )

(S := {x1 , . . . , xn+1 } ∈ Xn ).

˜ n f ∞ := sup{|n f (S)| : S ∈ Xn } we obviously have n f ∞ = Setting ˜ n f ∞ . For a finite subset S ⊂ X having at least two points, we define its diameter dS by max{|x − y| : x, y ∈ S}. Then 0 < dS < ∞. Lemma 2.5.86 Let n ≥ 2, Sn ∈ Xn . Then there exists an Sn−1 ∈ Xn−1 such that Sn−1 ⊂ Sn and ˜ n f (Sn ) ≤ d −1 ˜ n−1 f (Sn−1 ) . Sn Proof. Let x, y ∈ Sn be such that |x − y| = dSn . Then writing Sn = {x, y, x1 , . . . , xn−1 } we obtain ˜ n f (Sn ) = |n f (x, y, x1 , . . . , xn−1 )| = |x − y|−1 |n−1 f (x, x1 , . . . , xn−1 ) − n−1 f (y, x1 , . . . , xn−1 )| ≤ dS−1 max (|n−1 f (x, x1 , . . . , xn−1 )| , |n−1 f (y, x1 , . . . , xn−1 )|) n −1 ˜ = dS n−1 f (Sn−1 ) , n

where Sn−1 ∈ Xn−1 is either {x, x1 , . . . , xn−1 } or {y, x1 , . . . , xn−1 }.

Lemma 2.5.87 Let n ≥ 1. Then n f ∞ ≤n f . Proof. For n = 1 the formula (even with an equality sign) holds trivially, so let n ≥ 2 and Sn ∈ Xn . By using 2.5.86 repeatedly we arrive at sets Sn ⊃ Sn−1 ⊃ · · · ⊃ S1 , where Si ∈ Xi for i ∈ {1, . . . , n}, for which ˜ 1 f (S1 ) . ˜ n f (Sn ) ≤ dS−1 · · · dS−1 (2.8) n 2

2.5 Examples of Banach spaces

75

≤ Now let S1 := {x, y}. From |x − y| = dS1 ≤ dS2 ≤ · · · ≤ dSn we obtain dS−1 i −1 |x − y| for i ∈ {2, . . . , n} so that (2.8) yields the further estimate ˜ n f (Sn ) ≤ |x − y|n−1 |1 f (x, y)| = |x − y|−n |f (x) − f (y)| ≤n f , ˜ n f ∞ ≤n f . and n f ∞ =

Corollary 2.5.88 For all n ≥ 0 f n ≤ max(f ∞ , nf ). Proof. We have to prove that i f ∞ ≤ max(f ∞ , nf ) for all i ∈ {0, 1, . . . , n}. To this end we may assume that i ≥ 1. Let x, y ∈ X, x = y. If |x − y| ≥ 1 then |x − y|−i |f (x) − f (y)| ≤ |f (x) − f (y)| ≤ f ∞ , whereas, if |x − y| < 1, we have |x − y|−i |f (x) − f (y)| ≤ |x − y|−n |f (x) − f (y)| ≤n f . We see that i f ≤ max(f ∞ , nf ), and by 2.5.87 we find i f ∞ ≤ max(f ∞ , nf ). For locally constant f we have equality. Theorem 2.5.89 Let f : X−→K be locally constant. Then for n ∈ {0, 1, . . .} we have f n = max(f ∞ , nf ). Proof. By 2.5.88 and the fact that f ∞ ≤ f n we only need to prove that nf ≤ f n . By the Taylor Formula (2.5.80) and the fact that Di f = 0 for all i ≥ 1 we get f (x) = f (y) + (x − y)n n f (x, y, y, . . . , y)

(x, y ∈ X),

so that for x = y |f (x) − f (y)| = n f (x, y, y, . . . , y) ≤ n f ∞ ≤ f n , n |x − y| which proves the theorem.

The next step will be the following weak version of 2.5.85. Lemma 2.5.90 Let BC n (X) be of countable type. Then X is precompact (i.e., the closure of X in K is compact). Proof. The case n = 0 is trivial. Suppose X is not precompact. Then, for some r > 0, the balls in X of radius r form an infinite covering of X, say (Bi )i∈I . Choose ai ∈ Bi for each i ∈ I . Let D be the space of all bounded functions X−→K that are constant on each Bi . We claim that D ⊂ BC n (X) and that D, equipped with the induced topology, is linearly homeomorphic to ∞ (I ). (Then

76

Normed spaces

we have a contradiction since by 2.5.15 ∞ (I ) is not of countable type, proving the lemma.) Clearly D ⊂ C n (X) (in fact, D ⊂ C ∞ (X)). Further, if f ∈ D, x, y ∈ X, (y) x = y we have f (x)−f = 0 if x, y ∈ Bi for some i ∈ I . Whereas if x ∈ Bi , (x−y)n y ∈ Bj for i = j then |x − y| ≥ r and |f (x) − f (y)| ≤ f ∞ = sup{|f (ai )| : i ∈ I }. So we find nf

= sup x=y

|f (x) − f (y)| ≤ r −n f ∞ , |x − y|n

implying by 2.5.89 that f ∈ BC n (X) and that . ∞ is equivalent to . n on D. Then it is clear that f → (f (ai ))i∈I is a surjective linear homeomorphism D−→ ∞ (I ), which finishes the proof. Proof of 2.5.85. Again the case n = 0 is trivial. Suppose X is not compact; we derive a contradiction. By 2.5.90, X is precompact, so X is not closed in K, let a ∈ X \ X. By compactness of X the set {|x − a| : x ∈ X} is discrete with 0 as an accumulation point, say {r1 , r2 , . . .}, where r1 > r2 > · · · and limm rm = 0. For m ∈ N, let Rm := {x ∈ X : |x − a| = rm }. Then R1 , R2 , . . . is an infinite clopen covering of X. Choose am ∈ Rm for each m. Let D be the space of all f ∈ BC n (X) that are constant on each Rm and for which limm f (am ) = 0. Let λ1 , λ2 , . . . ∈ K be such that |λm | = rm−n for each m. As ∞ is not of countable type (2.5.15), we are done once we can prove that T

f −→ (λ1 f (a1 ), λ2 f (a2 ), . . .) is a linear homeomorphism of D onto ∞ . First, we are going to see that for, f ∈ D, sup m

|f (am )| |f (am )| ≤ f n ≤ max(r1n , 1) sup rmn rmn m

(which shows that T maps D homeomorphically into ∞ ). To prove the first m )| ≤ f n . We may suppose f (am ) = 0 inequality, let m ∈ N; we show that |f (a rmn f (aj ) < |f (am )|, so that so, since lim j f (aj ) = 0, there is a j > m for which |f (am )| = f (am ) − f (aj ). Also, am − aj = max(|am − a| , aj − a ) = |f (a )−f (a )| m )| |am − a| = rm . Thus, we obtain |f (a = am −a nj , which, by 2.5.89, is rmn | m j| ≤ f n . For the second inequality observe that, for f ∈ D, |f (am )| rmn |f (ak )| |f (am ) − f (ak )| ≤ sup max , . nf = sup rkn rmn rkn rkn m>k m>k

2.5 Examples of Banach spaces

rmn ≤ 1 so that the previous expression is ≤ rkn |f (am )| supm r n . Further, m

Now

f ∞ = sup m

77

supm>k max

|f (am )| |f (ak )| , rn rmn k

≤

|f (am )| n |f (am )| n rm ≤ sup r1 . n rm rmn m

Hence, by using 2.5.89, f n ≤ max(r1n , 1) sup m

|f (am )| , rmn

and we obtain the desired second inequality. To finish the proof it has only to be shown that T is surjective, i.e., we have to show that if (µ1 , µ2 , . . .) is a bounded sequence in K then the function f that has the value µλmm on each Rm , lies in D. Clearly f ∈ C ∞ (X), hence it is in C n (X). Also, limm f (am ) = limm µλmm = 0 n as (µ1 , µ2 , . . .) is bounded and λ−1 m = rm → 0. It remains to see that n f ∈ BC (X), i.e., that f n < ∞. Now f is clearly bounded, so by 2.5.89 it suffices to prove that nf is bounded. Let x, y ∈ X, x = y. We may suppose x ∈ Rm , y ∈ Rk with m > k. Then |x − y|n = rkn and |f (x) − f (y)| ≤ max(|f (ak )| , |f (am )|). So we get |f (ak )| |f (am )| rmn |f (x) − f (y)| ≤ max , |x − y|n rkn rmn rkn µm |f (am )| ≤ sup = sup λm = sup |µm | < ∞. n rm λm m m m We conclude this section by showing that, for a disk B, P S(B) can be characterized within C ∞ (B). Theorem 2.5.91 Let K be algebraically closed. Let B = B(0, r) for some r ∈ K × . Then P S(B) ⊂ C ∞ (B). In fact, if f : B−→K, then f ∈ P S(B) if and only if limn r n n f ∞ = 0. Proof. Suppose limn r n n f ∞ = 0. Then all n f are bounded functions, hence f ∈ C ∞ (B) (2.5.83), and we have the Taylor formula f (x) = f (0) + D1 f (0) x + · · · + Dn−1 f (0) x n−1 + n f (x, 0, 0, . . . , 0) x n (2.9)

78

Normed spaces

for n ∈ N, x ∈ B. Now n f (x, 0, 0, . . . , 0) x n ≤ r n n f ∞ → 0, so (2.9) yields f (x) =

∞

Dn f (0) x n

(x ∈ B),

n=0

from which f ∈ P S(B). m Conversely, let f ∈ P S(B) with expansion f (x) = ∞ m=0 λm x (x ∈ B). By using the product rule V we can prove inductively that n (Xm )∞ ≤ r m−n if n ≤ m and n (Xm ) = 0 if n > m. So λm n (Xm )∞ ≤ r n max |λm | r m−n = max |λm | r m , r n n f ∞ = r n m≥n

m≥n

m≥n

and by 2.5.53 we have limn maxm≥n |λm | r m = limn |λn | r n = 0.

In the same vein we have the next result, the proof of which is left to the reader. Theorem 2.5.92 Let K be algebraically closed. Let B = B(0, r − ) for some r > 0. Then BP S(B) ⊂ C ∞ (B). In fact, if f : B−→K, then f ∈ BP S(B) if and only if n → r n n f ∞ is bounded.

2.6 Notes Most of the basic theory of this chapter (except Section 2.4 and Subsections 2.5.4–2.5.7) can be found in [193], to which book we refer also for deeper study of Banach spaces. The non-existence of infinite-dimensional Hilbert spaces (Section 2.4) has been known for quite some time, but as analytic proofs were hard to find in the literature, we have presented the details here. Of course, despite of this, it can still be useful to study continuous bilinear or Hermitian forms on non-Archimedean normed spaces, see e.g. [7], [9], [19], [27], [42], [43], [44], [109], [138]. Surprisingly, if we allow the scalar field to have a more general so-called Krull valuation (see the notes to Chapter 1), Hilbert-like spaces appear! In fact, it was Keller who constructed in 1980, [136], an infinite-dimensional space E over a Krull valued field, having an inner product < , > in the spirit of 2.4.1, satisfying the Cauchy–Schwarz inequality and such that for each subspace D of E we have D = D ⊥⊥ ⇐⇒ D ⊕ D ⊥ = E.

(2.10)

2.6 Notes

79

The map x → |< x, x >| (x ∈ E) behaves like the square of a norm and induces a topology on E in the usual way. For a subspace D of E we have D is closed ⇐⇒ D = D ⊥⊥ .

(2.11)

Form (2.10) and (2.11) we may conclude that Keller’s space E really deserves to be called “Hilbert space”. It generated new interest in Functional Analysis over Krull valued fields, see e.g. [166]. Returning to 2.5.9, it was proved in (i) that every non-zero subspace D of c0 (I ), D of countable type, has an orthonormal base. However, one can even prove that each non-zero subspace of c0 (I ) has an orthonormal base. For the (difficult!) proof, see [193], 5.9. Part (ii) of 2.5.9 holds for arbitrary closed D if K is discretely valued (2.5.4). On the other hand, if the valuation of K is dense, (ii) may fail. In fact, by 2.5.6 there is a set I and a closed subspace D of c0 (I ) such that ∞ c0 (I )/D. In 8.2.2 we will prove that ∞ does not have a base. Suppose D has a complement D1 . Then ∞ is linearly homeomorphic to D1 by 2.1.17. Again, thanks to [193], 5.9, D1 has a base, hence also ∞ has one, a contradiction. Thus, D is not complemented in c0 (I ). The orthonormal bases of C(X) for compact X we constructed in 2.5.22 consist of characteristic functions; let us mention an important example of a base consisting of polynomials, the so-called Mahler base for C(Zp ) (K ⊃ Qp ). It is given by e0 , e1 , . . . , where x x(x − 1) · · · (x − n + 1) (n ∈ N ∪ {0}). en (x) = := n! n It is proved in [190] and [195] that it is, indeed, an orthonormal base of C(Zp ), useful for number theoretically oriented problems. The classical Banach–Stone Theorem, see e.g. [40], VI.2.1, states that if X, Y are compact Hausdorff spaces and T : C(X)−→C(Y ) is a surjective isometry, then T is a so-called Banach–Stone map, i.e., there is a homeomorphism h : Y −→X and a function g ∈ C(Y ) with |g(y)| = 1 for all y ∈ Y , such that T (f )(y) = g(y) f (h(y)) f or all f ∈ C(X), y ∈ Y (here C(X) and C(Y ) denote the sup-normed Banach spaces of real – or complex – valued functions on X and Y respectively). The non-Archimedean version of this theorem does not hold. In fact, let X be as in Subsection 2.5.3. In 1987, Beckenstein and Narici proved, [23], that if X is not rigid (i.e., the identity is not the only homeomorphism of X onto X), there are surjective isometries C(X)−→C(X) that are not Banach–Stone maps. In 1990, Araujo and Mart´ınez-Maurica, [16], extended this result to any X with more than one point, see also [10]. This fact has given rise to an interesting theory about properties and characterizations of the Banach–Stone maps in

80

Normed spaces

order to obtain certain weaker versions of the Banach–Stone Theorem in the non-Archimedean case. The above authors are the main ones responsible for the development of this theory, of which one can find a survey, up to 1997, in [11]. For later results, see e.g. [12], [13], [14], [15], [24], [25], [26]. The Banach spaces of continuous functions considered in Subsection 2.5.3 are ones of the most important examples of non-Archimedean Banach algebras. The definition of these algebras is the natural translation of the classical one, but the theory about them presents several deviations from its Archimedean counterpart. To avoid to make this chapter longer we have omitted the study of Banach algebras. For details and references on the subject we refer to [22], [54] and [193]. The Banach space theory of field extensions (Subsection 2.5.4) is new as far as we know. It is not necessary for the theory of locally convex spaces, yet we found it enough illustrative to include in this chapter. There has been quite some activity around building models in theoretical physics, where the base field is not R or C but a non-Archimedean valued field, see [47] [137], [138], [139], [140], [228], [229], for an impression. With this in mind it is natural to consider the commutation relation AB − BA = I

(2.12)

for operators A, B (where I is the identity), which plays a central role in quantum mechanics. It is well-known that, for bounded operators A, B on a non-zero Banach space over R or C, (2.12) does not hold. It is quite remarkable that in the non-Archimedean context (2.12) may have bounded solutions! In fact, let A, B : c0 −→c0 be given by A : (λ1 , λ2 , . . .) −→ (λ2 , λ3 , . . .) B : (λ1 , λ2 , . . .) −→ (0, λ1 , 2 λ2 , . . .). One checks immediately that AB − BA = I and that A ≤ 1, B ≤ 1. This example was discovered independently in [8] and [143].

3 Locally convex spaces

In this chapter we develop the basics of the main subject of this book, i.e., the theory of locally convex spaces over K. The reader will notice that Sections 3.1−3.6 contain some material that looks familiar to a classical analyst. However, we felt it convenient to give full proofs; it reveals which classical proofs can be translated and what modifications need to be made. In Section 3.1 we do not immediately consider topologies on our spaces, but introduce seminorms for which we require the strong triangle inequality, and convex sets in an algebraic way. Typical non-Archimedean features here are the solidity of a seminorm (3.1.1) and edged sets (3.1.5, 3.1.13). We prove in 3.1.11 and 3.1.14 that a convex set and a point outside it can be separated by a seminorm (implying that, contrary to the classical situation, convex sets in K n are closed, 3.4.22(i), 3.4.24). Section 3.2 is a preparation for Chapter 8 and reading of it may be postponed until that chapter is tackled. In Section 3.3 we define locally convex spaces in two equivalent ways, one by means of seminorms (3.3.7) and one that requires a neighbourhood base at 0 that consists of convex sets (3.3.16). In Section 3.4 we consider subspaces (3.4.3), quotients (3.4.6), products (3.4.9), locally convex direct sums (3.4.15), and projective (3.4.29) and inductive (3.4.32) limits of locally convex spaces. We show that every Hausdorff locally convex space can be embedded in a product of Banach spaces (3.4.10), which we use to construct completions, and prove some hereditary properties for completeness. The strongest locally convex topology is discussed in 3.4.22. Section 3.5 contains familiar-looking results on metrizable and Fr´echet spaces, including the Open Mapping Theorem 3.5.10. Bounded sets appear in Section 3.6; we then introduce quasicompleteness and sequential completeness in 3.6.7.

81

82

Locally convex spaces

At this point the theory acquires a more non-Archimedean flavour. In Section 3.7 we present various examples of locally convex spaces, thereby considering the question under what conditions they are normable, metrizable, (quasi)complete respectively. In Subsection 3.7.1 we study subspaces of C(X) equipped with a “W -topology”, where X is a non-empty zerodimensional Hausdorff topological space and W is a collection of “weight functions” (3.7.12). By varying W we obtain the topologies of pointwise convergence (3.7.1), of uniform convergence on compact sets (3.7.3), and the strict topology (3.7.14). Spaces of analytic functions with a natural locally convex topology appear in Subsection 3.7.2, among which are the space of “overconvergent power series” on a disk, the space of germs of analytic functions, and the space of rigid analytic functions (3.7.27−3.7.33). For a nonempty subset X of K without isolated points we study in Subsection 3.7.3 various spaces of C n , respectively C ∞ -functions on X; again W -topologies appear. In Section 3.8 we return to the main theory and treat compactoidity. By “convexifying” the notion of precompactness we arrive at the concept of compactoid set, which is useful for all (not only locally compact) K (3.8.1). We then can prove an Ascoli Theorem (3.8.2) and a Riesz Theorem (3.8.6) in terms of compactoid sets, show basic properties (3.8.4), and describe compactoids in products (3.8.14) and in direct sums (3.8.17). It is natural to expect that complete convex compactoids will take over the role played by the compact convex sets in classical Functional Analysis. However, this turns out to be partially true; without restrictions on K we have the following “compact like hereditary properties” only for metrizable compactoids. (Improvements can be obtained if K is spherically complete; this is the subject of Chapter 6.) If A is a complete metrizable absolutely convex compactoid and T is a continuous linear map then {λ T (A) : λ ∈ K, |λ| > 1} is complete (3.8.29(i)), but T (A) need not be (3.8.31(i)). In the same vein, if λ ∈ K, |λ| > 1, y1 , y2 , . . . ∈ T (A), yn → 0 then there is a sequence x1 , x2 , . . . in λ A tending to 0 such that T (xn ) = yn for each n (3.8.35), but the conclusion fails if λ = 1 (3.8.36). In any case, it follows that for every Hausdorff locally convex topology τ1 , weaker than the initial one τ , we have τ = τ1 on A (3.8.38). The aim of Section 3.9 is to derive a characterization of compactoidity (3.9.16) in terms of “orthogonality” (a notion that is a natural extension of “orthogonality”, 2.2.11, in the normed case). It says that any bounded subset X of a locally convex space is a compactoid if and only if each “orthogonal” sequence in X tends to 0. The technical Section 3.10 gives another characterization for compactoids in normed spaces, which will be needed in Chapter 8.

3.1 Seminorms and convexity

83

3.1 Seminorms and convexity In this section we set up an algebraic preparation for the sequel. Recall that vector spaces are over K. Definition 3.1.1 A seminorm on a vector space E is a map p : E−→[0, ∞) such that (i) p(x + y) ≤ max(p(x), p(y)), (ii) p(λ x) = |λ| p(x), for all x, y ∈ E, λ ∈ K. The seminorm is called solid if p(E) ⊂ |K| (see 2.1.10). If the valuation of K is dense every seminorm p is solid; if the valuation of K is discrete then p is solid if and only if p(E) ⊂ |K|. For a seminorm p we have p(0) = 0 but p(x) is allowed to be 0 for non-zero x. Ker p := {x ∈ E : p(x) = 0} is a subspace of E. Each norm is a seminorm that vanishes only at 0. A typical example of a seminorm is the function f −→ max{|f (y)| : y ∈ Y }

(f ∈ C(X)),

where X is a zero-dimensional non-compact Hausdorff space and Y ⊂ X is compact. Definition 3.1.2 Let p, q be seminorms on a vector space E. For x ∈ E set (p ∨ q)(x) := max(p(x), q(x)) (p ∧ q)(x) := inf{max(p(y), q(z)) : y, z ∈ E, x = y + z}. One verifies easily that p ∨ q and p ∧ q are seminorms, that p ∨ q is the smallest seminorm that is ≥ p and ≥ q, and that p ∧ q is the largest seminorm that is ≤ p and ≤ q. Thus, in a natural way, the set of all seminorms on E is a lattice. If p, q are solid then so are p ∨ q and p ∧ q. Our second object of study in this section is absolute convexity. Definition 3.1.3 A subset A of a vector space E is called absolutely convex if it is a BK -submodule of E; in other words, if 0 ∈ A and λ x + µ y ∈ A for all λ, µ ∈ BK , x, y ∈ A. Clearly the intersection of any non-empty collection of absolutely convex sets in E is absolutely convex. Linear subspaces are absolutely convex.

84

Locally convex spaces

Definition 3.1.4 Let X be a subset of a vector space E. Its absolutely convex hull, aco X, is the intersection of all absolutely convex sets containing X. If E is a topological vector space we write aco X instead of aco X. We have aco ∅ = {0}, and if X = ∅, aco X = {λ1 x1 + · · · + λn xn : n ∈ N, λ1 , . . . , λn ∈ BK , x1 , . . . , xn ∈ X}. The union of a nested collection of absolutely convex sets is absolutely convex. The image and inverse image of an absolutely convex set under a linear map are absolutely convex. Finite sums of absolutely convex sets are absolutely convex. The absolutely convex subsets of K are {0}, K, B(0, r), B(0, r − ), where r ∈ (0, ∞). The fact that for a seminorm p on E and r > 0 the p-balls {x ∈ E : p(x) ≤ r} and {x ∈ E : p(x) < r} are absolutely convex reveals a connection between seminorms and absolutely convex sets. We need some styling to shape it into a 1 − 1 correspondence. Definition 3.1.5 For an absolutely convex subset A of a vector space we define the edged hull Ae of A by Ae := {λ A : λ ∈ K, |λ| ≤ r}. r>1

In other words, A = A if the valuation of K is discrete and Ae = {λ A : λ ∈ K, |λ| > 1} if the valuation of K is dense. A is called edged if A = Ae . Mainly for later use we list the following facts, leaving the easy proofs to the reader. e

Theorem 3.1.6 Let A, B be absolutely convex subsets of a vector space. Then we have the following: (i) (ii) (iii) (iv) (v)

Ae is absolutely convex. Aee = Ae . (A ∩ B)e = Ae ∩ B e . (A + B)e ⊃ Ae + B e . (A + B)e = (Ae + B e )e = (Ae + B)e .

Remarks 3.1.7 (a) For an example where the inclusion in (iv) is strict, see 3.8.32. (b) As for the behaviour of edged sets under linear maps T : E−→F (E, F vector spaces) we have the following obvious facts. If A ⊂ E, B ⊂ F are absolutely convex then T −1 (B e ) = T −1 (B)e and T (Ae ) ⊂ T (A)e .

3.1 Seminorms and convexity

85

This last inclusion is not always an equality (however, see 6.2.3). In fact, let K have dense valuation. Let A := Bc0 and let T : c0 −→K, (λ1 , λ2 , . . .) → ∞ e n=1 λn µn , where 0 < |µ1 | < |µ2 | < . . . , limn |µn | = 1. Then T (A ) = − e T (A) = BK = BK = T (A) . In K all balls of the form B(0, r) (r > 0) are edged. If there is a ball B about 0 − not having × this form thene the valuation of K is dense and B = B(0, r ), where r ∈ K . In this case B = B(0, r). For any seminorm p on a vector space E and r > 0 the p-ball {x ∈ E : p(x) ≤ r} is edged. We need one more notion, which will occur frequently in the sequel. Definition 3.1.8 An absolutely convex subset A of a vector space E is called absorbing if for every x ∈ E there is a λ ∈ K such that x ∈ λ A, i.e., if E= {λ A : λ ∈ K} = [A]. Note that if A, B ⊂ E are absorbing then so are A ∩ B, A + B and (A + B)e . Now we are able to describe the announced 1 − 1 correspondence. Theorem 3.1.9 Let E a vector space. Then we have the following: (i) For each seminorm p on E its “closed unit ball” Ap := {x ∈ E : p(x) ≤ 1} is absolutely convex, absorbing and edged. (ii) For each absorbing absolutely convex set A ⊂ E its associated “Minkowski function” or “gauge” x −→ pA (x) := inf{|λ| : λ ∈ K, x ∈ λ A}

(x ∈ E)

is a solid seminorm on E. (iii) The map A → pA is a 1 − 1 correspondence between the absorbing edged subsets of E and the solid seminorms on E. Its inverse is p → Ap . Proof. Direct verification yields (i) and (ii); we now prove (iii). If the valuation of K is discrete and pA (x) > 0 the infimum in (ii) is a minimum, so that for each absorbing absolutely convex A ⊂ E A = {x ∈ E : pA (x) ≤ 1} whereas, in case the valuation of K is dense, {x ∈ E : pA (x) < 1} ⊂ A ⊂ {x ∈ E : pA (x) ≤ 1} implying Ae = {x ∈ E : pA (x) ≤ 1}.

86

Locally convex spaces

It follows that A → pA is injective. To prove surjectivity, let p be a solid seminorm on E and set A := {x ∈ E : p(x) ≤ 1}. Then, for each x ∈ E, pA (x) = inf{|λ| : λ ∈ K, x ∈ λ A} = inf{|λ| : λ ∈ K, p(x) ≤ |λ|}, which equals p(x) since p is solid. Thus, p = pA and we are done. The following obvious rules connect the lattice structure of the set of solid seminorms and the family of (edged) absolutely convex sets. Theorem 3.1.10 Let A, B be absorbing absolutely convex subsets of a vector space E. Then pA∩B = pA ∨ pB and pA+B = pA ∧ pB = p(A+B)e . As a further justification for choosing 3.1.3 for absolute convexity, we will show that every absolutely convex set is the intersection of balls of the form {x : p(x) < 1}, where p runs through some collection of (solid) seminorms, and also, similarly, that each edged set is the intersection of balls of the form {x : p(x) ≤ 1}. Theorem 3.1.11 Let A be an absolutely convex subset of a vector space E, let x ∈ E, x ∈ A. Then we have the following: (i) There is a seminorm p on E with p(E) ⊂ |K| such that p(y) < 1 for each y ∈ A and p(x) = 1. (ii) If, in addition, A is edged then p can be chosen such that p(y) ≤ 1 for each y ∈ A and p(x) > 1. Proof. (i). By using Zorn’s Lemma we obtain the existence of an absolutely convex subset U of E that is maximal with respect to the properties A ⊂ U and x ∈ U . Then BK− x + U is absolutely convex, contains U , and x ∈ BK− x + U (if x ∈ λ x + U for some λ ∈ BK− then (1 − λ)−1 ∈ BK and x ∈ (1 − λ)−1 U , so that x ∈ U , a contradiction). So, by maximality, BK− x ⊂ U.

(3.1)

Now let y ∈ E, y = 0. To prove that some non-zero multiple of y lies in U we may assume Ky + U U . Hence x ∈ Ky + U , i.e., x = λ y + u for some λ ∈ K × , u ∈ U . Choose ρ ∈ K, 0 < |ρ| < 1 . We see that ρ λ y = ρ x − ρ u ∈ U by (3.1). Thus, U is absorbing. Let pU be the Minkowski function associated to U . Then, since {z ∈ E : pU (z) < 1} ⊂ U ⊂ {z ∈ E : pU (z) ≤ 1}, and by (3.1), we have pU (λ x) ≤ 1 for all λ ∈ BK− .

(3.2)

3.1 Seminorms and convexity

87

To continue the proof of (i), we first assume that the valuation of K is discrete. Then A is edged and clearly pU (E) ⊂ |K|. Further, U = {z ∈ E : pU (z) ≤ 1} and since x ∈ U we have pU (x) > 1. Also, by (3.2) pU (x) = |ρ|−1 , with ρ a uniformizing element. Then take p := |ρ| pU . Observe also that the seminorm pU satisfies the requirements of (ii). Next assume that K is densely valued. Then it follows from (3.2) that pU (x) ≤ 1. But pU (x) < 1 would imply x ∈ U , a contradiction. So pU (x) = 1. Also, U = {z ∈ E : pU (z) < 1}. In fact, suppose pU (u) = 1 for some u ∈ U . Then Ku + U is an absolutely convex set that properly contains U . By maximality of U , x ∈ Ku + U . There is a λ ∈ K for which x − λ u ∈ U . Since x ∈ U we must have |λ| > 1. As pU (x − λ u) ≤ 1 and pU (λ u) = |λ| > 1 we arrive at pU (x) = pU (λ u) > 1, which is a contradiction. To finish the proof of (i) for the densely valued case we only have to see that pU (E) ⊂ |K| (then p := pU satisfies the requirements). Take z ∈ E, pU (z) = 0. With the same reasoning as before we conclude that the set Kz + U must contain x. Thus x = λ z + u for some λ ∈ K × , u ∈ U . So pU (x − λ z) = pU (u) < 1 = pU (x), which implies that pU (x) = pU (λ z) = 1. Hence, pU (z) = |λ|−1 ∈ |K|. Next we prove (ii). If the valuation of K is discrete the result has already been obtained in the proof of (i). Now suppose that the valuation of K is dense. Since x ∈ A = Ae there is a λ ∈ K with |λ| > 1 such that x ∈ λ A. Applying (i), we find a seminorm q on E with q(E) ⊂ |K| such that q(λ A) < 1 and q(x) = 1. Then p := |λ| q satisfies the conditions required in (ii). Absolutely convex sets always contain 0. We now define the general nonArchimedean version of convexity. Definition 3.1.12 A subset C of a vector space E is called convex if it is either empty or an additive coset of an absolutely convex set, i.e., of the form x + A, where x ∈ E, A ⊂ E absolutely convex. Note that a convex set is absolutely convex if and only if it contains 0. If A is absolutely convex, x, y ∈ E and x + A = y + A then x − y ∈ A ⊂ Ae , so x + Ae = y + Ae . This makes the following notion meaningful.

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Definition 3.1.13 Let C be a convex subset of a vector space E. If C = ∅, set C e := ∅. Otherwise, C = x + A where x ∈ E, A ⊂ E absolutely convex. Then set C e := x + Ae . C is called edged if C = C e . The following is an immediate consequence of 3.1.11. Corollary 3.1.14 Let C be a convex subset of a vector space E, let x ∈ C. Then there is a collection P of (solid) seminorms on E such that C= {y ∈ E : p(y − x) < 1}. p∈P

If, moreover, C is edged we can select P in such a way that C= {y ∈ E : p(y − x) ≤ 1}. p∈P

A more direct way of defining convexity, resembling the classical definition is presented in the following (see also (3.4)). Theorem 3.1.15 Let C be a subset of a vector space E. Then C is convex if and only if x, y, z ∈ C, λ, µ, ν ∈ BK , λ + µ + ν = 1 =⇒ λ x + µ y + ν z ∈ C. (3.3) Proof. Let C be convex. Then (3.3) holds if C = ∅, so assume C = a + A, where a ∈ E and A ⊂ E is absolutely convex. To get (3.3) write x = a + v1 , y = a + v2 , z = a + v3 , where v1 , v2 , v3 ∈ A. Then λ x + µ y + ν z = λ (a + v1 ) + µ (a + v2 ) + ν (a + v3 ) = a + λ v1 + µ v2 + ν v3 ∈ a + A, which proves (3.3). Conversely, suppose (3.3) holds and assume C = ∅. Choose a ∈ C, set A := −a + C; we prove that A is absolutely convex. In fact, 0 ∈ A and for x = −a + c1 , y = −a + c2 , where c1 , c2 ∈ C and λ, µ ∈ BK we have λ x + µ y = −(λ + µ) a + λ c1 + µ c2 = −a + (1 − λ − µ) a + λ c1 + µ c2 ∈ −a + C = A, and we are done. The intersection of any collection of convex subsets of E is convex, which enables us to define the following. Definition 3.1.16 Let X be a subset of a vector space E. Its convex hull, co X, is the intersection of all convex sets containing X. If E is a topological vector space we write co X instead of co X.

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It is easily checked that co X= λ1 x1 + · · · + λn xn : n ∈ N, λ1 , . . . , λn ∈ BK , x1 , . . . , xn ∈ X,

n

λi = 1 .

i=1

The union of a nested collection of convex sets is convex. The image and inverse image of a convex set under a linear map are convex. Finite sums of convex sets are convex, linear manifolds are convex. The convex sets of K are ∅, singleton sets, balls, K. With the classical definition of convexity in mind one may wonder whether the version of (3.3) involving only two points, x, y ∈ C, λ, µ ∈ BK , λ + µ = 1 =⇒ λ x + µ y ∈ C,

(3.4)

implies convexity of C. This question may not be so important but the answer is rather peculiar. Theorem 3.1.17 The following are equivalent: (α) For each vector space E and subset C ⊂ E, (3.4) implies convexity of C. (β) The residue class field k of K has at least three elements. Proof. (α) =⇒ (β). Suppose k = 2; we produce a non-convex set C ⊂ K 2 satisfying (3.4). In fact, let C := B(0, 1− ) ∪ B(e1 , 1− ) ∪ B(e2 , 1− ), where the balls are with respect to the canonical maximum norm on K 2 and e1 , e2 are the unit vectors. One checks readily that the union of any two of these three balls is convex, so (3.4) holds. But 0, e1 , e2 ∈ C, (−1) 0 + e1 + e2 = e1 + e2 ∈ C, so C is not convex. (β) =⇒ (α). Let k ≥ 3 and suppose C ⊂ E satisfies (3.4). To prove convexity we may assume (translates of C also satisfy (3.4)) that 0 ∈ C. Let x, y ∈ C, λ, µ ∈ BK ; we show λ x + µ y ∈ C. First, notice that for each α ∈ BK we have α x = α x + (1 − α).0 ∈ C, similarly α y ∈ C. By assumption k has an element different from 0 or 1, so there is a β ∈ BK such that |β| = |1 − β| = 1. Then λ β −1 x and µ (1 − β)−1 y are in C and by (3.4) also λ x + µ y = β (λ β −1 x) + (1 − β) (µ (1 − β)−1 y) ∈ C.

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3.2 Absolutely convex sets of countable type This section will be needed in Chapter 8, see 8.6.5. An absolutely convex subset A of a K-vector space is called c-countably generated if there exists a countable set X ⊂ A such that A = aco X. Lemma 3.2.1 One-dimensional absolutely convex sets are c-countably generated. Proof. Let A be a one-dimensional absolutely convex set. We may assume A ⊂ K. We distinguish three cases: (i) A = BK (0, r) where r ∈ |K|. Then A = aco {λ}, with λ ∈ K, |λ| = r. (ii) A is bounded but not of the form (i). Then the valuation of K is dense and A = BK (0, r − ) for some r > 0. Let λ1 , λ2 , . . . ∈ K, |λ1 | < |λ2 | < · · · , limn |λn | = r. It is easily seen that A = aco {λ1 , λ2 , . . .}. (iii) A = K. Let λ1 , λ2 , . . . ∈ K, limn |λn | = ∞. Then A = aco {λ1 , λ2 , . . .}. Lemma 3.2.2 Finite-dimensional absolutely convex sets are c-countably generated. Proof. Let E be a finite-dimensional K-vector space, let A ⊂ E be absolutely convex. We will proceed by induction with respect to n := dim E. The case n = 1 is covered by 3.2.1. For the induction step n − 1 → n, let dim E = n. We may assume A = {0}; let x ∈ A, x = 0, let π : E−→E/Kx be the canonical map. Then π(A) is absolutely convex and by the induction hypothesis there is a countable set X1 ⊂ A such that π(A) = π (aco X1 ). By 3.2.1 there is a countable set X2 ⊂ Kx ∩ A such that Kx ∩ A = aco X2 . One verifies directly that then A = aco (X1 ∪ X2 ). Now we extend 3.2.2. Theorem 3.2.3 Let E a K-vector space of countable dimension. Then every absolutely convex subset of E is c-countably generated. Proof. Let D1 ⊂ D2 ⊂ · · · be subspaces of E such that dim Dn = n for each n and n∈N Dn = E. Let A ⊂ E be absolutely convex. Then, for each n, Dn ∩ A = aco Xn for some countable Xn ⊂ Dn ∩ A (by 3.2.2). Now from A = n∈N Dn ∩ A it then follows that A = aco ( n∈N Xn ) and we are done. The topological version of the above result for normed spaces of countable type is covered by the following.

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Theorem 3.2.4 Let E be a normed space of countable type. Then, for each absolutely convex set A ⊂ E there is a countable X ⊂ A such that A ⊂ aco X. For the proof we introduce the following notion. Let B ⊂ A ⊂ E, where E is a normed space. Let ε > 0. We say that B is ε-dense in A if for each x ∈ A there is a y ∈ B such that x − y < ε. Lemma 3.2.5 Let A be an absolutely convex subset of a normed space E of countable type. Let ε > 0. Then there exists a countable set X ⊂ A such that aco X is ε-dense in A. Proof. We may assume that E is infinite-dimensional. Let D be a subspace of countable dimension that is dense in E. Let Aε := A + B(0, ε− ). Then Aε is absolutely convex and open so that D ∩ Aε is dense in Aε . By 3.2.3, D ∩ Aε = aco Y for some countable Y . For each y ∈ Y , choose a y ∈ A such that y − y < ε, and let X := {y : y ∈ Y }. Then X is countable and X ⊂ A. To show that aco X is ε-dense in A, let x ∈ A. There is an x1 ∈ D ∩ Aε such that x − x1 < ε (since D ∩ Aε is dense in Aε ). x1 has the form ni=1 λi yi where n ∈ N, λi ∈ BK , yi ∈ Y . Put x2 :=

n

λi yi .

i=1

Then x2 ∈ aco X and x2 − x ≤ max(x2 − x1 , x1 − x) = max(

n

λi (yi − yi ), x1 − x)

i=1

≤ max( max yi − yi , x1 − x) < ε 1≤i≤n

and we are done.

Proof of 3.2.4. Let A ⊂ E be absolutely convex. For each n ∈ N we can, by 3.2.5, find a countable Xn ⊂ A such that aco Xn is n1 -dense in A. Then obviously, by taking X := n∈N Xn we have that aco X is dense in A and X is countable.

3.3 Definition of a locally convex space Recall that a topological vector space (over K) is a vector space E together with a topology τ such that the vector operations are continuous.

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Definition 3.3.1 Let E be a vector space, let p be a seminorm on E. (i) A subset U of E is called p-open if for each x ∈ U there is an r > 0 such that {y ∈ E : p(y − x) < r} ⊂ U . The collection of all p-open sets is called the p-topology. (ii) Let πp : E−→E/Ker p be the natural map, write Ep := E/Ker p. The assignment πp (x) −→ p(x)

(x ∈ E)

defines a norm on Ep which we denote by p. The following two theorems are classical. Theorem 3.3.2 Let p be a seminorm on a vector space E. Then the p-topology makes E into a topological vector space. The sets {x ∈ E : p(x) < r}, where r > 0, form a neighbourhood base of 0 for the p-topology and p is continuous. The p-topology is Hausdorff if and only if p is a norm. In general, Ker p is the closure of {0} in the p-topology. Theorem 3.3.3 Let p, q be seminorms on a vector space E. Then the following are equivalent: (α) (β) (γ ) (δ)

q is p-continuous. The q-topology is weaker than the p-topology. There is a constant C > 0 such that q ≤ C p. Ker p ⊂ Ker q and the natural map Ep −→Eq given by πp (x) → πq (x) (x ∈ E) is a continuous linear map of the normed space (Ep , p) to the normed space (Eq , q).

Definition 3.3.4 Two seminorms p, q on a vector space are called equivalent if the p-topology and the q-topology coincide. Clearly, p, q are equivalent if and only if there exist positive constants c, C such that c p ≤ q ≤ C p (compare with 2.1.7). Lemma 3.3.5 Every seminorm is equivalent to a solid seminorm. Proof. If the valuation of K is dense every seminorm is solid, so there is nothing to prove. If the valuation of K is discrete and p is a seminorm on a vector space E, as in the proof of 2.1.9, we set q(x) := max{|λ| : λ ∈ K, |λ| ≤ p(x)}

(x ∈ E).

One checks easily that q is a seminorm on E with q(x) ∈ |K| for all x ∈ E and that |ρ| p ≤ q ≤ p, where ρ is a uniformizing element.

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In the sequel we will be interested in the topology induced by seminorms rather than in the seminorms themselves. Then, by the above lemma, it will cause no harm to restrict ourselves to solid seminorms only. In doing so we will avoid needless technical difficulties. From now on in this book (semi)norms are assumed to be solid. The proof of the following theorem is basically classical and left to the reader. Theorem 3.3.6 Let P be a collection of seminorms on a vector space E. For a topology τ on E the following are equivalent: (α) τ is the weakest topology on E making all p ∈ P continuous. (β) τ is the union of all p-topologies where p ∈ P. (γ ) A subset U ⊂ E is τ -open if and only if for each x ∈ U there exist p1 , . . . , pn ∈ P and r > 0 such that {y ∈ E : max pi (y − x) < r} ⊂ U. 1≤i≤n

Definition 3.3.7 A topology τ on a vector space E is called a locally convex topology if there exists a collection P of seminorms that induces τ , or generates τ , i.e., such that (α)–(γ ) of the above theorem hold. The pair (E, τ ) is called a locally convex space (over K). Frequently we will write E instead of (E, τ ). Now let (E, τ ) be a locally convex space with a generating family P of seminorms. The set of all τ -continuous seminorms may be strictly bigger than P. But for a τ -continuous seminorm q, the set U := {x ∈ E : q(x) ≤ 1} is a τ -zero neighbourhood. So there are p1 , . . . , pn ∈ P and r > 0 such that {x ∈ E : max1≤i≤n pi (x) < r} ⊂ U . It follows easily that q ≤ r −1 max1≤i≤n pi (recall that seminorms are solid!). This observation leads to the following. Definition 3.3.8 A collection P of seminorms on a vector space E is called saturated if: (i) p, q ∈ P implies p ∨ q ∈ P, (ii) p ∈ P, c ∈ |K| implies c p ∈ P, (iii) If p ∈ P, if q is a seminorm, q ≤ p then q ∈ P. The next theorem can be proved in a straightforward manner. Theorem 3.3.9 For any collection P of seminorms on a vector space E the smallest saturated collection P ∼ that contains P exists. It is the set of all seminorms q on E for which q ≤ C max(p1 , . . . , pn ) for certain C > 0,

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p1 , . . . , pn ∈ P. Let τ be the locally convex topology on E induced by P. Then P ∼ is the largest collection of seminorms inducing τ ; it is the collection of all τ -continuous seminorms. The map assigning to a locally convex topology the collection of its continuous seminorms, is a 1 − 1 correspondence between all locally convex topologies on E and all saturated sets of seminorms on E. Of some importance is also the notion “base of continuous seminorms”. Definition 3.3.10 A collection P of continuous seminorms on a locally convex space E is called a base of continuous seminorms if for every continuous seminorm p there exist a q ∈ P and a C > 0 such that p ≤ C q. Let P be any collection of seminorms on a vector space E inducing a locally convex topology τ . Then the set P := {max(p1 , . . . , pn ) : n ∈ N, p1 , . . . , pn ∈ P} is a base of τ -continuous seminorms, and the sets {x ∈ E : p(x) < r}, where p ∈ P , r > 0 form a τ -neighbourhood base at 0. We now will establish an alternative definition of local convexity that at the same time explains the terminology (3.3.16). To this end we derive some facts on convexity and seminorms on arbitrary topological vector spaces. Theorem 3.3.11 Let E be a topological vector space. Let C ⊂ E be convex. Then we have the following: (i) The closure C of C is convex. (ii) The interior int C of C is convex. Much more than that, if int C = ∅ then C is clopen. (iii) C e ⊂ (C)e . (iv) C e is closed if and only if C ⊂ C e . Proof. By using translations it is enough to prove the statements for absolutely convex C. (i). Let x, y ∈ C, λ, µ ∈ BK . There exist nets (xi )i , (yi )i in C converging to x, resp. y. By continuity of vector operations we have λ x + µ y = limi (λ xi + µ yi ) ∈ C. (ii). Assume int C = ∅. Let U be an open non-empty subset of C, choose x ∈ C. By absolute convexity x + U ⊂ C and, since E is a topological vector space, x + U is open. This proves that C is open. Hence, all its cosets are open and so is {x + C : x ∈ E \ C}, which equals E \ C because C is an additive group. We conclude that C is closed. (iii). By (i), C is absolutely convex, so (C)e is defined, and clearly C e ⊂ (C)e . Since (C)e is closed we have C e ⊂ (C)e .

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(iv). If C e is closed then from C ⊂ C e we obtain C ⊂ C e . Conversely, if C ⊂ C e then (C)e ⊂ C e and by (iii) C e ⊂ C e . Remark 3.3.12 Let the valuation of K be dense. There exist non-closed absolutely convex sets A such that Ae is closed. In fact, let E := c0 and choose λ1 , λ2 , . . . ∈ K with limn |λn | = ∞. Set A := {(µ1 , µ2 , . . .) ∈ c0 : sup |λn µn | < 1}. n∈N

Then A = {(µ1 , µ2 , . . .) ∈ c0 : supn∈N |λn µn | ≤ 1}, so it is a closed subset of c0 . However, A is not closed. To see this, choose µ1 , µ2 , . . . ∈ K such that |λ1 µ1 | < |λ2 µ2 | < · · · , limn |λn µn | = 1. Then x := (µ1 , µ2 , . . .) ∈ A but x = limn (µ1 , . . . , µn , 0, 0, . . .) ∈ A. e

e Problem Does there exist an absolutely convex set A with Ae = A ? Equivalently, does there exist an absolutely convex edged set B such that B is not edged? Lemma 3.3.13 Let E be a topological vector space. For a seminorm p on E the following assertions are equivalent: (α) (β) (γ ) (δ)

p is continuous. p is continuous at 0. For each r > 0 the set {x ∈ E : p(x) < r} is open. {x ∈ E : p(x) < 1} is open.

Proof. The implications (α) =⇒ (β) =⇒ (γ ) =⇒ (δ) are evident. To prove (δ) =⇒ (α), let x ∈ E, r > 0. We produce a neighbourhood V of x such that |p(v) − p(x)| < r for all v ∈ V . To this end, set U := {y ∈ E : p(y) < 1}, choose λ ∈ K, 0 < |λ| < r. Then, since vector operations are continuous, V := x + λ U is open. Now let v = x + λ u ∈ V , where u ∈ U . Then |p(v) − p(x)| ≤ p(v − x) = p(λ u) = |λ| p(u) < |λ| < r. Corollary 3.3.14 For a continuous seminorm p on a topological vector space E the sets {x ∈ E : p(x) ≤ r} and {x ∈ E : p(x) < r} (r > 0) are clopen. Proof. Use the lemma and 3.3.11(ii).

Theorem 3.3.15 Let E be a topological vector space. Then we have the following: (i) Every zero neighbourhood is absorbing. (ii) If A ⊂ E is absolutely convex and absorbing then pA is continuous if and only if A is a zero neighbourhood.

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Proof. Continuity of K−→E, λ → λ x at 0 for each x ∈ E implies (i). To prove (ii) let A ⊂ E be absolutely convex and absorbing. If pA is continuous then, by 3.3.13, BA := {x ∈ E : pA (x) < 1} is a zero neighbourhood, hence so is A. Conversely, suppose A is a zero neighbourhood. Since BA contains a multiple of A, it follows from 3.3.11(ii) that BA is open. Thus, pA is continuous, again by 3.3.13. Now we arrive at the desired characterization. Corollary 3.3.16 A topological vector space (E, τ ) is locally convex if and only if it has a base (of zero neighbourhoods) consisting of (absolutely) convex sets. Proof. We only need to prove the “if”. So let the τ -open absolutely convex sets form a base B of zero neighbourhoods. Let P be the collection of all τ continuous seminorms on E and let τ be the locally convex topology induced by P. By construction τ ≤ τ . Also, using 3.3.15 we obtain that for each U ∈ B its Minkowski function pU is in P and so U is a τ -zero neighbourhood. Hence τ ≤ τ . Corollary 3.3.17 Let E be a locally convex space. The map U → pU is a 1 − 1 correspondence between the absolutely convex edged zero neighbourhoods of E and the continuous seminorms on E.

Proof. 3.1.9(iii) and 3.3.15.

Any normed space is, of course, a locally convex space with respect to the norm topology. Conversely, a locally convex space is called normable if its topology is induced by a single norm. Examples of typically non-normable spaces will be treated in Section 3.7. There exist topological vector spaces that are not locally convex; it is not difficult to find somewhat pathological examples in the spirit of classical analysis, see [41], [110]. However, the space below may be more illustrative to a classical analyst. Note 2 is not locally convex. 2 Let 2 be the set of all sequences (λ1 , λ2 , . . .) ∈ K N for which ∞ n=1 |λn | < ∞. 2 It is easily seen that is a vector space under coordinatewise operations and that ∞ |λn |2 x = (λ1 , λ2 , . . .) −→ N(x) := n=1

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97

defines an A-norm (see 2.1.2) on 2 . This A-norm N induces a topology τ on 2 in the usual way making 2 into a topological vector space. We now show that τ is not locally convex. Suppose it was. Then the N -unit ball contains an absolutely convex zero neighbourhood U and, in its turn, U contains an N -ball about 0. Thus, there exists a ρ ∈ K, 0 < |ρ| < 1 such that {x ∈ 2 : N(x) ≤ |ρ|} ⊂ U ⊂ {x ∈ 2 : N(x) ≤ 1}. Apparently, the vectors x1 := (ρ, 0, 0, . . .), x2 = (0, ρ, 0, . . .), . . . are in U and by absolute convexity x1 + · · · + xn ∈ U for each n ∈ N. But N (x1 + · · · + √ xn ) = |ρ| n conflicting the N-boundedness of U . Remark 3.3.18 A similar procedure works for p , where p ∈ R, 1 ≤ p < ∞.

3.4 Basic facts and constructions Definition 3.4.1 Let E, F be locally convex spaces. The collection of all continuous linear maps E−→F form in a natural way a vector space denoted by L(E, F ). We write L(E) := L(E, E), E := L(E, K). E is called the dual space (or simply dual) of E. By E ∼ F we mean that there is a linear homeomorphism of E onto F and then we say that E is isomorphic to F . An embedding E−→F is a linear homeomorphism of E onto the image. Theorem 3.4.2 Let E, F be locally convex spaces. Let P be a collection of seminorms on F generating its topology. For a linear map T : E−→F the following are equivalent: (α) T ∈ L(E, F ). (β) T is continuous at 0. (γ ) For every q ∈ P there is a continuous seminorm p on E such that q(T (x)) ≤ p(x) for all x ∈ E. (δ) For every q ∈ P the function q ◦ T is a continuous seminorm on E. Proof. Straightforward.

We now consider subspaces and quotients of locally convex spaces. When talking about subspaces of locally convex spaces we shall, unless explicitly stated otherwise, assume that their topology is the inherited one. It is induced by the restrictions of all continuous seminorms. It is useful to know that this way one obtains all continuous seminorms on the subspace.

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Theorem 3.4.3 Let D be a subspace of a locally convex space E. Then every continuous seminorm on D can be extended to a continuous seminorm on E. Proof. Let q be a continuous seminorm on D. The family {p|D : p is a continuous seminorm on E} generates the topology on D and is closed with respect to the forming of finite maxima and multiplication by numbers in |K|, so it is a base of continuous seminorms and there is a continuous seminorm q1 on E such that q ≤ q1 on D. The formula p(x) := inf{max(q(d), q1 (x − d)) : d ∈ D} defines a seminorm p on E for which p ≤ q1 (so p is continuous) and p = q on D. We now look at quotients. Definition 3.4.4 Let p be a seminorm on a vector space E, let D be a subspace of E, let π : E−→E/D be the canonical map. Then the formula pπ (π (x)) := inf{p(x − d) : d ∈ D} defines a seminorm pπ on E/D called the quotient seminorm of p on E/D. Theorem 3.4.5 Let (E, τ ) be a locally convex space, let D ⊂ E be a subspace, let π : E−→E/D be the canonical map. For a topology ν on E/D the following are equivalent: (α) For every collection P of seminorms on E generating τ , ν is the locally convex topology on E/D generated by {pπ : p ∈ P}. (β) ν is generated by the family of all seminorms q on E/D such that q ◦ π is τ -continuous. (γ ) ν is the locally convex topology on E/D having {π(U ) : U is an absolutely convex τ -zero neighbourhood in E} as a neighbourhood base of 0. (δ) ν is the strongest (locally convex) topology on E/D making π continuous. (ε) π : (E, τ )−→(E/D, ν) is continuous and open. Proof. Classical and straightforward.

Definition 3.4.6 (Quotients of locally convex spaces) The topology ν described above will be denoted by τπ and will be referred to as the quotient topology on E/D. Henceforth, unless explicitly stated otherwise, we will assume that E/D is equipped with the quotient topology.

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We now investigate under which circumstances E/D is Hausdorff. First an obvious remark. Theorem 3.4.7 A locally convex space E is Hausdorff if and only if for each non-zero x ∈ E there is a continuous seminorm p on E such that p(x) = 0. Theorem 3.4.8 Let D be a subspace of a locally convex space E. Then E/D is Hausdorff if and only if D is closed. Proof. If x ∈ D then, with π : E−→E/D the canonical map, pπ (π (x)) = 0 for all continuous seminorms p on E, hence q(π(x)) = 0 for all continuous seminorms q on E/D. Thus, if E/D is Hausdorff then π (x) = 0, i.e., x ∈ D and it follows that D is closed. Conversely, let D be closed. Let z ∈ E/D, z = 0. There is an x ∈ E with π(x) = z. Then x ∈ D, so there is a continuous seminorm p on E and an r > 0 such that {y ∈ E : p(x − y) < r} ∩ D = ∅. Then p(x − d) ≥ r for all d ∈ D, hence pπ (z) = pπ (π (x)) = 0. We see that E/D is Hausdorff. Next we consider the forming of products. Definition 3.4.9 (Products of locally convex spaces) Let I be an index set and let, for each i ∈ I , a locally convex space Ei be given. Then by the product {Ei : i ∈ I } we mean the space i∈I Ei equipped with the product topology. For this product we use again the same notation i∈I Ei . If Ei = E for each i we also write E I instead of i∈I Ei . If I = {1, . . . , n} for some n ∈ N we often write E1 × · · · × En instead of i∈I Ei . It is not hard to see that i∈I Ei is locally convex and its topology is induced by the seminorms (xi )i∈I −→ pj (xj ), where j runs through I and where pj runs through the collection of all con tinuous seminorms on Ej . Also notice that i∈I Ei is Hausdorff if and only if each Ei is Hausdorff. Products occur in the following representation theorem. Theorem 3.4.10 Every Hausdorff locally convex space can linearly and homeomorphically be embedded into a product of Banach spaces. Proof. Let E be a Hausdorff locally convex space and let P be a collection of seminorms on E defining the topology. For each p ∈ P consider πp

jp

E −→ Ep −→ Ep∧

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(see 3.3.1), where jp is the inclusion of Ep into its completion Ep∧ . Define T : E−→ p∈P Ep∧ by T (x) := (jp (πp (x)))p∈P . The Hausdorff property implies that the linear map T is injective. For a net (xi )i in E we have limi T (xi ) = 0 if and only if limi jp (πp (xi )) = 0 for each p ∈ P, i.e., limi πp (xi ) = 0 for each p ∈ P, i.e., limi p(πp (xi )) = 0 for each p ∈ P, i.e., limi p(xi ) = 0 for each p ∈ P, i.e., limi xi = 0, which proves that T is a homeomorphism. Products can be employed when constructing completions (3.4.13). Definition 3.4.11 Let E be a locally convex space. A net (xi )i in E is called Cauchy net if for each zero neighbourhood U in E there exists an i0 such that for i, j ≥ i0 we have xi − xj ∈ U or, equivalently, if limi,j p(xi − xj ) = 0 for all continuous seminorms p on E. A subset X of a Hausdorff locally convex space is called complete if each Cauchy net in X converges to a limit that lies in X. A completion of a Hausdorff locally convex space E is a complete locally convex space E∧ together with a linear homeomorphism i : E−→E ∧ such that i(E) is dense in E ∧ . Completions are unique in the sense that if (j, E ∼ ) is another completion of E then there exists a unique bijective linear homeomorphism T : E ∧ −→E ∼ making the diagram i / E∧ EB BB z z BB zz B zz T j BB z }z E∼

commute. Also, it is not difficult to see that the notions of completeness and completion coincide with the metric ones in the normed case. Lemma 3.4.12 Let {Ei : i ∈ I } be a collection of Hausdorff locally convex spaces. Then i∈I Ei is complete if and only if each Ei is complete. Proof. Left to the reader.

Theorem 3.4.13 Each Hausdorff locally convex space E has a completion. Proof. From 3.4.10 it follows that there is a linear homeomorphism T of E into a product of Banach spaces which is complete by the above lemma. Then also T (E) is complete and the pair (T , T (E)) is a completion of E.

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In view of this theorem and the uniqueness property we may from now on speak about “the” completion of a Hausdorff locally convex space E. We will denote it by E ∧ and often will think of the map i : E−→E ∧ as an inclusion. The following generalization of 2.1.4 has a straightforward proof (and has been used already in the uniqueness proof!); we quote it for reference. Theorem 3.4.14 Let E, F be Hausdorff locally convex spaces, let T ∈ L(E, F ). Then T extends uniquely to a T ∧ ∈ L(E ∧ , F ∧ ). Next we turn to direct sums. First, let us observe that the “coordinate maps” Ei −→Ej (j ∈ I ), πj : i∈I

given by (xi )i∈I −→ xj , where Ei are locally convex spaces, are continuous. Even more, the topology on i∈I Ei is the weakest one making all the coordinate maps continuous. In the next definition we consider a “dual” situation. Definition 3.4.15 (Locally convex direct sums) Let I an index set and let, for each i ∈ I , a locally convex space Ei be given. Then, by the locally convex alg direct sum of {Ei : i ∈ I } we mean the algebraic direct sum E := i∈I Ei , equipped with the so-called locally convex direct sum topology: this is the strongest locally convex topology such that all natural injections φj : Ej −→E (j ∈ I ) are continuous. We denote this locally convex direct sum by i∈I Ei . If Ei = E for each i we also write E (I ) instead of i∈I Ei . A seminorm p on i∈I Ei is continuous if and only if p ◦ φj is continuous for each j ∈ I . The coordinate maps πj : i∈I Ei −→Ej are continuous and πj ◦ φj is the identity on Ej whereas πj ◦ φi = 0 for i ∈ I , i = j . It follows easily that i∈I Ei is Hausdorff if and only if each Ei is Hausdorff. The study of locally convex direct sums is somewhat more involved than the one of products. A useful tool will be the following notion that will occur at several other places in this book as well. Definition 3.4.16 Let τ1 ≤ τ2 be locally convex topologies on a vector space E. We say that τ1 is closely related to τ2 if there exists a τ2 -neighbourhood base of 0 consisting of τ1 -closed absolutely convex sets. Lemma 3.4.17 Let τ1 ≤ τ2 be closely related locally convex topologies on a vector space E. Let (xi )i be a Cauchy net in (E, τ2 ) that converges to an x ∈ E with respect to τ1 . Then (xi )i converges to x with respect to τ2 .

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Proof. Let U be a τ2 -neighbourhood of 0. Without loss, assume U is τ1 -closed. There exists an i0 ∈ I such that for i, j ∈ I , i, j ≥ i0 xi − xj ∈ U. Thus, τ1 -limi (xi − xj ) = x − xj ∈ U , because U is τ1 -closed. Hence x − xj ∈ U for j ≥ i0 , which proves the theorem. We now return to the locally convex direct sums. Let I be an index set, let, for each i ∈ I , Ei be a locally convex space. In 3.4.18−3.4.20 we will write E := i∈I Ei . For each x ∈ E we have x= (φi ◦ πi )(x) i∈I

(only finitely many summands are non-zero). Theorem 3.4.18 The family of all seminorms of the form maxi∈I pi ◦ πi , where, for each i, pi is a continuous seminorm on Ei , is a base of continuous seminorms on E. Proof. For each x ∈ E there are only finitely many i ∈ I for which πi (x) = 0, so the expression maxi∈I pi ◦ πi makes sense and defines a seminorm p on E. Let j ∈ I . Then p ◦ φj = maxi∈I pi ◦ πi ◦ φj = pj ◦ πj ◦ φj = pj , so p is continuous. Now let q be an arbitrary continuous seminorm on E. Then for each x ∈ E q(x) = q (φi ◦ πi )(x) ≤ max(q ◦ φi )(πi (x)). i∈I

i∈I

So q is dominated by a seminorm of the chosen collection, and we are done. Corollary 3.4.19 The topology on E of coordinatewise convergence (i.e., the restriction to E of the product topology) is closely related to the locally convex direct sum topology on E. Proof. Let P be the base of continuous seminorms on E constructed above. For each p = maxi∈I pi ◦ πi ∈ P and r > 0, U := {x ∈ E : p(x) ≤ r} = i∈I {x ∈ E : (pi ◦ πi )(x) ≤ r} is an intersection of sets that are closed with respect to the product topology. We now use these observations to prove the following. Theorem 3.4.20 The locally convex direct sum of a collection of complete (Hausdorff) locally convex spaces is complete.

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103

Proof. We use the notations as before. Let (x γ )γ be a Cauchy net in E. Then, for each i ∈ I , γ → πi (x γ ) is a Cauchy net in Ei . By completeness it converges to an element xi ∈ Ei . It suffices to prove that M := {i ∈ I : xi = 0} is finite. (Then we can define x := i∈I φi (xi ) ∈ E and (x γ )γ converges coordinatewise to x and we use 3.4.17 and 3.4.19 to conclude that (x γ )γ converges to x in the locally convex direct sum topology.) To prove finiteness of M, choose for each i ∈ M a continuous seminorm pi on Ei for which pi (xi ) = 1, and choose pi := 0 if i ∈ I \ M. Then p := maxi∈I pi ◦ πi is a continuous seminorm on E. There is a γ0 ∈ such that for γ , γ ≥ γ0 we have p(x γ − x γ ) ≤ 1/2. Hence pi (πi (x γ ) − πi (x γ0 )) ≤

1 2

(i ∈ M, γ ≥ γ0 ).

By convergence of γ → πi (x γ ) to xi we get pi (xi − πi (x γ0 )) ≤

1 2

(i ∈ M).

Since pi (xi ) = 1 we obtain pi (πi (x γ0 )) = 1

(i ∈ M).

But {i ∈ I : πi (x γ0 ) = 0} is finite. It follows that M must be finite.

Remark 3.4.21 The converse of the above theorem is also true, i.e., if {Ei : i ∈ I } is a collection of Hausdorff locally convex spaces then completeness of of each Ei . (In fact, for each i ∈ I , φi (Ei ) is a i∈I Ei implies completeness closed subspace of i∈I Ei and φi is a homeomorphism.) A special locally convex direct sum is worth mentioning; it is useful for constructing (counter) examples. Theorem 3.4.22 (The strongest locally convex topology) Let E be a vector space equipped with the strongest locally convex topology (i.e., the locally convex topology induced by all seminorms on E). Then E has the following properties: (i) Every convex subset of E is closed. (ii) Every linear map of E to any locally convex space F is continuous. (iii) Every subspace and every quotient carry also the strongest locally convex topology.

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(iv) Let (ei )i∈I be an algebraic base of E. Then the map (λi )i∈I −→ λi ei i∈I

is a linear surjective homeomorphism K (I ) −→E. (v) E is complete. (vi) Every convergent sequence lies in a finite-dimensional subspace. (vii) If E is infinite-dimensional then E is not metrizable (see 3.5.1). Proof. Statement (i) follows from 3.1.14 and 3.3.14. Also, (ii), (iii) are straightforward. The map defined in (iv) is a linear bijection. By definition of the locally convex direct sum the space K (I ) has the strongest locally convex topology which proves (iv), and (v) follows from (iv) and the previous theorem. We proceed to prove (vi). Let x1 , x2 , . . . be a convergent sequence and suppose [x1 , x2 , . . .] is infinite-dimensional. Then we could find a subsequence y1 , y2 , . . . of linearly independent vectors. The formula ∞ λi yi := max |λ2n | p i=1

n∈N

defines a (continuous) seminorm on [y1 , y2 , . . .] and can be extended to a seminorm q on E (3.4.3). But the sequence q(y1 ), q(y2 ), . . . does not converge, a contradiction. Finally, we prove (vii). Let x1 , x2 , . . . be linearly independent elements of E. Set Dn := [x1 , . . . , xn ]. By (i) and (v) D := n∈N Dn is complete, and the union of countably many closed subsets with empty interior. By the Baire Category Theorem D cannot be metrizable. Remark 3.4.23 Almost everything of the above is classical except for (i), which is typically non-Archimedean. In 2.1.14(iii) we have seen that finite-dimensional spaces have only one norm topology. We now prove that this topology is also unique in the locally convex setting. Theorem 3.4.24 Let E be a finite-dimensional vector space. Then the strongest locally convex topology on E is the unique Hausdorff locally convex topology. Proof. It is easily seen that, with respect to the norm topology on E, each seminorm is continuous, so the norm topology equals the strongest locally convex topology. To prove uniqueness we proceed by induction on n := dim E. For n = 1 the conclusion is obvious. For the induction step, let τ be a Hausdorff

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105

locally convex topology on an n-dimensional space E. Choose x ∈ E, x = 0 and a τ -continuous seminorm p with p(x) = 0 (3.4.7). Then dim Ker p < n, so by the induction hypothesis there is a τ -continuous norm on Ker p that can, by 3.4.3, be extended to a τ -continuous seminorm q on E. Then p ∨ q is easily seen to be a norm on E. Thus, E has a τ -continuous norm, so that τ is equal to the strongest locally convex topology. Next we give some consequences of the above results. Corollary 3.4.25 Let E be a finite-dimensional Hausdorff locally convex space. Then, for any locally convex space F , every linear map of E to F is continuous. Proof. Apply 3.4.22(ii) and 3.4.24.

We obtain the following generalization of 2.1.13. Corollary 3.4.26 Let E, F be locally convex spaces. Then every linear map of E to F whose kernel is closed and finite-codimensional is continuous. In particular, if f ∈ E ∗ then f is continuous if and only if Ker f is closed. Proof. Let T : E−→F be linear and suppose that Ker T is closed and finitecodimensional. Let π : E−→E/Ker T be the (continuous) quotient map and let S : E/Ker T −→F , π (x) → T (x). By assumption and 3.4.8, E/Ker T is finitedimensional and Hausdorff. Applying 3.4.25 we obtain that S is continuous, hence so is T = S ◦ π. By using 3.4.25 we also have the following result on complementation of subspaces. Observe that the notions of continuous projection, complemented subspace and complement given in 2.2.5 and in the comments thereafter carry over to the locally convex setting. Corollary 3.4.27 Every closed finite-codimensional subspace of a locally convex space E is complemented. Proof. Let D be a closed finite-codimensional subspace of E and let π : E−→E/D be the quotient map. From Algebra we know that there exists a linear map ϕ : E/D−→E such that π ◦ ϕ equals the identity on E/D. Then I − ϕ ◦ π is easily seen to be a projection onto D. Now to prove continuity it suffices to show that ϕ is continuous. This follows from 3.4.25, because E/D is Hausdorff and finite-dimensional. The reason why complementation is fundamental lies in the following fact.

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Theorem 3.4.28 Let D be a complemented subspace of a locally convex space E. Then for any complement D1 of D, the space D × D1 is isomorphic to E through the map (d, d1 ) → d + d1 . Proof. Let h : D × D1 −→E, (d, d1 ) → d + d1 . Clearly h is linear and bijective. Also, addition in E is continuous, hence so is h. To prove continuity of h−1 we have to show that the maps d + d1 → d and d + d1 → d1 are continuous. But it is clear, since for any continuous projection P with Im P = D and Ker P = D1 , we have d = P (d + d1 ) and d1 = (I − P )(d + d1 ). Closely related to products and direct sums are the projective and inductive limits. They can be defined in a very general way (see [144], Section 19) but in this book we need only the following “countable” version. Definition 3.4.29 A projective system is a sequence (En , πn )n∈N where, for each n, En is a locally convex space, and where πn ∈ L(En+1 , En ): π2

π1

· · · · −→E3 −→ E2 −→ E1 . In case each πn is injective we will interpret them as inclusions and call (En , πn )n∈N a projective sequence. The projective limit of the projective system (En , πn )n∈N , denoted by lim(En , πn ), or simply by lim En , is the subspace of all (x1 , x2 , . . .) ∈ n∈N En ←− ←− such that πn (xn+1 ) = xn for all n. Note that lim En is a closed subspace of n∈N En . By 3.4.12 we obtain that the ←− projective limit of complete spaces is complete. The projective limit has the following Functorial Property. Let (Fn , ρn )n∈N be a second projective system and let, for each n ∈ N, Tn ∈ L(En , Fn ) be such that the diagram En+1 Tn+1

πn

Tn

Fn+1

/ En

ρn

/ Fn

commutes. Then the natural linear map T : n∈N En −→ n∈N Fn induced by Tn is continuous and maps lim En into lim Fn . ←−

←−

By taking in the above Fn = En∧ , ρn := πn∧ (3.4.14), Tn : En −→En∧ the canonical map, we easily derive the following.

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107

Theorem 3.4.30 Let (En , πn )n∈N be a projective system of Hausdorff locally convex spaces En . Then lim En is also Hausdorff and (lim En )∧ is naturally isomorphic to lim En∧ .

←−

←−

←−

For projective sequences we have a much simpler interpretation of the projective limits. Theorem 3.4.31 Let (E1 , τ1 ), (E2 , τ2 ), . . . be locally convex spaces such that E1 ⊃ E2 ⊃ · · · and τn |En+1 ≤ τn+1 for each n. Let E := n∈N En be equipped with the union τ of all τn |E (n ∈ N). Then the map (E, τ )−→ lim En given by ←−

x −→ (x, x, x, . . .) is a linear surjective homeomorphism.

Proof. Straightforward.

In other words, the projective limit of a projective sequence (En , πn )n∈N is the space E := n∈N En equipped with the topology induced by n∈N {p|E : p is a continuous seminorm on En }. It is the weakest (locally convex) topology on E for which all inclusions E−→En are continuous. In a similar way we will introduce inductive limits. Definition 3.4.32 An inductive system is a sequence (En , in )n∈N where, for each n, En is a locally convex space, and where in ∈ L(En , En+1 ): i1

i2

E1 −→ E2 −→ E3 −→ · · · · In case each in is injective we will interpret them as inclusions and call (En , in )n∈N an inductive sequence. The inductive limit of the inductive system (En , in )n∈N , denoted by lim(En , in ), or simply by lim En , is the quotient space ( n∈N En )/D, where −→ −→ D is the subspace generated by all elements of the form (x1 , −i1 (x1 ), 0, 0, . . .) (x1 ∈ E1 ), (0, x2 , −i2 (x2 ), 0, 0, . . .) (x2 ∈ E2 ), . . . . The above spaces E1 , E2 , . . . are often called informally the “steps”. It is not difficult to see that inductive limits have functorial properties similar to the ones of projective limits. We leave formulation and details to the reader. Remark 3.4.33 The reader may expect here a statement concerning completeness of inductive limits in the spirit of 3.4.30. But it will turn out later (11.2.5, 11.4.10(a)) that this is a delicate matter. The same goes for completeness of quotients. There is even no guarantee that inductive limits of Hausdorff spaces are Hausdorff (see 11.2.2).

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In many cases we will have to deal with inductive sequences. For these we have, like in the case of projective sequences, a simple description of its inductive limits as follows. Theorem 3.4.34 Let (E1 , τ1 ), (E2 , τ2 ), . . . be locally convex spaces such that E1 ⊂ E2 ⊂ · · · and τn+1 |En ≤ τn for each n. Let E := n∈N En be equipped with the strongest locally convex topology τ for which the inclusions (En , τn )−→(E, τ ) are continuous. For each m, let um be the canonical map Em −→

En −→ lim En . −→

n∈N

Then um+1 |Em = um for all m, so the um define a map u, u

(E, τ ) −→ lim En −→

which is a linear surjective homeomorphism. Proof. Direct verification.

In other words, we can view the inductive limit of an inductive sequence (En , in )n∈N as the space E := n∈N En equipped with the topology induced by n∈N {p : p seminorm on E, p|En is continuous on En }. It is the strongest locally convex topology on E for which all inclusions En −→E are continuous. Remark 3.4.35 We will encounter inductive limits in this book several times. Chapter 11 is devoted to this subject.

3.5 Metrizable and Fr´echet spaces Definition 3.5.1 A locally convex space is called metrizable if there exists a metric inducing the topology. A complete metrizable locally convex space is called a Fr´echet space. A metric d on a vector space E is called invariant if d(x + z, y + z) = d(x, y) for all x, y, z ∈ E. Theorem 3.5.2 For a locally convex space E the following are equivalent: (α) E is metrizable. (β) E is Hausdorff. There is a countable base of (absolutely convex) zero neighbourhoods. (γ ) E is Hausdorff. There is a countable collection of seminorms generating the topology. (δ) There is an invariant ultrametric on E inducing the topology.

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109

Proof. (α) =⇒ (β). Let d be a metric on E inducing the topology. The sets {x ∈ E : d(x, 0) < n1 } (n ∈ N) form a countable base of neighbourhoods of 0. By local convexity we have a countable base of absolutely convex zero neighbourhoods. (β) =⇒ (γ ). Let {Un : n ∈ N} be a countable base of absolutely convex zero neighbourhoods. Then, for each n, pn := pUn is continuous (3.3.15). We claim that p1 , p2 , . . . generate the topology. In fact, let p be a continuous seminorm. Then there is an m such that x ∈ Um implies p(x) < 1. Now pm (x) < 1 implies x ∈ Um and hence p(x) < 1. It follows that p ≤ pm . (γ ) =⇒ (δ). Let {p1 , p2 , . . .} be a countable set of seminorms inducing the topology. The formula d(x, y) := max min(pn (x − y), 2−n ) n∈N

(x, y ∈ E)

defines an invariant ultrametric on E. One verifies directly that a net (xi )i in E converges to x ∈ E if and only if limi d(x, xi ) = 0. (δ) =⇒ (α). Trivial. Remark 3.5.3 For a metrizable locally convex space E, a subset is closed if and only if it is sequentially closed, and E is complete if and only if each Cauchy sequence converges. If d is an invariant ultrametric inducing the topology ((δ) above) then a sequence x1 , x2 , . . . in E is Cauchy if and only if limn d(xn , xn+1 ) = 0. Corollary 3.5.4 Let E be a metrizable locally convex space. Then: (i) there is an increasing sequence p1 ≤ p2 ≤ · · · of seminorms forming a base of continuous seminorms, (ii) there is a decreasing sequence U1 ⊃ U2 ⊃ · · · of absolutely convex sets forming a neighbourhood base of 0. Proof. By the previous theorem there is a countable collection {q1 , q2 , . . .} of seminorms generating the topology. Then (i) is established by taking pn := max(q1 , . . . , qn ), and by defining Un := {x ∈ E : pn (x) ≤ n1 } we arrive at (ii). Clearly (closed) subspaces of metrizable (Fr´echet) spaces are metrizable (Fr´echet). Theorem 3.5.5 Countable products and projective limits of metrizable locally convex spaces are metrizable. The quotient of a metrizable locally convex space by a closed subspace is metrizable.

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Proof. Let E1 , E2 , . . . be metrizable locally convex spaces. Thanks to 3.5.2 there is, for each n, an (invariant) ultrametric dn on En inducing the topology of En . We may assume dn ≤ 1. Now, for x = (xn )n∈N , y = (yn )n∈N in n∈N En , set d(x, y) := maxn∈N 2−n dn (xn , yn ). It is easily seen that d is an ultrametric inducing the product topology. Thus, n∈N En is metrizable, hence so are projective limits (3.4.29). Now let D be a closed subspace of a metrizable locally convex space E, let π : E−→E/D be the canonical map, let P be a countable collection of seminorms on E inducing the topology (3.5.2). Then by 3.4.5 the countable set {pπ : p ∈ P} induces the quotient topology. Also, by 3.4.8 E/D is Hausdorff. Remark 3.5.6 From 3.4.22(vii) we see that the space K (N) is not metrizable. So countable direct sums of metrizable spaces need not be metrizable. The same goes for inductive limits, as we will show in 11.2.8. Corollary 3.5.7 Countable products and projective limits of Fr´echet spaces are Fr´echet. Proof. Combine 3.5.5 with 3.4.12 and 3.4.30.

The completeness of quotients is somewhat more involved. We need the following lemma. Lemma 3.5.8 Let D be a subspace of a metrizable locally convex space E, let π : E−→E/D be the canonical map. Then, if y1 , y2 , . . . is a sequence in E/D converging to 0, there is a sequence x1 , x2 , . . . in E, converging to 0, such that π(xn ) = yn for each n. Proof. By 3.5.4(ii) there are absolutely convex U1 ⊃ U2 ⊃ · · · forming a neighbourhood base of 0 in E. Then (3.4.5) π(U1 ), π (U2 ), . . . is a neighbourhood base of 0 in E/D. There are m1 < m2 < · · · such that for each k m ≥ mk =⇒ ym ∈ π (Uk ). Now let n ∈ N. If n < m1 choose an arbitrary xn ∈ E with π (xn ) = yn . If n ≥ m1 there is a unique k such that mk ≤ n < mk+1 . Then yn ∈ π (Uk ), so we can find an xn ∈ Uk such that π(xn ) = yn . Clearly, limn xn = 0. Theorem 3.5.9 Let D be a closed subspace of a Fr´echet space E. Then E/D is Fr´echet. Proof. Since E/D is metrizable (3.5.5) we can use Cauchy sequences to prove completeness. Let again π : E−→E/D be the canonical map. Let y1 , y2 , . . .

3.5 Metrizable and Fr´echet spaces

111

be a Cauchy sequence in E/D. Then y1 , y2 − y1 , y3 − y2 , . . . tends to 0, so by the lemma above there are x1 , x2 , . . . in E, tending to 0, such that π(x1 ) = y1 , π(xn ) = yn − yn−1 for n ≥ 2. Since E is Fr´echet, x1 , x2 , . . . is summable, let ∞ x := ∞ n=1 xn . Then π (x) = y1 + n=2 (yn − yn−1 ) = limm ym . We see that y1 , y2 , . . . converges to π (x). We now prove the familiar Open Mapping Theorem. Theorem 3.5.10 (Open Mapping Theorem for Fr´echet spaces) Let E, F be Fr´echet spaces and let T ∈ L(E, F ) be surjective. Then T is an open mapping. Proof. (Not materially different from the proof of 2.1.17.) Let U ⊂ E be absolutely convex, open. We first prove that T (U ) is open. In fact, let λ1 , λ2 , . . . ∈ K × , limn |λn | = ∞. Then, since U is absorbing (3.3.15(i)), E = n∈N λn U , so F = n∈N λn T (U ). By completeness, metrizability and the Baire Category Theorem, there is an n such that int (λn T (U )) = ∅, so by 3.3.11(ii) λn T (U ), hence T (U ), is open. Next we show that T (U ) itself is open. There are absolutely convex U := U1 ⊃ U2 ⊃ · · · forming a neighbourhood base of 0 in E. By the first part of the proof T (U1 ) ⊃ T (U2 ) ⊃ · · · are clopen, hence form a neighbourhood base of 0 in F . Let us prove T (U1 ) ⊂ T (U ) (then, by 3.3.11(ii), T (U ) is open). For that, let y ∈ T (U1 ). The set y − T (U1 ) meets T (U2 ), so there is an x1 ∈ U1 such that y − T (x1 ) ∈ T (U2 ). By the same token there is an x2 ∈ U2 such that y − T (x1 ) − T (x2 ) ∈ T (U3 ), etc. Inductively, we arrive at a sequence x1 , x2 , . . . , where xn ∈ Un ⊂ U and y − T (x1 + · · · + xn ) ∈ T (Un ) for each n, so limn T (x1 + · · · + xn ) = y. Now limn xn = 0, so by complete ness of E, x := ∞ n=1 xn exists and lies in U because U is closed (again by 3.3.11(ii)). We see that y = T (x) ∈ T (U ). Remark 3.5.11 Like in the Banach case (2.1.19) we can derive a Closed Graph Theorem. We leave the details to the reader. This theorem will be extended to a larger class of spaces in 11.1.10. For a generalization of the Uniform Boundedness Principle 2.1.20 and the Banach–Steinhaus Theorem 2.1.21 we refer to 7.1.8. We have one more, peculiar, consequence. Corollary 3.5.12 Let E, F be Fr´echet spaces, let T ∈ L(E, F ) be such that codim T (E) < ∞. Then T (E) is closed. Proof. We may assume that T is injective (3.5.9). Let D be an algebraic complement of T (E). D is a Fr´echet space equipped with the unique Hausdorff

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locally convex topology (3.4.24). The map u : E × D−→F given by u : (x, d) −→ T (x) + d

(x ∈ E, d ∈ D)

is a continuous linear bijection between the Fr´echet spaces E × D and F , so it is a homeomorphism by 3.5.10. Thus, T (E) = u(E × {0}) is closed.

3.6 Bounded sets Boundedness is defined just like in the classical case. Definition 3.6.1 A subset X of a locally convex space E is called bounded (in E) if it is absorbed by every zero neighbourhood, i.e., if for every zero neighbourhood U there is a λ ∈ K such that X ⊂ λ U . Accordingly, a sequence x1 , x2 , . . . in E is called bounded if {x1 , x2 , . . .} is bounded. Clearly a subset X of a locally convex space E is bounded in E if and only if it is bounded in [X], which enables us to speak about boundedness without referring to the embedding space. A subset X of E is bounded if and only if each continuous seminorm (equivalently, each seminorm of a collection generating the topology) is bounded on X, showing that, in case of a normed space E, boundedness coincides with norm boundedness. Theorem 3.6.2 A Hausdorff locally convex space is normable if and only if it has a bounded zero neighbourhood. Proof. We prove that a Hausdorff locally convex space E with a bounded zero neighbourhood U is normable. In fact, the Minkowski function pU is continuous (3.3.15). Let p be a continuous seminorm on E. Then by boundedness of U there is a λ ∈ K such that p(x) ≤ |λ| for x ∈ U . This implies p ≤ |λ| pU . So {pU } generates the topology and, since E is Hausdorff, pU is a norm. We leave verification of the next theorem to the reader. Theorem 3.6.3 Let E be a locally convex space. Then we have the following: (i) Subsets, finite unions and sums, (absolutely) convex hulls, closures, continuous linear images of bounded sets in E are bounded. (ii) If C ⊂ E is a convex bounded set then so is C e . (iii) (Pre)compact subsets of E are bounded. (iv) X ⊂ E is bounded if and only if each countable subset of X is bounded.

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113

Theorem 3.6.4 Let I be an index set and let, for each i ∈ I , a locally convex space Ei be given. Then we have the following (where, as in Section 3.4, the πi are the coordinate maps): (i) A subset X of i∈I Ei is bounded if and only if each πi (X) is bounded. (ii) A subset X of i∈I Ei is bounded if and only if πi (X) is bounded in Ei for each i and πi (X) ⊂ {0} for all but finitely many i. Proof. (i). We only need to prove the “if” part; the “only if” follows from (i) of the above theorem. So let πi (X) be bounded in Ei for each i. Now P := {pi ◦ πi : i ∈ I , pi is a continuous seminorm on Ei } generates the product topology on i∈I Ei and each member of P is bounded on X. Thus, X is bounded. (ii). Suppose X is bounded. Then, again by (i) of the above theorem, πi (X) is bounded in Ei , for each i. If we would have mutually distinct i1 , i2 , . . . ∈ I and x1 , x2 , . . . ∈ X with πin (xn ) ∈ {0} for all n, choose λ1 , λ2 , . . . ∈ K such that limn |λn | = ∞ and continuous seminorms pin on Ein with pin (πin (xn )) = |λn |. For i ∈ I \ {i1 , i2 , . . .}, choose pi := 0. Then, by 3.4.18, maxi∈I pi ◦ πi is a continuous seminorm on i∈I Ei , not bounded on X, a contradiction. Conversely, let J ⊂ I be a finite set such that πi (X) ⊂ {0} for i ∈ I \ J . Then (with φi as in 3.4.15) X ⊂ i∈J φi (πi (X)) + i∈I \J φi (πi (X)). Ei

E

Now, for i ∈ I \ J we have πi (X) ⊂ {0} , so that φi (πi (X)) ⊂ {0} , hence E i∈I \J φi (πi (X)) ⊂ {0} is bounded. Also, i∈J φi (πi (X)) is bounded and then so is X (3.6.3(i)). Corollary 3.6.5 A subset X of the projective limit of a projective system (En , πn )n∈N of locally convex spaces is bounded if and only if each πn (X) is bounded. Remarks 3.6.6 (a) Let E be a vector space equipped with the strongest locally convex topology. Applying 3.4.22(iv) and 3.6.4(ii) we obtain that bounded subsets of E are finite-dimensional. (b) The characterization of bounded sets in inductive limits lim En in terms of −→ the steps En is a delicate matter, see Chapter 11. (c) In 9.8.2(i) we will construct a Fr´echet space E, a closed subspace D ⊂ E and a bounded set Y ⊂ E/D which is not the image under the quotient map E−→E/D of any bounded subset of E. Definition 3.6.7 A Hausdorff locally convex space E is called quasicomplete if every closed bounded subset of E is complete, or equivalently, if every bounded

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Cauchy net in E is convergent. Also, E is called sequentially complete if every Cauchy sequence in E is convergent. Clearly every complete space is quasicomplete and every quasicomplete space is sequentially complete (note that every Cauchy sequence is bounded). These three notions coincide when E is metrizable. For examples of sequentially complete spaces that are not quasicomplete and of quasicomplete spaces that are not complete, see 3.7.10 and 5.4.7 respectively.

3.7 Examples of locally convex spaces 3.7.1 Spaces of continuous functions In 2.5.17–2.5.40 we considered examples of Banach spaces of continuous functions. In the same spirit we will study now spaces of continuous functions whose natural topologies are locally convex but, in general, not normable. Throughout Subsection 3.7.1 we assume that X is a non-empty zerodimensional Hausdorff topological space. Recall that C(X) is the space of all continuous functions X−→K . Notation For a non-empty subset Y ⊂ X and f : X−→K we set (admitting the value ∞) pY (f ) := sup{|f (x)| : x ∈ Y }. For convenience we put p∅ (f ) := 0. Notice that pY (f ) < ∞ if Y is compact, f ∈ C(X). We first study the topologies of simple convergence (3.7.1) and compact convergence (3.7.3) on C(X). Definition 3.7.1 The topology of simple (or, pointwise) convergence, τs , on C(X) is the topology generated by the seminorms p{x} , where x ∈ X. Thus, the seminorms pY , where Y runs through the class of finite subsets of X, form a base of τs -continuous seminorms. A net (fi )i in C(X) converges to f ∈ C(X) in τs if and only if fi → f pointwise, i.e., limi fi (x) = f (x) for each x ∈ X. Clearly, τs is Hausdorff. Theorem 3.7.2 (i) (C(X), τs ) is normable if and only if X is finite. (ii) (C(X), τs ) is metrizable if and only if X is countable. (iii) (C(X), τs ) is (quasi)complete if and only if X is discrete.

3.7 Examples of locally convex spaces

115

(iv) (C(X), τs ) is sequentially complete if and only if the intersection of countably many clopen sets in X is again clopen. Proof. (i). Suppose (C(X), τs ) is normable. Then there is a finite set Y ⊂ X such that pY is a norm generating the topology. If Y = X, let x ∈ X, x ∈ Y . There is a clopen U containing x, but U ∩ Y = ∅. Then ξU = 0 but pY (ξU ) = 0, a contradiction. So X = Y is finite. The converse is obvious. (ii). Suppose (C(X), τs ) is metrizable. Then (3.5.2) τs is induced by pY1 , pY2 , . . . , where Y1 , Y2 , . . . are finite sets. Without loss, assume Y1 ⊂ Y2 ⊂ · · · . Let x ∈ X. Then p{x} ≤ C pYn for some C > 0 and n ∈ N. By the same argument as in (i) we find x ∈ Yn . Hence X = n∈N Yn is countable. Conversely, if X is countable from the natural embedding (C(X), τs )−→K X and 3.5.5 we obtain metrizability of (C(X), τs ). (iii). Suppose (C(X), τs ) is quasicomplete. Let x ∈ X. Then {ξU : U ⊂ X clopen, x ∈ U } forms in a natural way a bounded Cauchy net in τs . So it converges and its limit has to be ξ{x} . Thus, {x} is clopen, and X is discrete. Conversely, if X is discrete then (C(X), τs ) = K X with the product topology, which is complete by 3.4.12. (iv). Suppose (C(X), τs ) is sequentially complete. Let U1 , U2 , . . . ⊂ X be clopen, let U := n∈N Un . To prove that U is clopen we may assume U1 ⊃ U2 ⊃ · · · . The sequence of characteristic functions ξU1 , ξU2 , . . . converges pointwise to ξU . So ξU ∈ C(X), i.e., U is clopen. Conversely, assume that the intersection of countably many clopen sets in X is clopen. Let f1 , f2 , . . . ∈ C(X) be a τs -Cauchy sequence in C(X), i.e., f (x) := limn fn (x) exists for every x ∈ X. To prove sequential completeness we have to show that f ∈ C(X). Let x ∈ X, ε > 0. Let us see that U := {y ∈ X : |f (y) − f (x)| < ε} is open. For each n ∈ N, consider the clopen set Un := {y ∈ X : |fn (y) − fn (x)| < ε}. Then U = m∈N Vm , where V1 := n≥1 Un , V2 := n≥2 Un , . . . are clopen in X. Hence U is open. Next we turn to compact convergence. Definition 3.7.3 The topology of uniform convergence on compact sets, or the compact open topology, τc , on C(X) is the topology generated by the seminorms pY , where Y ⊂ X, Y compact. Thus, the seminorms pY , where Y runs through the class of compact subsets of X, form a base of τc -continuous seminorms. Since singleton sets are compact we have τs ≤ τc , so τc is Hausdorff. A net (fi )i in C(X) converges to f ∈ C(X)

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Locally convex spaces

in τc if and only if limi fi (x) = f (x) uniformly on x ∈ Y , for each compact Y ⊂ X. For the characterizations of 3.7.9 we need some facts and notions on settheoretic topology. Definition 3.7.4 A fundamental sequence of compact sets in X is a sequence Y1 , Y2 , . . . of compact sets in X such that for each compact Y ⊂ X there is an n for which Y ⊂ Yn . X is called hemicompact if X possesses a fundamental sequence of compact sets. Remarks 3.7.5 (a) If Y1 , Y2 , . . . is a fundamental sequence of compact sets in X then so is Y1 , Y1 ∪ Y2 , Y1 ∪ Y2 ∪ Y3 , . . . . (b) There is no relation between hemicompactness and local compactness. In fact, an uncountable discrete space is locally compact but not hemicompact. As for the converse, let X := c00 be over Qp with the strongest locally convex topology. To prove that X is hemicompact consider, for m, n ∈ N, Ym,n := p−m {(λ1 , . . . , λn , 0, 0, . . .) : λ1 , . . . , λn ∈ Zp }. Every compact subset of X is bounded, so (3.6.6(a)) is finite-dimensional, and hence it is contained in some Ym,n . In the same way, if U were a compact neighbourhood of 0 in X then U is finite-dimensional, which conflicts with the fact that U is absorbing (3.3.15(i)). Therefore, X is not locally compact. (c) Clearly every hemicompact space is σ -compact, but the converse does not hold. To see this, consider Q with the usual topology inherited from R. The clopen subsets {x ∈ Q : a < x < b}, where a, b ∈ R \ Q, a < b form a base of the topology, so Q is zero-dimensional (even ultrametrizable, [193], p. 39). By countability Q is σ -compact. To prove that Q is not hemicompact, let Y1 ⊂ Y2 ⊂ · · · be compact sets in Q. Let n ∈ N. Then Yn cannot contain [− n1 , n1 ] ∩ Q (otherwise, by completeness it would contain irrational numbers), so we can find an an ∈ Q, an ∈ Yn , |an | ≤ n1 . The set Y := {0, a1 , a2 , . . .} is compact but Y ⊂ Yn for no n. (d) It is easily seen that if X is both σ -compact and locally compact then X is hemicompact. In the next theorem, notice that (α) is a property of the space X whereas in the formulation of (β) the field K is involved.

3.7 Examples of locally convex spaces

117

Theorem 3.7.6 The following statements on X are equivalent: (α) A subset U ⊂ X is clopen if and only if U ∩ Y is clopen in Y for each compact Y ⊂ X. (β) A function f : X−→K is continuous if and only if its restriction to each compact set is continuous. Proof. (α) =⇒ (β). Let f : X−→K be such that the restriction of f to each compact set is continuous. To prove continuity of f , let S ⊂ K be clopen. For each compact Y ⊂ X, f −1 (S) ∩ Y = (f |Y )−1 (S) is clopen in Y so, by (α), f −1 (S) is clopen, and f is continuous. Conversely, assume (β) and let U ⊂ X be such that U ∩ Y is clopen in Y for each compact Y . Then ξU |Y is continuous for each compact Y ⊂ X, so by (β) ξU is continuous, i.e., U is clopen. Definition 3.7.7 We say that X is a k0 -space if it satisfies (α) or (β) of the preceding theorem. Theorem 3.7.8 Every metrizable or locally compact X is a k0 -space. Proof. We only prove (β) of 3.7.6 for metrizable X. Let f : X−→K be such that f |Y is continuous for each compact Y ⊂ X. Let x1 , x2 , . . . ∈ X, limn xn = x ∈ X. Then {x, x1 , x2 , . . .} is compact, so by assumption limn f (xn ) = f (x). Thus, f is sequentially continuous, hence continuous by metrizability. Now we arrive at the desired characterizations. Theorem 3.7.9 (i) (C(X), τc ) is normable if and only if X is compact. (ii) (C(X), τc ) is metrizable if and only if X is hemicompact. (iii) (C(X), τc ) is (quasi)complete if and only if X is a k0 -space. Proof. (i). Suppose (C(X), τc ) is normable. The same reasoning as in the proof of 3.7.2(i), but replacing “finite” by “compact” leads to compactness of X. The converse is obvious. (ii). Let (C(X), τc ) be metrizable. Then the topology τc is induced by countably many seminorms (3.5.2); we may assume that they have the form pY1 , pY2 , . . . , where Y1 ⊂ Y2 ⊂ · · · are compact. Now let Y ⊂ X be compact. Then pY ≤ C pYn for some C > 0 and n ∈ N. Suppose Y ⊂ Yn ; we derive a contradiction. Choose y ∈ Y , y ∈ Yn , y has a clopen neighbourhood U that does not meet Yn . Then 1 = pY (ξU ) ≤ C pYn (ξU ) = 0, a contradiction. Hence, Y1 , Y2 , . . . is a fundamental sequence of compact sets. Conversely, if X is hemicompact, let Y1 ⊂ Y2 ⊂ · · · be a fundamental sequence of compact sets. It can be seen almost immediately that pY1 , pY2 , . . . generate the topology τc and metrizability follows.

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Locally convex spaces

(iii). Suppose (C(X), τc ) is quasicomplete. Let U ⊂ X be such that U ∩ Y is clopen in Y for each compact Y ⊂ X; we prove that U is clopen. For each compact Y ⊂ X the function ξU |Y is continuous, so by 2.5.23 there is an fY ∈ BC(X) with fY ∞ ≤ 1 such that fY |Y = ξU |Y . Now {fY : Y compact, Y ⊂ X} forms in a natural way a bounded Cauchy net in τc and by assumption it converges to some f ∈ C(X). But limY fY = ξU pointwise, so ξU = f is continuous, i.e., U is clopen. To prove the converse, let (fi )i be a Cauchy net in (C(X), τc ). Then there is an f : X−→K such that limi fi = f uniformly on compact subsets. Apparently, f |Y is continuous for each compact Y . By assumption f is continuous. Hence, (C(X), τc ) is complete. We do not know of a characterization of sequential completeness of (C(X), τc ). However, we have the following. Example 3.7.10 There is an X such that both (C(X), τs ) and (C(X), τc ) are sequentially complete but not quasicomplete. Proof. Take the space X of 2.5.33. If U1 , U2 , . . . are open in X then so is n∈N Un . (If 0 ∈ Un for some n then 0 ∈ n∈N Un , so n∈N Un is open. If 0 ∈ Un for all n then X \ Un is countable for all n, hence so is the set X \ ( n∈N Un ) = n∈N (X \ Un ), and then n∈N Un is open.) Thus, from 3.7.2(iv), we obtain that (C(X), τs ) is sequentially complete. Nevertheless, as X is not discrete, (C(X), τs ) is not quasicomplete by 3.7.2(iii). As for the compact open topology, since the compact subsets of X are finite, we have τs = τc , and the result follows. We now will define a family of topologies having τs and τc as particular cases. Definition 3.7.11 For f, φ : X−→K we set, admitting the value ∞, pφ (f ) := sup{|f (x) φ(x)| : x ∈ X}. Let W be a non-empty family of (so-called) weight functions X−→K. Let BW (X) := {f : X−→K : pφ (f ) < ∞ for each φ ∈ W }, and BCW (X) := BW (X) ∩ C(X). It is easy to see that BW (X) is a linear space under pointwise operations and that pφ is a (solid!) seminorm on BW (X) for each φ ∈ W . Definition 3.7.12 The W -topology τW on BW (X) (BCW (X)) is the locally convex topology generated by the seminorms {pφ : φ ∈ W }.

3.7 Examples of locally convex spaces

119

Let W be as above. Set W := {φ : X−→K : there are φ1 , . . . , φn ∈ W and C > 0 such that |φ| ≤ C max(|φ1 | , . . . , |φn |)}. Then BW (X) = BW (X) and τW = τW . It is also clear that τW is Hausdorff if and only if for each x ∈ X there is a φ ∈ W such that φ(x) = 0. From now on we assume that τW is Hausdorff. Lemma 3.7.13 (BW (X), τW ) is complete. Proof. Let (fi )i be a Cauchy net in (BW (X), τW ). Let x ∈ X, choose ϕ ∈ W with ϕ(x) = 0. Then from |g(x)| ≤ |ϕ(x)|−1 pϕ (g)

(g ∈ BW (X))

it follows that f (x) := limi fi (x) exists. Let φ ∈ W . Then there is an i0 and M > 0 such that pφ (fi ) ≤ M for i ≥ i0 . From |fi (x)| |φ(x)| ≤ M

(x ∈ X, i ≥ i0 )

it then follows that |f (x)| |φ(x)| ≤ M

(x ∈ X)

or pφ (f ) < ∞, i.e., f ∈ BW (X). Now the topology of pointwise convergence on BW (X) is closely related to τW , so by 3.4.17 we have limi fi = f in the topology τW . It is easy to see that (BCW (X), τW ) = (C(X), τs ) if W is the collection of all φ : X−→K with finite support. If we choose W := {φ : X−→K : φ bounded, φ has compact support} then (BCW (X), τW ) = (C(X), τc ). Below we discuss two other choices for W . Definition 3.7.14 (The strict topology) A function φ : X−→K is said to vanish at infinity if for each ε > 0 the set {x ∈ X : |φ(x)| ≥ ε} is contained in a compact set (compare with 2.5.20). Let W0 be the collection of all bounded functions φ : X−→K that vanish at infinity. The topology τW0 on BW0 (X) is called the strict topology, usually denoted by β0 . Clearly each bounded function X−→K is in BW0 (X). Also the converse holds. Lemma 3.7.15 Each f ∈ BW0 (X) is bounded. Proof. Suppose f ∈ BW0 (X) is unbounded; we derive a contradiction. There are x1 , x2 , . . . ∈ X such that |f (x1 )| < |f (x2 )| < · · · , limn |f (xn )| = ∞. There

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Locally convex spaces

√ are λ1 , λ2 , . . . ∈ K such that n → |λn | |f (xn )| is bounded and bounded away from zero. Define φ : X−→K by λn if n ∈ N, x = xn φ(x) := 0 if x ∈ X \ {x1 , x2 , . . .}. Then, since limn λn = 0 we have φ ∈ W0 , so f φ is bounded, which is in conflict with lim |f (xn ) φ(xn )| = lim(|λn | |f (xn )|) |f (xn )| = ∞. n

n

Corollary 3.7.16 (BCW0 (X), τW0 ) = (BC(X), β0 ). We now study the strict topology β0 in the spirit of 3.7.2 and 3.7.9. Theorem 3.7.17 The following are equivalent: (α) (BC(X), β0 ) is normable. (β) (BC(X), β0 ) is metrizable. (γ ) X is compact. Proof. We only need to prove (β) =⇒ (γ ). By 3.5.2 there are φ1 , φ2 , . . . ∈ W0 such that pφ1 , pφ2 , . . . generate β0 . Without loss, assume |φ1 | ≤ |φ2 | ≤ · · · . Suppose X is not compact; we derive a contradiction. Choose λ1 , λ2 , . . . ∈ K such that 1 > |λ1 | > |λ2 | > · · · , limn λn = 0. For n ∈ N set Yn := {x ∈ X : |φn (x)| ≥ |λn |2 }. Then each Yn is compact, Y1 ⊂ Y2 ⊂ · · · . Since X \ Y1 , X \ Y2 , . . . are infinite we can find x1 , x2 , . . . ∈ X such that x1 ∈ Y1 , xn+1 ∈ Yn+1 ∪ {x1 , . . . , xn } for each n. Choose clopen sets Un for which xn ∈ Un , Un ∩ Yn = ∅ for all n. Set λn if n ∈ N, x = xn φ(x) := 0 if x ∈ X \ {x1 , x2 , . . .}. Then φ ∈ W0 , so there is a constant C > 0 and a k ∈ N for which pφ ≤ C pφk .

(3.5)

For all n we have pφ (ξUn ) = sup{|φ(x)| : x ∈ Un } ≥ |φ(xn )| = |λn |, whereas, for n ≥ k, pφk (ξUn ) = sup{|φk (x)| : x ∈ Un } ≤ sup{|φk (x)| : x ∈ Yn } ≤ sup{|φn (x)| : x ∈ Yn } ≤ |λn |2 . So (3.5) entails that |λn | ≤ C |λn |2 for n ≥ k, an impossibility since λn → 0. Theorem 3.7.18 (BC(X), β0 ) is (quasi)complete if and only if X is a k0 -space.

3.7 Examples of locally convex spaces

121

Proof. From quasicompleteness of (BC(X), β0 ) we can deduce that X is a k0 -space with essentially the same proof as in 3.7.9(iii). Conversely, let X be a k0 -space. By 3.7.13 completeness of (BC(X), β0 ) follows as soon as we prove that BC(X) is a closed subspace of (BW0 (X), τW0 ). For that, let (fi )i be a net in BC(X) converging to some f in (BW0 (X), τW0 ). Then f is bounded by 3.7.15. On the other hand, the compact open topology on BW0 (X) is weaker than β0 , so limi fi = f with respect to τc and f ∈ C(X) by 3.7.9(iii). We see that f ∈ BC(X), and we are done. We now introduce a second choice for W . Definition 3.7.19 Let Wc be the set of all functions X−→K that are bounded on compact sets. Further, denote by Cc (X) the space of all continuous functions X−→K with compact support. Clearly Cc (X) ⊂ BCWc (X). For metrizable spaces we prove the converse. Theorem 3.7.20 Let X be metrizable. Then each f ∈ BWc (X) is bounded and has compact support. In particular, Cc (X) = BCWc (X). Proof. Since φ := 1 belongs to Wc we have that f = f φ is bounded. Now let Z := {x ∈ X : f (x) = 0}. Suppose Z is not compact; we derive a contradiction. There is a sequence x1 , x2 , . . . in Z without convergent subsequences. Letting d be a metric inducing the topology on X we can choose y1 , y2 , . . . ∈ Z such that d(xn , yn ) < n1 for each n. Then y1 , y2 , . . . does not have convergent subsequences. Without loss, assume yn = ym whenever n = m. Choose λ1 , λ2 , . . . ∈ K such that |λn | → ∞ and define φ : X−→K by λn f (yn )−1 if n ∈ N, x = yn φ(x) := 0 if x ∈ X \ {y1 , y2 , . . .}. Therefore f φ is unbounded, so to arrive at a contradiction it suffices to show that φ ∈ Wc . Let Y ⊂ X be compact. If Y ∩ {y1 , y2 , . . .} were infinite then y1 , y2 , . . . would have a convergent subsequence, so Y ∩ {y1 , y2 , . . .} is finite, showing that φ takes on Y only finitely many non-zero values and it follows that φ is bounded on Y . Apart from τWc , the space Cc (X) has a natural topology of its own. Definition 3.7.21 For each compact Y ⊂ X, let GY := {f ∈ C(X) : supp f ⊂ Y }. The inductive topology τi on Cc (X) is the strongest locally convex topology for which all the inclusions (GY , . ∞ )−→Cc (X) (Y ⊂ X, Y compact) are continuous.

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Thus, a seminorm p on Cc (X) is τi -continuous if and only if for each compact Y ⊂ X there is a CY ≥ 0 such that p(f ) ≤ CY f ∞ for all f ∈ Cc (X) with support in Y . Clearly every τWc -continuous seminorm is τi -continuous. For locally compact X we now prove the converse. Theorem 3.7.22 Let X be locally compact. Then on Cc (X) the topologies τi and τWc coincide. Proof. Let p be a τi -continuous seminorm; we shall prove that there is a φ ∈ Wc such that p(f ) ≤ pφ (f ) for all f ∈ Cc (X). To this end, for each open compact U ⊂ X, we set CU := min{C ≥ 0 : p(f ) ≤ C f ∞ for all f ∈ Cc (X) with supp f ⊂ U }. Trivially, if V ⊂ U are open compact, CV ≤ CU , and we can define ϕ : X−→[0, ∞) by ϕ(x) := inf{CU : U open compact, x ∈ U }. For each open compact U one verifies ϕ(x) ≤ CU for all x ∈ U , so that ϕ is bounded on (open) compact sets. Next we prove that for f ∈ Cc (X), p(f ) ≤ sup{|f (x)| ϕ(x) : x ∈ X}.

(3.6)

There is an open compact U ⊂ X such that supp f ⊂ U . Let ε > 0. Each x ∈ U has an open compact neighbourhood Vx ⊂ U such that ϕ(x) ≥ CVx − ε, |f (y) − f (x)| < ε for y ∈ Vx .

(3.7)

By compactness there are x1 , . . . , xn ∈ U such that Vx1 , . . . , Vxn cover U . Setting S1 := Vx1 , Si := Vxi \ Vxj (i ∈ {2, . . . , n}) j
we obtain an open compact partition S1 , . . . , Sn of U . For each i ∈ {1, . . . , n} we obtain, using the fact that supp f ξSi ⊂ Si ⊂ Vxi and (3.7), p(f ξSi ) ≤ CVxi f ξSi ∞ ≤ CVxi f ξVxi ∞ ≤ (ϕ(xi ) + ε) (|f (xi )| + ε), so that for f = ni=1 f ξSi we obtain p(f ) ≤ max p(f ξSi ) ≤ max (ϕ(xi ) + ε) (|f (xi )| + ε) 1≤i≤n

1≤i≤n

≤ sup{|f (x)| ϕ(x) : x ∈ X} + ε f ∞ + ε sup{ϕ(x) : x ∈ U } + ε2

3.7 Examples of locally convex spaces

123

and, since this holds for each ε > 0, (3.6) follows. Finally, let ρ ∈ K, 0 < |ρ| < 1. For each r ∈ [0, ∞) choose a λ(r) ∈ K such that r < |λ(r)| ≤ r |ρ|−1 . Then φ := λ ◦ ϕ is a map X−→K, bounded on compact sets and p(f ) ≤ sup{|f (x)| |φ(x)| : x ∈ X} (f ∈ Cc (X)), which completes the proof. Theorem 3.7.23 Let X be locally compact and metrizable. Then (Cc (X), τi ) = (BCWc (X), τWc ) is complete, and the following are equivalent: (α) (Cc (X), τi ) is normable. (β) (Cc (X), τi ) is metrizable. (γ ) X is compact. Proof. The equality (Cc (X), τi ) = (BCWc (X), τWc ) follows from 3.7.20 and 3.7.22. Now we prove completeness. By 3.7.13 we have that (BWc (X), τWc ) is complete. If (fi )i is a net in BCWc (X) converging to some f ∈ BWc (X) then since φ := 1 is a member of Wc , fi → f uniformly, so that f is continuous. It follows that BCWc (X), being a closed subspace of BWc (X), is complete. To prove the equivalences it suffices to consider only (β) =⇒ (γ ). Suppose (Cc (X), τi ) = (Cc (X), τWc ) is metrizable. By 3.5.2 there exist φ1 , φ2 , . . . ∈ Wc such that the seminorms pφ1 , pφ2 , . . . induce τWc on Cc (X). Without loss, assume |φ1 | ≤ |φ2 | ≤ · · · . Now suppose X is not compact; we derive a contradiction. By metrizability of X there exists a sequence x1 , x2 , . . . in X, with xn = xm whenever n = m, without convergent subsequences. Hence, by local compactness we can construct open compact sets U1 , U2 , . . . mutually disjoint, such that xn ∈ Un for every n. Set Mn := sup{|φn (x)| : x ∈ Un }, and choose λn ∈ K with |λn | > n Mn for each n. Define φ : X−→K by φ(x) :=

λn 0

if n ∈ N, x = xn if x ∈ X \ {x1 , x2 , . . .}.

Then φ ∈ Wc (with the same trick as in 3.7.20), so there are a C > 0 and a k ∈ N such that pφ ≤ C pφk on Cc (X). Applied to ξUn for n ≥ k we find n Mn < |λn | = |φ(xn )| ≤ pφ (ξUn ) ≤ C pφk (ξUn ) = C sup{|φk (x)| : x ∈ Un } ≤ C sup{|φn (x)| : x ∈ Un } = C Mn . It follows that n < C for n ≥ k, an absurdity.

Remark 3.7.24 We do not know of interesting results in the spirit of 3.7.20– 3.7.23 for more general X.

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3.7.2 Spaces of analytic functions Throughout Subsection 3.7.2 we assume that K is algebraically closed, so the valuation of K is dense. Recall (2.5.56) that for every disk of the type B(0, r − ), where r ∈ (0, ∞), there are unbounded functions in P SB(0, r − ). But P SB(0, r − ) has a natural locally convex topology as follows. Definition 3.7.25 For r ∈ (0, ∞], let A(r) be the space P SB(0, r − ) of all functions f : B(0, r − )−→K that can be expanded into a power series, equipped with the locally convex topology induced by the norms f −→ f s := max{|f (x)| : |x| ≤ s}, where 0 < s < r. By viewing for 0 < s1 < s2 < r the restriction map BP S(B(0, s2 ))−→BP S(B(0, s1 )) (which is a continuous linear injection) as an inclusion we can write BP S(B(0, s)). A(r) = 0<s
The map f −→ (λ0 , λ1 , . . .)

(f (x) =

∞

λn x n , |x| < r)

n=0

sends A(r) isomorphically onto the space {(λ0 , λ1 , . . .) ∈ K N∪{0} : limn |λn | s n = 0 for each 0 < s < r} = {(λ0 , λ1 , . . .) ∈ K N∪{0} : n → |λn | s n is bounded for each 0 < s < r} equipped with the locally convex topology induced by the norms given by (λ0 , λ1 , . . .)s := max |λn | s n , n∈N∪{0}

where 0 < s < r. (Notice that these norms . s are solid, see 2.5.53 and 2.5.58.) Observe that A(∞) is the space of entire functions. By choosing 0 < s1 < s2 < . . . , limn sn = r we have the projective sequence · · · · −→BP S(B(0, s2 ))−→BP S(B(0, s1 ))

3.7 Examples of locally convex spaces

125

and it is easily verified that A(r) = lim BP S(B(0, sn )), so we arrive at the ←− following theorem. Theorem 3.7.26 For each r ∈ (0, ∞] the space A(r) is complete, metrizable, not normable. Proof. By 3.5.7 A(r) is Fr´echet. If it were normable there would be an s ∈ (0, r) such that . s is equivalent to . t for each t ∈ [s, r). But it is easy to see that this is not the case. By a “dual” approach we arrive at the following. Definition 3.7.27 For r ∈ [0, ∞), let A† (r) := s>r BP S(B(0, s)), equipped with the strongest locally convex topology making all inclusions BP S(B(0, s))−→A† (r) continuous. By choosing ∞ > s1 > s2 > · · · , limn sn = r we therefore have (see the comments after 3.4.34) A† (r) = lim BP S(B(0, sn )). −→

Similarly as for A(r) there is an interpretation of A† (r) as the space {(λ0 , λ1 , . . .) ∈ K N∪{0} : n → |λn | s n is bounded for some s > r} = {(λ0 , λ1 , . . .) ∈ K N∪{0} : limn |λn | s n = 0 for some s > r}. If r > 0, A† (r) can be viewed as the space of all f ∈ BP S(B(0, r)) whose power series converges on some disk with radius > r, and therefore A† (r) is sometimes called the space of overconvergent power series on B(0, r). If r = 0 then A† (r) can be considered as the collection of all functions that can be expanded into a power series on some disk about 0, modulo the equivalence relation ≡ given by f ≡ g if f, g coincide on some disk about 0. A† (0) is sometimes called the space of germs of analytic functions at 0. Remark 3.7.28 Both A(r) and A† (r) are examples of a so-called sequence space, i.e., a locally convex space that is algebraically isomorphic to a subspace of K N . We will return to A(r) and A† (r) within the framework of systematic study of sequence spaces in Chapter 9. In particular we will see that A† (r) is complete (9.7.5(ii)) but non-metrizable (9.7.7(a)). A further study of A† (r) will appear in Chapter 11. A third way of defining locally convex spaces related to analytic functions is the following. Recall (2.5.64) that for closed, bounded A ⊂ K the space H (A) of analytic elements is a Banach space, but that in general the uniform topology on H (A) is awkward, even may not be a vector topology (2.5.65). This leads

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to the following more natural approach. Recall (2.5.62) that R(A) is the space of rational functions A−→K (having no poles in A). Definition 3.7.29 Let A ⊂ K. We say that f : A−→K is a rigid analytic function if there exists a net (fi )i in R(A) such that limi fi = f uniformly on all complete bounded subsets of A. With the topology induced by the seminorms f −→ f B := sup{|f (x)| : x ∈ B}, where B runs through the collection of all complete bounded subsets of A, the space O(A) of all rigid analytic functions on A is a locally convex space. A standard argument shows that O(A) is complete and that O(A) = H (A) as soon as A is closed and bounded. Notice that O(A) = A(r) in case A = B(0, r − ), where 0 < r < ∞, but O(K) = A(∞). Turning now to metrizability we introduce the following natural concept. Definition 3.7.30 Let A ⊂ K. A sequence Y1 ⊂ Y2 ⊂ · · · of subsets of A is called a fundamental sequence of complete bounded subsets of A if each Yn is complete and bounded and if for each complete bounded Y ⊂ A we have Y ⊂ Yn for some n. Theorem 3.7.31 If A ⊂ K has a fundamental sequence of complete bounded subsets then O(A) is a Fr´echet space. Proof. It is almost obvious that the norms . Yn , where Y1 , Y2 , . . . is a fundamental sequence of complete bounded sets, generate the topology. Then apply 3.5.2. Clearly closed subsets of K have a fundamental sequence of complete bounded sets. We even have the following. Theorem 3.7.32 Let A := B \ X, where B, X ⊂ K are closed, and such that bounded closed subsets of X are compact. Then A has a fundamental sequence of complete bounded sets. Proof. We may assume X = ∅. Set Yn := {x ∈ B : |x| ≤ n, dist (x, X) ≥ n1 }. Then each Yn is a bounded, complete subset of A; let Y ⊂ A be complete and bounded. If Y ⊂ Yn for no n then, by boundedness of Y , for large n there is yn ∈ Y with dist (yn , X) < n1 , so there are xn ∈ X such that limn (yn − xn ) = 0. By boundedness of Y the sequence x1 , x2 , . . . is bounded and by assumption {x1 , x2 , . . .} is compact, so there is a subsequence xn1 , xn2 , . . . converging to x ∈ X. But then yn1 , yn2 , . . . also converges to x and so x ∈ Y by completeness. Hence X ∩ Y = ∅, a contradiction.

3.7 Examples of locally convex spaces

127

Example 3.7.33 The Fr´echet space O(Cp \ Qp ) (where K = Cp ) is called the space of rigid analytic functions on the “upper half plane” Cp \ Qp of Cp , see [210], I.5.D.

3.7.3 Spaces of differentiable functions In the remaining part of Section 3.7 X is a non-empty subset of K without isolated points. We will use the terminology of Subsection 2.5.7. For a non-empty subset Y of X, n ∈ N ∪ {0}, f ∈ C n (X) we set, admitting the value ∞, pYn (f ) := max i f Y , 0≤i≤n

where

i f Y := sup{i f (t) : t ∈ Y i+1 }.

If also g ∈ C n (X) then pYn (f + g) ≤ max(pYn (f ), pYn (g)) and pYn (f g) ≤ pYn (f ) pYn (g) (where 0.∞ = 0). This last formula is a direct consequence of rule V of 2.5.78. We need the following lemma. Lemma 3.7.34 Let Y1 ⊂ Y2 be non-empty subsets of X. Then for each f ∈ C n (X), n ∈ {0, 1, . . .}, pYn1 (f ) ≤ pYn2 (f ). If, in addition, Y2 \ Y1 has no isolated points and d := dist (Y1 , Y2 \ Y1 ) > 0 we have for each f ∈ C n (X) that is 0 on Y2 \ Y1 and n ∈ {0, 1, . . .}, pYn2 (f ) ≤ d n pYn1 (f ), where d := max(1, d1 ). Proof. Only the second statement needs a proof, which we shall carry out by induction on n. The case n = 0 is obvious. For the induction step n − 1 → n first observe that pYn2 (f ) = max(n f Y2 , pYn−1 (f )), and by the 2 n−1 n−1 n n (f ) ≤ d p (f ) ≤ d p (f ). It therefore sufinduction hypothesis, pYn−1 Y1 Y1 2 fices to prove that for (x1 , . . . , xn+1 ) ∈ Y2n+1 n f (x1 , . . . , xn+1 ) ≤ d n pn (f ). (3.8) Y1 Now (3.8) is true if all x1 , . . . , xn+1 are in Y1 , and also if all x1 , . . . , xn+1 are in Y2 \ Y1 (n f (x1 , . . . , xn+1 ) = 0 if (x1 , . . . , xn+1 ) ∈ n+1 (Y2 \ Y1 ), so by

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density of n+1 (Y2 \ Y1 ) in (Y2 \ Y1 )n+1 and continuity we have n f = 0 on (Y2 \ Y1 )n+1 ). For the remaining case we may, by symmetry of n f (2.5.78,II), assume that x1 ∈ Y2 \ Y1 , x2 ∈ Y1 . Then |x1 − x2 | ≥ d , so |x1 − x2 |−1 ≤ d1 ≤ d and n f (x1 , . . . , xn+1 ) f (x , x , . . . , x ) − f (x , x , . . . , x ) 1 3 n+1 n−1 2 3 n+1 n−1 = x1 − x2 (f ) ≤ d d n−1 pYn−1 (f ) ≤ d n pYn1 (f ), ≤ d n−1 f Y2 ≤ d pYn−1 2 1 and the proof is complete.

In a special case we have a more precise estimate. Lemma 3.7.35 Let B ⊂ X and suppose 0 < dist (B, X \ B) < 1. Then, for each n ∈ {0, 1, . . .}, n (ξB ) = n (ξB )X = dist (B, X \ B)−n . pX

Proof. It suffices to prove the second equality. B and X \ B are clopen, so they have no isolated points. From the previous lemma we obtain n (ξB )X ≤ n (ξB ) ≤ dist (B, X \ B)−n pBn (ξB ) = dist (B, X \ B)−n . Conversely, since B pX is clopen in X we have ξB ∈ C ∞ (X) and by using the Taylor formula 2.5.80 for x ∈ B, y ∈ X \ B we arrive at n ξB (x, x, . . . , x, y) = |ξB (x) − ξB (y)| = |x − y|−n , |x − y|n showing that n (ξB )X ≥ sup{|x − y|−n : x ∈ B, y ∈ X \ B} = dist (B, X \ B)−n . For f ∈ C n (X) and non-empty compact Y the continuous functions f, 1 f, . . . , n f are bounded on Y, Y 2 , . . . , Y n+1 respectively, so pYn (f ) < ∞. This leads to a Hausdorff locally convex topology on C n (X), which for n = 0 coincides with the topology τc on C(X) = C 0 (X) (3.7.3). Definition 3.7.36 Let n ∈ {0, 1, . . .}. The locally convex topology on C n (X) generated by the seminorms {pYn : ∅ = Y ⊂ X, Y compact} is denoted τcn . It is called the topology of uniform convergence on compact sets of difference quotients of order 0, 1, . . . , n. On C ∞ (X) we define the topology τc∞ of uniform convergence on compact sets of all difference quotients to be the union of all τcn , n ∈ {0, 1, . . .}.

3.7 Examples of locally convex spaces

129

Indeed, it is almost obvious to verify that a net (fi )i in C n (X) (C ∞ (X)) converges to f ∈ C n (X) (C ∞ (X)) with respect to τcn (τc∞ ) if and only if k fi converges, uniformly on compact subsets of X k+1 , to k f for each k ∈ {0, 1, . . . , n} (k ∈ {0, 1, . . .}). If X is compact, n ∈ {0, 1, . . .} the topology τcn on C n (X) = BC n (X) can n be described by the single norm pX (which is the norm . n of 2.5.82); the n norms {pX : n ∈ {0, 1, . . .}} induce the metrizable topology τc∞ . In general we have the following results. Compare with 3.7.9. Theorem 3.7.37 The spaces (C n (X), τcn ) (n ∈ {0, 1, . . .}) and (C ∞ (X), τc∞ ) are complete. Proof. Let n ∈ {0, 1, . . .} and let (fi )i be a Cauchy net in (C n (X), τcn ). For each k ∈ {1, . . . , n + 1} we define a function hk : Xk −→K by hk := limi k−1 fi , uniformly on compact subsets of X k . By metrizability and 3.7.6, 3.7.8 the functions hk are continuous. By taking pointwise limits we have for (x1 , . . . , xn+1 ) ∈ n+1 X: hn+1 (x1 , . . . , xn+1 ) = n h1 (x1 , . . . , xn+1 ). So, by definition of a C n -function, h1 ∈ C n (X). Hence, (fi )i converges to h1 in τcn . The statement for (C ∞ (X), τc∞ ) is now trivial. Theorem 3.7.38 Let n ∈ {0, 1, . . .}. Then (C n (X), τcn ) is normable if and only if X is compact. Proof. We prove that normability of (C n (X), τcn ) implies compactness of X. There exists a non-empty compact Y ⊂ X such that pYn is a norm generating the topology. If Y = X, let U be a clopen subset of X such that Y ⊂ U X. Then 0 = ξX\U ∈ C ∞ (X) but pYn (ξX\U ) = 0, a contradiction. Theorem 3.7.39 (C ∞ (X), τc∞ ) is not normable. Proof. Suppose (C ∞ (X), τc∞ ) is normable; we derive a contradiction. There are an n ∈ N and a non-empty compact Y ⊂ X such that τc∞ is generated by the norm pYn . As in the previous proof we may conclude that Y = X is compact and therefore we have n (f ) n+1 f X ≤ C pX

(f ∈ C ∞ (X))

(3.9)

for some C > 0. Let B be a ball in X such that 0 < dist (B, X \ B) < 1. Then from 3.7.35 and (3.9) for f = ξB we deduce dist (B, X \ B)−n−1 ≤ C dist (B, X \ B)−n ,

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Locally convex spaces

implying dist (B, X \ B) ≥ C −1 . But X has no isolated points, so we can choose B such that dist (B, X \ B) is small. Contradiction. Before characterizing metrizability of (C n (X), τcn ) we prove first a topological fact. Lemma 3.7.40 An ultrametric space is hemicompact if and only if it is locally compact and separable. Proof. The “if” part is covered by 2.5.35(γ ) =⇒ (α) and 3.7.5(d). To prove the converse, let Z be an ultrametric hemicompact space, let Y1 ⊂ Y2 ⊂ · · · be a fundamental sequence of compact subsets of Z. Suppose there is an x ∈ Z without compact neighbourhood; we derive a contradiction. In fact, for n ∈ N, B(x, n1 ) ⊂ Yn (otherwise B(x, n1 ) would be compact), hence there exists an xn ∈ B(x, n1 ) \ Yn . Then xn → x, so {x, x1 , x2 , . . .} is compact but is contained in no Yn , contradiction. Thus, Z is locally compact. Separability now follows from 2.5.35(α) =⇒ (γ ). Theorem 3.7.41 Let n ∈ {0, 1, . . .}. Then the following are equivalent: (α) (β) (γ ) (δ)

(C n (X), τcn ) is metrizable. (C ∞ (X), τc∞ ) is metrizable. X is hemicompact. X is locally compact and separable.

Proof. By 3.7.40 we only have to concentrate on (α), (β) and (γ ). (γ ) =⇒ (α). The spaces X, X2 , . . . , Xn+1 are hemicompact, so by 3.7.9(ii) the spaces (C(X k ), τc ), k ∈ {1, . . . , n + 1}, are metrizable and so is E := k 1≤k≤n+1 (C(X ), τc ) (3.5.5). Now the map f → (f, 1 f, . . . , n f ) sends n n (C (X), τc ) linearly and homeomorphically into E and metrizability follows. (γ ) =⇒ (β). By the above, (γ ) implies (α) for each n ∈ {0, 1, . . .}. Now apply 3.5.2 to arrive at (β). To prove (α) =⇒ (γ ) and (β) =⇒ (γ ), let F be either (C n (X), τcn ) or ∞ (C (X), τc∞ ) and suppose that F is metrizable. By 3.5.2 there exist compact Y1 , Y2 , . . . ⊂ X and m1 , m2 , . . . ∈ {0, 1, . . .} such that {pYmkk : k ∈ N} induce the topology. Without loss, assume Y1 ⊂ Y2 ⊂ · · · and m1 ≤ m2 ≤ · · · . Now let Y ⊂ X be compact. Then pY0 ≤ C pYmkk for some C > 0 and k ∈ N. If Y ⊂ Yk , let x ∈ Y , x ∈ Yk and let B a ball in X about x not meeting Yk . Then ξB ∈ F and 1 = |ξB (x)| ≤ pY0 (ξB ) ≤ C pYmkk (ξB ) = 0, a contradiction.

3.7 Examples of locally convex spaces

131

In the spirit of 3.7.1 one can also think of introducing a C n -version of pointwise convergence as follows. Definition 3.7.42 Let n ∈ {0, 1, . . .}. The locally convex topology on C n (X) defined by the seminorms pYn , where Y runs through the collection of all nonempty finite subsets of X, is called τsn . On C ∞ (X) we define the topology τs∞ as the union of all τsn |C ∞ (X). Clearly τs0 coincides with τs introduced in 3.7.1. There is an easier way to describe τsn in terms of Hasse derivatives (see 2.5.79). Theorem 3.7.43 Let n ∈ {0, 1, . . .}. Then on C n (X) the seminorms f → |Dk f (x)| (k ∈ {0, 1, . . . , n}, x ∈ X) generate the topology τsn . Proof. Let Y ⊂ X be finite, Y = ∅. It suffices to prove that there is a constant C > 0 such that pYn (f ) ≤ C max{|Dk f (y)| : 0 ≤ k ≤ n, y ∈ Y }

(3.10)

for all f ∈ C n (X). This is obvious if Y is a singleton set (we can choose C := 1), so assume Y ≥ 2. We now proceed by induction on n. The case n = 0 is trivial. For the step n − 1 → n we use an idea, similar to the one of the proof of 3.7.34. In fact, let d := min{y − y : y, y ∈ Y, y = y } and put d := max(1, d1 ). We will prove (3.10) with C := d n . By the induction hypothesis, it suffices to show that n f Y ≤ d n max{|Dk f (y)| : 0 ≤ k ≤ n, y ∈ Y }.

(3.11)

Let x1 , . . . , xn+1 ∈ Y . If they are not all equal we may suppose by symmetry that x1 = x2 . Then n f (x1 , . . . , xn+1 ) f (x , x , . . . , x ) − f (x , x , . . . , x ) n−1 1 3 n+1 n−1 2 3 n+1 = ≤ d n−1 f Y , x1 − x2 which is, again by the induction hypothesis, ≤ d d n−1 max{|Dk f (y)| : 0 ≤ k ≤ n − 1, y ∈ Y } ≤ d n max{|Dk f (y)| : 0 ≤ k ≤ n, y ∈ Y }. If all x1 , . . . , xn+1 are equal we have n f (x1 , . . . , xn+1 ) = |Dn f (x1 )| , which is obviously smaller than or equal to the right-hand side of (3.11).

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Remark 3.7.44 It follows from the above that a net (fi )i in C n (X) is Cauchy (resp. converges to f ∈ C n (X)) in τsn if and only if (Dk fi )i is pointwise Cauchy (resp. Dk fi → Dk f pointwise) for all k ∈ {0, 1, . . . , n}. The proof of the next characterization is in the same vein as the one of 3.7.41. Theorem 3.7.45 (i) The spaces (C n (X), τsn ) (n ∈ {0, 1, . . .}) and (C ∞ (X), τs∞ ) are neither normable nor (sequentially) complete. (ii) Let n ∈ {0, 1, . . .}. Then the following are equivalent: (α) (C n (X), τsn ) is metrizable. (β) (C ∞ (X), τs∞ ) is metrizable. (γ ) X is countable. Proof. (i). Suppose, for some n, (C n (X), τsn ) or (C ∞ (X), τs∞ ) is normable. Then there is a non-empty finite set Y ⊂ X and an m such that the norm pYm generates the topology. Since X has no isolated points we have Y= X, so there is a non-empty clopen set U ⊂ X that does not meet Y . Then ξU = 0 but pYm (ξU ) = 0, a contradiction. To prove non-sequential completeness, let x ∈ X. Consider the sequence i → ξB(x, 1 ) . For each y ∈ X and k ∈ {0, 1, . . .}, i lim Dk ξB(x, 1 ) (y) − Dk ξB(x, 1 ) (y) = 0, i

i+1

i

so that by 3.7.44 the sequence is Cauchy. But it converges pointwise to ξ{x} and this function is not even continuous, so the above sequence does not converge in τsn or τs∞ . (ii). If X is countable then the collections f −→ |Dk f (x)|

(k ∈ {0, 1, . . . , n}, x ∈ X)

f −→ |Dk f (x)|

(k ∈ {0, 1, . . .}, x ∈ X)

and

are countable, so by 3.5.2 and 3.7.43, (C n (X), τsn ) and (C ∞ (X), τs∞ ) are metrizable, which proves (γ ) =⇒ (α) and (γ ) =⇒ (β). To prove (α) =⇒ (γ ) and (β) =⇒ (γ ), let F be either (C n (X), τsn ) or ∞ (C (X), τs∞ ) and suppose that F is metrizable. By 3.5.2 there exist finite sets ∅ = Y1 ⊂ Y2 ⊂ . . . and m1 , m2 , . . . ∈ {0, 1, . . .} with m1 ≤ m2 ≤ . . . such 0 ≤ C pYmkk for that {pYmkk : k ∈ N} induce the topology. Now let x ∈ X. Then p{x} some C > 0 and k ∈ N. Suppose x ∈ Yk , let B be a ball in X about x not meeting Yk . Then ξB ∈ F and 0 1 = |ξB (x)| = p{x} (ξB ) ≤ C pYmkk (ξB ) = 0, a contradiction. Hence X = k∈N Yk is countable.

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133

In 2.5.82 and 2.5.84 we considered the Banach spaces (BC n (X), . n ). Now we look at the “locally convex intersection”. Definition 3.7.46 Let BC ∞ (X) be the space n∈N∪{0} BC n (X) equipped with the locally convex topology defined by the family of norms { . n : n ∈ {0, 1, . . .}}. Theorem 3.7.47 BC ∞ (X) is a Fr´echet space. Proof. Metrizability follows from 3.5.2. Also, from completeness of each BC n (X) it is easily seen that BC ∞ (X) is complete. Next we study weighted topologies on spaces of differentiable functions in the spirit of 3.7.12. To be able to define these topologies properly we will assume that our weight functions are in C ∞ (X). To obtain interesting results we will require that X is locally compact. Definition 3.7.48 Let X ⊂ K be locally compact. Let W be a family of functions in C ∞ (X), and suppose for each x ∈ X there is a locally constant φ ∈ W with φ(x) = 0.

(3.12)

n For n ∈ {0, 1, . . .}, let BCW (X) be the space of all f ∈ C n (X) for which n n (X) is the locally pX (f φ) < ∞ for each φ ∈ W . The W -topology τWn on BCW convex topology induced by the seminorms n f −→ f nφ := pX (f φ)

n (f ∈ BCW (X)),

∞ where φ runs through W . The W -topology τW∞ on BCW (X) := n n BC (X) is the union of all τ , n ∈ {0, 1, . . .}. n∈N∪{0} W W Obviously the W -topologies are Hausdorff.

From now until Section 3.8 we will assume that X is locally compact and that W ⊂ C ∞ (X) satisfies (3.12). The next lemma express the fact that τcn and τc∞ are – in some sense – minimal among the W -topologies. n Lemma 3.7.49 For each n ∈ {0, 1, . . .} τWn is stronger than τcn |BCW (X), and ∞ ∞ ∞ τW is stronger than τc |BCW (X). For

W := {f : X−→K : f is locally constant, f has compact support} n ∞ (X), τWn ) = (C n (X), τcn ) and (BCW (X), τW∞ ) = (C ∞ (X), τc∞ ). we have (BCW

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Proof. Let n ∈ {0, 1, . . .}. Consider := {U ⊂ X : U open compact, pUn is τWn -continuous}. Clearly clopen subsets of elements of are in . We next prove that covers X. Let x ∈ X. By assumption there is a locally constant φ ∈ W with φ(x) = 0. Let U be an open compact neighbourhood of x on which φ is constant. Then n (X) we have for f ∈ BCW n pX (f φ) ≥ pUn (f φ) = |φ(x)| pUn (f ),

showing that U ∈ . Now let Y ⊂ X be compact. To prove that pYn is τWn continuous, let U1 , . . . , Um be a finite covering of Y by members of ; we may assume the Ui to be disjoint. It suffices to show that pYn ≤ C max1≤i≤m pUn i for some C > 0. To this end, let B := U2 ∪ · · · ∪ Um . Then, using 3.7.34, for f ∈ C n (X) pYn (f ) ≤ pUn 1 ∪B (f ) = pUn 1 ∪B (f ξU1 + f ξB ) ≤ max(pUn 1 ∪B (f ξU1 ), pUn 1 ∪B (f ξB )) ≤ max(d n pUn 1 (f ), d n pBn (f )), where d := max(1, dist (B, U1 )−1 ). Now by using the decomposition B = U2 ∪ U3 ∪ · · · ∪ Um we can derive in the same way that ˜ n max(p n (f ), pn ˜ pBn (f ) ≤ (d) U2 U3 ∪...∪Um (f )), where d := max(1, dist (U2 , U3 ∪ −1 · · · ∪ Um ) ), and so on. After finitely many steps we arrive at the desired inequality pYn (f ) ≤ C max pUn i (f ) 1≤i≤m

is τWn -continuous. That τc∞ ≤ τW∞ is now for some C > 0, showing that obvious. Now choose W as indicated. We only need to prove that τWn ≤ τcn for n ∈ {0, 1, . . .}. So let φ ∈ W ; we prove that . nφ is τcn -continuous. Write φ = m i=1 λi ξUi , where λ1 , . . . , λm ∈ K and U1 , . . . , Um are open compact. For f ∈ C n (X) we have pYn

n n f nφ = pX (f φ) ≤ max |λi | pX (f ξUi ). 1≤i≤m

By 3.7.34 we find constants di (1 ≤ i ≤ m), depending only on φ, such that n pX (f ξUi ) ≤ din pUn i (f ξUi ) = din pUn i (f ),

so combined with the previous estimate one gets f nφ ≤ C max pUn i (f ), 1≤i≤m

where C :=

max1≤i≤m din

|λi |, which finishes the proof.

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135

Remarks 3.7.50 n (X) = C n (X). Obviously, we have τWn ≤ (a) If X is compact then clearly BCW τcn . The above lemma tells us that τWn ≥ τcn . Hence, for any choice of W we n n have (BCW (X), τWn ) = (C n (X), τcn ), where τcn is defined by the norm pX (called . n in 2.5.82). (b) It is not hard to derive from 3.7.49 that τsn and τs∞ (see 3.7.42) are not W -topologies. This also follows directly from the next result (recall that by 3.7.45 the spaces (C n (X), τsn ) and (C ∞ (X), τs∞ ) are not complete). n ∞ (X), τWn ) (n ∈ {0, 1, . . .}) and (BCW (X), Theorem 3.7.51 The spaces (BCW τW∞ ) are complete. n (X), τWn ) for n ∈ {0, 1, . . .}. Proof. It suffices to prove completeness of (BCW n (X), τWn ). Then by the previous lemma Let (fi )i be a Cauchy net in (BCW n n it is Cauchy in (C (X), τc ), so by 3.7.37 there is an f ∈ C n (X) such that n (fi φ) is eventually bounded, fi → f in τcn . Now let φ ∈ W . Then i → pX n n say pX (fi φ) ≤ M for i ≥ i0 . It follows easily that also pX (f φ) ≤ M. Hence n f ∈ BCW (X). The usual argument now shows that limi fi = f in τWn .

We do not have general results on metrizability of τWn and τW∞ . To end this section we discuss a few other choices for W . First of all the n (X), τWn ) equals choice W := {1} leads to spaces we already met. In fact, (BCW ∞ n the Banach space (BC (X), . n ) (see 2.5.82) and (BCW (X), τW∞ ) equals the Fr´echet space BC ∞ (X) (see 3.7.46). Now we move to the other extreme, i.e., W := C ∞ (X). For this we need a lemma on ultrametrics. Lemma 3.7.52 Let Z be an ultrametric space, let U ⊂ Z be open, U not compact. Then there exists a clopen partition U0 , U1 , . . . of Z such that ∅ = Un ⊂ U for all n ≥ 1. Proof. U is not compact, so as in the proof of 3.7.20 there is a sequence y1 , y2 , . . . in U without convergent subsequences. Without loss, assume yn = ym whenever n = m. Then, for each n ∈ N, there exists a ball Bn about yn with Bn ⊂ U and ym ∈ Bn for m = n. By the mercury drop behaviour the balls Bn are mutually disjoint. Next, in order to find the desired partition U0 , U1 , . . . of Z, we distinguish (the only possible) two cases for the diameters of the Bn . First assume that limn diam Bn = 0. Then take Un := Bn for n ≥ 1 and U0 := Z \ n∈N Un . We only have to show that n∈N Un is closed. For that, let x ∈ Z and let x1 , x2 , . . . ∈ n∈N Un be such that xi → x. For each i ∈ N there is a unique ni such that xi ∈ Uni . Letting d be the ultrametric of Z we

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have d(xi , yni ) ≤ diam Bni for each i ∈ N. If {n1 , n2 , . . .} is unbounded then by applying the above inequality to a subsequence of {n1 , n2 , . . .} tending to ∞ and using xi → x, we find a subsequence of yn1 , yn2 , . . . converging to x, which is a contradiction. Hence, {n1 , n2 , . . .} is bounded, so there is an m such that xi ∈ U1 ∪ · · · ∪ Um for all i ∈ N. Since U1 ∪ · · · ∪ Um is closed we obtain that x ∈ U1 ∪ · · · ∪ Um ⊂ n∈N Un , and we are done in this case. Now assume that there is an r > 0 such that the set {n ∈ N : diam Bn ≥ r} is infinite, that is, this set is of the form {m1 , m2 , . . .}, m1 < m2 < · · · . Then it is easily seen that Un := Bmn for n ≥ 1 and U0 := Z \ n∈N Un satisfy the required conditions. Example 3.7.53 Let W := C ∞ (X). Then we have the following: n (X) equals (i) Let n ∈ {0, 1, . . .}. Then BCW

Ccn (X) := {f ∈ C n (X) : f has compact support}. ∞ (X) equals Also, BCW

Cc∞ (X) := {f ∈ C ∞ (X) : f has compact support}. (ii) Let n ∈ {0, 1, . . .}. Then τWn is the strongest locally convex topology making the obvious embeddings (C n (U ), . n )−→Ccn (X) (U ⊂ X, U open n compact) continuous. In other words, a seminorm p on BCW (X) is τWn continuous if and only if for each open compact U ⊂ X there is a CU ≥ 0 such that p(f ) ≤ CU pUn (f ) for all f ∈ Ccn (X) whose support is in U . (iii) τW∞ is the union of τWn , with n ∈ {0, 1, . . .}. (iv) For each open compact U ⊂ X, the obvious embeddings (C n (U ), . n )−→(Ccn (X), τWn )

(n ∈ {0, 1, . . .})

and (C ∞ (U ), τc∞ )−→(Cc∞ (X), τW∞ ) are homeomorphic. n (X) for n ∈ {0, 1, . . .}. (Then Proof. (i). We prove the equality Ccn (X) = BCW ∞ n the equality for BCW (X) is obvious). If f ∈ Cc (X) and φ ∈ C ∞ (X) then f φ ∈ n (f φ) < ∞, C n (X) (text following 2.5.79) and f φ has compact support so pX n n hence f ∈ BCW (X). Conversely, let f ∈ BCW (X). Suppose supp f is not compact; we derive a contradiction. Let U := {x ∈ X : f (x) = 0}. By 3.7.52

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137

there exists a clopen partition U0 , U1 , . . . of X such that ∅ = Un ⊂ U for all n ≥ 1. Choose x1 ∈ U1 , x2 ∈ U2 , . . . and λ1 , λ2 , . . . ∈ K with limn |λn | = ∞. Define φ : X−→K by f (xn )−1 λn if n ∈ N, x ∈ Un φ(x) := 0 if x ∈ U0 . Then φ is locally constant, hence φ ∈ C ∞ (X). But n → |f (xn ) φ(xn )| = |λn | n is unbounded, so f φ∞ does not exist and neither does pX (f φ), a contradiction. (ii). First, let p be a τWn -continuous seminorm, let U ⊂ X be open compact. We prove the existence of a CU ≥ 0 such that p(f ) ≤ CU pUn (f ) for all f ∈ Ccn (X) for which supp f ⊂ U . To this end we may assume that p has the form n f → pX (f φ) for some φ ∈ C ∞ (X). For f ∈ Ccn (X), supp f ⊂ U we have n (f φ) ≤ d n pUn (f φ), where d depends only on U . By thanks to 3.7.34 that pX the product rule (see the beginning of 3.7.3) we have pUn (f φ) ≤ pUn (f ) pUn (φ), n (f φ) ≤ CU pUn (f ). so by taking CU := d n pUn (φ) we find pX n Conversely, let p be a seminorm on Cc (X) such that for every open compact set U ⊂ X there is a CU > 0 such that p(f ) ≤ CU pUn (f ) for f ∈ Ccn (X), n supp f ⊂ U . We will find a C ∞ -function φ on X such that p(f ) ≤ pX (f φ) n for all f ∈ Cc (X). Let {Ui : i ∈ I } be a partition of X into open compact sets (1.1.3). For each i ∈ I , choose αi ∈ K such that |αi | CU−1i ≥ 1 and set, for x ∈ X, φ(x) := αi , where i ∈ I , x ∈ Ui . Then φ is locally constant, so φ ∈ C ∞ (X). Now let f ∈ Ccn (X). Then there are i1 , . . . , im ∈ I such that supp f ⊂ Ui1 ∪ · · · ∪ Uim . For each k ∈ {1, . . . , m} we have n pX (f φ) ≥ pUn i (f φ) = αik pUn i (f ) = αik pUn i (f ξUik ) k k k ≥ αik CU−1i p(f ξUik ) ≥ p(f ξUik ), k

so that

p(f ) = p f

m k=1

ξUik

n ≤ max p(f ξUik ) ≤ pX (f φ), 1≤k≤m

and we are done. (iii). Follows directly from the above. (iv). For n ∈ {0, 1, . . .} the continuity of the embedding is proved in (ii). This yields the continuity of the embedding in the second case. One also verifies immediately that the restriction maps (Ccn (X), τWn )−→(C n (U ), . n ) and (Cc∞ (X), τW∞ )−→(C ∞ (U ), τc∞ ) are continuous, showing that the above embeddings are open.

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Remark 3.7.54 If X is compact then (Ccn (X), τWn ) = (C n (X), . n ) is a Banach space (2.5.84). Since by 3.4.31 (Cc∞ (X), τW∞ ) is isomorphic to the projective limit of the projective sequence ((C n (X), . n ))n∈N∪{0} , we conclude from 3.5.7 that (Cc∞ (X), τW∞ ) = (C ∞ (X), τc∞ ) is a Fr´echet space. Let X be locally compact. Then (Cc∞ (X), τW∞ ) is not normable. In fact, if it would be normable then, by 3.7.53(iv), for each open compact U ⊂ X, (C ∞ (U ), τc∞ ) would be also normable, a contradiction with 3.7.39. Now let X be locally compact and σ -compact. Then it has a fundamental sequence of compact open sets U1 ⊂ U2 ⊂ · · · . With the obvious embeddings C n (U1 ) ⊂ C n (U2 ) ⊂ · · · ⊂ Ccn (X) we have, with W as above, (Ccn (X), τWn ) = lim(C n (Uj ), . n ). −→

(In 11.2.9(i), we will see that if X is not compact this inductive limit is not metrizable.) By taking n = 0 and observing that (Cc0 (X), τW0 ) = (Cc (X), τi ), we have (Cc (X), τi ) = lim(C(Uj ), . ∞ ). −→

This description as an inductive limit justifies the name “inductive” for the topology τi introduced in 3.7.21. Example 3.7.55 Let W be the collection of all polynomial funcn tions X −→ K. Let n ∈ {0, 1, . . .}. Then BCW (X) = {f ∈ C n (X) : n m pX (f X ) < ∞ for all m ∈ {0, 1, . . .}} (recall that the symbol X stands for the function x → x). The topology τWn is induced by countably many semin ∞ (X) is a Fr´echet space. Similarly, (BCW (X), τW∞ ) is a Fr´echet norms, so BCW space. Remarks 3.7.56 Let W be as in 3.7.55. ∞ (a) It is not hard to see that for every x ∈ X there exists an f ∈ BCW (X) with f (x) = 0. In fact, for each ball B ⊂ X the characteristic function ∞ (X). To see this first notice that ξB Xm ∈ C ∞ (X) for all ξB is in BCW m ∈ {0, 1, . . .}. Now let m, n ∈ {0, 1, . . .}. By 3.7.34 there is a constant n (ξB Xm ) ≤ d n pBn (ξB Xm ) (this also holds if B = X). d ≥ 1 for which pX n m Since pB (ξB X ) = pBn (Xm ), applying rule X of 2.5.78 we obtain that n pX (ξB Xm ) < ∞. n (X), τWn ) is the Banach space BC n (X) (= (b) Clearly, for compact X, (BCW ∞ C n (X)) of 2.5.82 and (BCW (X), τW∞ ) is the Fr´echet space BC ∞ (X) (= ∞ C (X)) of 3.7.46. n (X). (c) See 3.7.62 for an alternative description of BCW

3.7 Examples of locally convex spaces

139

Finally we look at “differentiable functions vanishing at infinity”. Recall 2.5.20, where we introduced C0 (X). Definition 3.7.57 Let n ∈ {0, 1, . . .}. We say that f ∈ C n (X) vanishes at infinity if i f ∈ C0 (X i+1 ) for i ∈ {0, 1, . . . , n}. The space of all C n -functions on X that vanish at infinity is called C0n (X). C0∞ (X) := n∈N∪{0} C0n (X). n The spaces C0n (X) are Banach spaces with respect to the norm pX , and C0∞ (X) n is a Fr´echet space with respect to the locally convex topology induced by {pX : n n ∞ ∞ n ∈ {0, 1, . . .}}. If X is compact then C0 (X) = C (X), C0 (X) = C (X). We now show that C0n (X) (C0∞ (X)) is an ideal in the ring BC n (X) (BC ∞ (X)) considered in 2.5.82 (3.7.46).

Theorem 3.7.58 Let n ∈ {0, 1, . . .}, let f ∈ C0n (X), g ∈ BC n (X). Then f g ∈ C0n (X). Also, f ∈ C0∞ (X) and g ∈ BC ∞ (X) imply f g ∈ C0∞ (X). Proof. It suffices to prove the first assertion, which will be carried out by induction on n. The case n = 0 being obvious (2.5.21) we proceed by proving the step n − 1 → n. Clearly f g ∈ C n (X). By the induction hypothesis it remains to be shown that n (f g) ∈ C0 (X n+1 ). Let ε > 0. There is a compact Y ⊂ X such that, for all i ∈ {0, 1, . . . , n}, i f (y) < ε (gn + 1)−1 , for each y ∈ Xi+1 \ Y i+1 . We prove that n (f g)(z) < ε for all z ∈ X n+1 \ Y n+1 . By symmetry we may assume z = (x1 , . . . , xn+1 ) with x1 ∈ Y . Then, for each i ∈ {0, 1, . . . , n}, we have (x1 , . . . , xi+1 ) ∈ Y i+1 and by rule V of 2.5.78 we obtain n n (f g)(z) = i f (x1 , . . . , xi+1 ) n−i g(xi+1 , . . . , xn+1 ) i=0

≤ ε (gn + 1)−1 gn < ε. In 3.7.57 we imposed heavy conditions on the spaces, so it is worth asking whether they contain non-trivial functions. The answer is given in the next theorem (compare with 2.5.32). Theorem 3.7.59 The following are equivalent: (α) There is a function f : X−→K such that f Xm ∈ C0∞ (X) for all m ∈ {0, 1, . . .} and f (x) = 0 for all x ∈ X. (β) For each x ∈ X there is an f ∈ C01 (X) with f (x) = 0. (γ ) Each ball in X is compact.

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Locally convex spaces

Proof. (α) =⇒ (β) is trivial. (β) =⇒ (γ ). Suppose there exists a ball B in X with centre x ∈ X and radius r > 0 that is not compact; we derive a contradiction. By (β) there is an f ∈ C01 (X) f (x) = 0. Take a compact set Y ⊂ X such that |f | < |f (x)| off Y with and 1 f < r −1 |f (x)| off Y × Y . Now Y ∩ B, being compact, is not equal to B, so there is a y ∈ B, y ∈ Y . Then |f (y)| < |f (x)| and hence |f (x) − f (y)| = |f (x)|. Also, (x, y) ∈ Y × Y , hence 1 f (x, y) < r −1 |f (x)|. But on the other hand we have |x − y| ≤ r. Therefore, 1 f (x, y) = f (x) − f (y) ≥ r −1 |f (x)| , x−y which is a contradiction. (γ ) =⇒ (α). Let us assume that X is not compact (otherwise, the constant function 1 does the job). Let {Pi : i ∈ I } be a partition of X into (compact) balls Pi in X of radius 1. By separability of X (apply 2.5.35) and non-compactness we may assume I = N. For i ∈ N, let δi := max{|x| : x ∈ Pi }. Choose nonzero λ1 , λ2 , . . . ∈ K such that limi |λi | δim = 0 for all m ∈ {0, 1, . . .} (e.g. for ρ 1 ≤ |λi | ≤ [δi +1]! , where ρ ∈ K, 0 < |ρ| < 1 each i, one can take λi with [δi +1]! and where [ ] indicates the entire part). Then define f : X−→K by f (x) := λi

if i ∈ N, x ∈ Pi .

The rest of the proof is devoted to show that f satisfies the conditions required in (α). Clearly f ∈ C ∞ (X) and f (x) = 0 for all x ∈ X. Now let m ∈ {0, 1, . . .}. In order to get that f Xm ∈ C0∞ (X) we shall prove by induction on n ∈ {0, 1, . . .} that for each ε > 0 there is a ball Bn in X such that (3.13) n (f Xm )(x1 , . . . , xn+1 ) < ε for all (x1 , . . . , xn+1 ) ∈ X n+1 \ B n+1 . n First, take n = 0 and ε > 0. Since for each i ∈ N and each x ∈ Pi , |f (x) x m | = |λi | |x m | ≤ |λi | δim , and limi |λi | δim = 0, there exists an i0 ∈ N such that, for i > i0 , |f Xm | < ε on Pi . So any ball B0 containing P1 ∪ · · · ∪ Pi0 does the job. Next we prove the step n − 1 → n. To this end let, for convenience, B be the collection of all sets in X of the form {x ∈ X : |x| ≤ r}, r ≥ 1. Clearly every ball in X is contained in some element of B. Now let ε > 0. By the veracity of (3.13) for n = 0 and by the induction hypothesis, there is a Bn−1 ∈ B such that: (i) |f (x)| |x|m < ε, (ii) n−1 (f Xm )(z) < ε,

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141

n . Then (ii) and the fact that f Xm ∈ for all x ∈ X \ Bn−1 and z ∈ Xn \ Bn−1 n−1 m C (X) imply that n−1 (f X )∞ < ∞. Hence there is a Bn ∈ B such that Bn ⊃ Bn−1 and dist (Bn , X \ Bn ) > ε −1 n−1 (f Xm )∞ . Let (x1 , . . . , xn+1 ) ∈ X n+1 \ Bnn+1 . We claim that (3.13) holds, i.e., n (f Xm )(x1 , . . . , xn+1 ) < ε.

We consider the two possible cases: (a) xj − xk ≤ 1 for all j, k ∈ {1, . . . , n + 1}. By symmetry we may assume that |x1 | = max1≤j ≤n xj . Then since (x1 , . . . , xn+1 ) ∈ Bnn+1 , there is a j ∈ {1, . . . , n + 1} such that xj ∈ Bn which, together with |x1 | ≥ xj , imply that x1 ∈ Bn , so x1 ∈ Bn−1 (in particular, |x1 | > 1). There exists a unique i ∈ N such that {x1 , . . . , xn+1 } ⊂ Pi , and f takes the constant value f (x1 ) on Pi . Thus, by rule X of 2.5.78 and (i) we have n (f Xm )(x1 , . . . , xn+1 ) = f (x1 ) n (Xm )(x1 , . . . , xn+1 ) ≤ |f (x1 )| |x1 |m < ε.

(b) xj − xk > 1 for some j, k ∈ {1, . . . , n + 1}. Without loss, suppose j = 1, k = 2. Now consider the identity n (f Xm )(x1 , . . . , xn+1 ) (f Xm )(x , x , . . . , x ) − (f Xm )(x , x , . . . , x ) 1 3 n+1 n−1 2 3 n+1 n−1 = . x1 − x 2 (3.14) If one of the points x3 , . . . , xn+1 is not in Bn or if both x1 and x2 are not in Bn ε we may apply (ii) to deduce that the right-hand side of (3.14) is < |x1 −x < 2| ε. Thus, it remains to discuss the case where x3 , . . . , xn+1 are all in Bn and, say, x1 ∈ Bn . But then x2 ∈ Bn (as otherwise (x1 , . . . , xn+1 ) ∈ Bnn+1 ). Hence, |x1 − x2 | ≥ dist (Bn , X \ Bn ) > ε−1 n−1 (f Xm )∞ , so again the Xm )∞ < ε. right-hand side of (3.14) is ≤ n−1|x(f 1 −x2 | Remark 3.7.60 Let K := Qp . Then X := Qp \ {0} is locally compact, without isolated points, but the ball {x ∈ X : |x − 1| ≤ 1} = Zp \ {0} is not compact. Theorem 3.7.61 (Compare with 2.5.21) Let balls in X be compact. Then, for each n ∈ {0, 1, . . .}, the space {f ∈ C0n (X) : f has compact support} is dense in C0n (X). Also, {f ∈ C0∞ (X) : f has compact support} is dense in C0∞ (X). Proof. To prove the first assertion, let n ∈ {0, 1, . . .}, f ∈ C0n (X), ε > 0. There k+1 \ B k+1 , is a ball B ⊂ X such that, for all k ∈ {0, 1, . . . , n} and z ∈ X k f (z) < ε. We may assume diam B ≥ 1. Then ξB j ≤ 1 for each j ∈

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{0, 1, . . .}, so that ξB ∈ BC ∞ (X). By 3.7.58 we have f ξB ∈ C0n (X). We prove n that pX (f − f ξB ) < ε. To this end, let i ∈ {0, 1, . . . , n}, (x1 , . . . , xi+1 ) ∈ i+1 X . If (x1 , . . . , xi+1 ) ∈ B i+1 then i (f ξX\B )(x1 , . . . , xi+1 ) = 0. Otherwise, we may assume by symmetry that x1 ∈ B. Since ξB j ≤ 1 for each j , we get by rule V of 2.5.78 that i (f ξX\B )(x1 , . . . , xi+1 ) i k f (x1 , . . . , xk+1 ) i−k ξX\B (xk+1 , . . . , xi+1 ) = k=0 ≤ max k f (x1 , . . . , xk+1 ) < ε. 0≤k≤i

n It follows that i (f ξX\B )X < ε for each i ∈ {0, 1, . . . , n}, i.e., pX (f − n f ξB ) = pX (f ξX\B ) < ε. Now let f ∈ C0∞ (X), n ∈ {0, 1, . . .}, ε > 0. By the above proof there is a n ball B ⊂ X such that ξB ∈ BC ∞ (X) and pX (f − f ξB ) < ε, and by 3.7.58 we ∞ have f ξB ∈ C0 (X). n ∞ We finish this section by giving a description of the space BCW (X) (BCW (X)) n ∞ of 3.7.55 as a part of C0 (X) (C0 (X)).

Theorem 3.7.62 Let balls in X be compact. Let W be the collection of all polynomial functions X−→K. Then, for each n ∈ {0, 1, . . .}, n BCW (X) = {f ∈ C0n (X) : f Xm ∈ C0n (X) for all m ∈ {0, 1, . . .}}.

Proof. For the inclusion “⊂” it suffices to show that each function in n (X) belongs to C0n (X). (Indeed, since for every m ∈ {0, 1, . . .} and every BCW n n (X) we have that f Xm ∈ BCW (X) then also, by the foregoing, f ∈ BCW n m f X ∈ C0 (X), and we are done.) The other inclusion follows easily from n (X) and C0n (X). the definitions of BCW n Let f ∈ BCW (X). Let us see that f ∈ C0n (X). Choose ε > 0. By using an induction reasoning it suffices to prove that there is a compact set Y ⊂ X such that n f (x1 , . . . , xn+1 ) < ε for all (x1 , . . . , xn+1 ) ∈ X n+1 \ Y n+1 . First, let n = 0. Since the function f X is bounded on X, we can take a C0 > 0 with f X∞ ≤ C0 . Then Y := {y ∈ X : |y| ≤ C0 ε−1 } is compact by assumption on X and, if x ∈ X \ Y , |f (x)| ≤ C|x|0 < C0Cε0−1 = ε. Now let n ≥ 1. Then n−1 f and n (f X) are bounded on Xn and X n+1 respectively. Let Cn > 0 be such that max(n−1 f ∞ , n (f X)∞ ) ≤ Cn . The set Y := {y ∈ X : |y| ≤ Cn ε−1 } is compact by assumption on X and, if (x1 , . . . , xn+1 ) ∈ Xn+1 \ Y n+1 , n f (x1 , . . . , xn+1 ) < ε. In fact, by symmetry we may assume

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|xn+1 | ≥ · · · ≥ |x1 |. Thus, xn+1 ∈ Y i.e., |xn+1 | > Cn ε−1 . Further, by rule V of 2.5.78 we have the identity n (f X)(x1 , . . . , xn+1 ) = n f (x1 , . . . , xn+1 ) xn+1 + n−1 f (x1 , . . . , xn ). n < ε. So we obtain n f (x1 , . . . , xn+1 ) ≤ |xCn+1 | Corollary 3.7.63 Let X, W be as above. Then ∞ BCW (X) = {f ∈ C0∞ (X) : f Xm ∈ C0∞ (X) for all m ∈ {0, 1, . . .}}.

Definition 3.7.64 Let balls in X be compact. For W as in 3.7.55, the Fr´echet ∞ (X), τW∞ ) is usually called S(X), the space of C ∞ -functions rapidly space (BCW decreasing at infinity. From 3.7.59 we see that there exist functions in S(X) that are non-zero everywhere!

3.8 Compactoids Recall that a subset X of a locally convex space E is called precompact if, for every zero neighbourhood U there is a finite set G ⊂ E such that X ⊂ U + G. Now assume that E is Hausdorff and that K is not locally compact. Any convex set in E containing at least two points x = y contains {x + λ (y − x) : λ ∈ BK } which is homeomorphic to BK , hence not precompact. So convex (pre)compact sets in E are trivial. To overcome this difficulty we “convexify” the notion of precompactness yielding the concept of compactoidity, see 3.8.1 below. It is not unreasonable to expect that (complete) convex compactoids will take over the role played by (compact) precompact convex sets in classical Functional Analysis. In the sequel we will see to what extent this belief comes true (see 3.8.29, 3.8.35 and 3.8.38). Compactoidity is useful for all scalar fields K. In Chapter 6 we will treat an alternative concept, c-compactness, which is of interest only when K is spherically complete. In 6.1.13 we will compare both notions. Definition 3.8.1 A subset X of a locally convex space E is called (a) compactoid in E if for every zero neighbourhood U in E there exists a finite set G in E such that X ⊂ U + aco G. As a first illustration of the usefulness of this concept we prove the ultrametric version of the classical Ascoli Theorem.

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Theorem 3.8.2 Let X be a compact topological space. For a subset F of C(X) the following are equivalent: (α) F is a compactoid in C(X). (β) F is pointwise bounded and equicontinuous. Proof. (α) =⇒ (β). Clearly F is bounded. To prove equicontinuity, let x ∈ X, r > 0. By compactoidity there exist f1 , . . . , fn ∈ C(X) such that F ⊂ B(0, r) + aco {f1 , . . . , fn }. There is a neighbourhood U of x such that |fi (y) − fi (x)| ≤ r for all y ∈ U and i ∈ {1, . . . , n}. Thus, the set F := {f ∈ C(X) : |f (y) − f (x)| ≤ r for all y ∈ U } contains {f1 , . . . , fn } and, obviously, B(0, r). By absolute convexity of F we have F ⊃ B(0, r) + aco {f1 , . . . , fn }, so F ⊃ F and the equicontinuity of F is proved. (β) =⇒ (α). Let r > 0. For x, y ∈ X define x ∼ y if |f (x) − f (y)| ≤ r for all f ∈ F. Then ∼ is an equivalence relation on X whose classes, due to equicontinuity, are open, and hence clopen. By compactness there are only finitely many equivalence classes, say, U1 , . . . , Un . Choose xi ∈ Ui for each i ∈ {1, . . . , n}. By pointwise boundedness there is a λ ∈ K such that |f (xi )| ≤ |λ| for each i ∈ {1, . . . , n} and f ∈ F. Now, for i ∈ {1, . . . , n}, let gi := λ ξUi . We claim that F ⊂ B(0, r) + aco {g1 , . . . , gn }. Indeed, for each f ∈ F we have g := ni=1 f (xi ) ξUi ∈ aco {g1 , . . . , gn } and f − g∞ ≤ r. To start with the basic theory of compactoids we prove that compactoidity and precompactness coincide if K is locally compact. Theorem 3.8.3 Let E be a locally convex space over a locally compact field K. Then each compactoid in E is precompact. If, in addition, E is Hausdorff each complete compactoid subset of E is compact. Proof. It suffices to prove the first assertion ([103], 3.5.3). Thus, let X ⊂ E be a compactoid in E and let U be an absolutely convex zero neighbourhood in E. There is a finite set G1 such that X ⊂ U + aco G1 . Let G1 := {x1 , . . . , xn }. The map K n −→E given by (λ1 , . . . , λn ) → ni=1 λi xi is continuous and maps the compact set BKn onto aco G1 , so aco G1 is compact. Thus, there is a finite set G ⊂ E such that aco G1 ⊂ U + G. Then X ⊂ U + U + G = U + G, proving that X is precompact. If K is not locally compact, BK is a compactoid but not precompact. The following result contains the basic properties of compactoids. Theorem 3.8.4 Let E, F be locally convex spaces. Then we have the following: (i) Precompact subsets of E are compactoids in E. (ii) If X ⊂ E is compactoid then so are X, aco X, and subsets of X.

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(iii) If x1 , x2 , . . . is a Cauchy sequence in E then aco {x1 , x2 , . . .} is a compactoid in E. (iv) If X is a compactoid in E and T ∈ L(E, F ) then T (X) is a compactoid in F . (v) If C is a convex compactoid in E then so is C e . (vi) Finite unions and sums of compactoids in E are compactoids. (vii) Compactoids in E are bounded. (viii) A subset X of E is a compactoid in E if and only if X is bounded and for every zero neighbourhood U in E there is a finite-dimensional subspace D ⊂ E such that X ⊂ U + D. In particular, if E is finite-dimensional the compactoids in E are just the bounded sets. (ix) If E is finite-dimensional then E has a compactoid zero neighbourhood. Proof. We only prove the “if” of the first part of (viii). We leave verification of the rest to the reader. Let U be an absolutely convex zero neighbourhood in E. By assumption, there is a finite-dimensional subspace D ⊂ E such that X ⊂ U + D, we may suppose that the dimension of D is minimal. The restriction pU |D of the Minkowski function pU associated to U is a norm on D. (If pU (d) = 0 for some d ∈ D, d = 0 then [d] ⊂ U and U + D = U + D1 , where D1 ⊂ D is an algebraic complement of [d] in D, conflicting minimality.) Now (X + U ) ∩ D is pU |D-bounded, so we have (X + U ) ∩ D ⊂ aco {d1 , . . . , dn } for some d1 , . . . , dn ∈ D, with n ≤ dim D. Hence X ⊂ U + ((X + U ) ∩ D) ⊂ U + aco {d1 , . . . , dn }, and X is a compactoid in E. For Hausdorff spaces the converse of (ix) holds. Theorem 3.8.5 A Hausdorff locally convex space is finite-dimensional if and only if it has a compactoid zero neighbourhood. Proof. We prove that a Hausdorff locally convex space E with a compactoid zero neighbourhood U is finite-dimensional. Let ρ ∈ K, 0 < |ρ| < 1. There is a finite-dimensional space D ⊂ E such that U ⊂ ρ U + D. Then ρ U ⊂ ρ 2 U + D, so U ⊂ ρ 2 U + D, etc. For each x ∈ U there is a sequence y1 + d1 , y2 + d2 , . . . , where yn ∈ ρ n U , dn ∈ D and x = yn + dn for each n. By boundedness of U we have limn yn = 0, so x = limn dn ∈ D. But D is normable (3.4.24) and complete (2.1.14(i)). Hence D = D and we conclude U ⊂ D, implying that E is finite-dimensional. As a corollary we obtain the ultrametric version of the classical Riesz Theorem.

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Theorem 3.8.6 The unit ball of a normed space E is a compactoid if and only if E is finite-dimensional. The concept of compactoidity as given in 3.8.1 depends on the embedding space E. But we will show that the finite set G can be chosen in [X]. To this end we need two lemmas. Lemma 3.8.7 Let A, B be absolutely convex subsets of a vector space E, let A ⊂ B + aco {x} for some x ∈ E. Let α ∈ K, α := 1 if the valuation of K is discrete, 0 < |α| < 1 otherwise. Then there is a y ∈ A such that α A ⊂ B + aco {y}. Proof. Let C := {µ ∈ BK : µ x ∈ A + B}. Then C is an absolutely convex subset of BK and it is easily seen that there is a c ∈ C for which α C ⊂ BK c ⊂ C. Then c x ∈ y + B for some y ∈ A. Now let z ∈ A. Then z = b + µ x for some b ∈ B, µ ∈ C, so α z = α b + α µ x ∈ B + aco {c x} ∈ B + aco (y + B) = B + aco {y}. Lemma 3.8.8 Let E, α be as above. Let A, B be absolutely convex subsets of E such that A ⊂ B + aco {x1 , . . . , xn } for some n ∈ N and x1 , . . . , xn ∈ E. Then there exist y1 , . . . , yn ∈ A such that α A ⊂ B + aco {y1 , . . . , yn }. Proof. Choose α1 , . . . , αn ∈ K such that αi := 1 for all i ∈ {1, . . . , n} if the of K is discrete, 0 < |αi | < 1 for all i ∈ {1, . . . , n} and |α| ≤ nvaluation αi otherwise. We have A ⊂ (B + aco {x2 , . . . , xn }) + aco {x1 }. So by i=1

3.8.7 there is a y1 ∈ A such that

α1 A ⊂ B + aco {x2 , . . . , xn } + aco {y1 } = (B + aco {y1 , x3 , . . . , xn }) + aco {x2 }. Again by 3.8.7 there is a y2 ∈ α1 A such that α2 α1 A ⊂ B + aco {y1 , y2 , x3 , . . . , xn }, and so on. Inductively we arrive at points y1 , . . . , yn , where y1 ∈ A and, for each i ≥ 2, yi ∈ αi−1 · · · α1 A ⊂ A, such that α A ⊂ α1 · · · αn A ⊂ B + aco {y1 , y2 , . . . , yn }. From the above lemma we derive the following important result. Notice that X is not assumed to be convex. Theorem 3.8.9 Let X be a compactoid in a locally convex space E. Let U be a zero neighbourhood in E. (i) If the valuation of K is discrete there is a finite set G ⊂ X such that X ⊂ U + aco G.

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(ii) If the valuation of K is dense then for each λ ∈ K, |λ| > 1 there is a finite set G ⊂ λ X such that X ⊂ U + aco G. Proof. For each Y ⊂ E and finite set G ⊂ aco Y there is a finite set G ⊂ Y such that aco G ⊂ aco G. This, together with 3.8.4(ii) shows that we may assume that X is absolutely convex. We also may assume that U is absolutely convex. Let λ ∈ K, λ := 1 if the valuation of K is discrete, |λ| > 1 otherwise. By compactoidity there is a finite set G1 ⊂ E such that X ⊂ λ−1 U + aco G1 . By 3.8.8 there exists a finite set G2 ⊂ X such that λ−1 X ⊂ λ−1 U + aco G2 , so X ⊂ U + aco G, where G := λ G2 ⊂ λ X. Remark 3.8.10 To show that (ii) is false if λ = 1 and the valuation of K is dense, consider X := BK− , which is a compactoid in the normed space K. It is easily seen that there is no finite set G in X such that X ⊂ B(0, 12 ) + aco G. Remark 3.8.11 Thanks to the above theorem we may speak from now on about “compactoids” without having to refer to the embedding space. As an application we prove the following (compare with 3.6.3(iv)). Theorem 3.8.12 A subset X of a locally convex space E is a compactoid if and only if each countable subset of X is a compactoid. Proof. We only have to prove the “if”. Suppose each countable subset of X is a compactoid. Then by 3.6.3(iv) X is bounded. Suppose X is not a compactoid; we construct a countable subset of X that is not a compactoid. By 3.8.4(viii) there is a zero neighbourhood U in E such that X ⊂ U + D for each finitedimensional space D ⊂ E. Inductively we can construct x1 , x2 , . . . ∈ X such that for each n xn+1 ∈ U + [x1 , . . . , xn ]. Then Y := {x1 , x2 , . . .} is a countable subset of X. If it were a compactoid then by 3.8.9 there is a finite-dimensional space D ⊂ [x1 , x2 , . . .] such that Y ⊂ U + D. Now D ⊂ [x1 , . . . , xn ] for some n ∈ N. Then Y ⊂ U + [x1 , . . . , xn ], conflicting xn+1 ∈ U + [x1 , . . . , xn ]. There is another application of 3.8.9. Theorem 3.8.13 Let τ1 and τ2 be two locally convex topologies on a K-vector space E such that τ1 = τ2 on an absolutely convex set A ⊂ E. If A is a τ1 compactoid then also A is a τ2 -compactoid. Proof. Let λ ∈ K, |λ| > 1. By the assumption, τ1 = τ2 on λ A. Let U be a τ2 -zero neighbourhood in E. Then U ∩ λ A is a τ2 -zero neighbourhood in λ A,

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hence a τ1 -zero neighbourhood. So there is a τ1 -zero neighbourhood V in E such that V ∩ λ A ⊂ U ∩ λ A. By τ1 -compactoidity and 3.8.9 there exists a finite set G ⊂ λ A such that A ⊂ V + aco G. Then A ⊂ V ∩ λ A + aco G ⊂ U ∩ λ A + aco G ⊂ U + aco G, proving that A is a τ1 -compactoid. We now consider compactoids in products. Theorem 3.8.14 Let I be an index set and let, for each i ∈ I , a locally convex space Ei be given. For each i ∈ I , let πi : i∈I Ei −→Ei be the coordinate map. Then, a set X ⊂ i∈I Ei is a compactoid if and only if each πi (X) is a compactoid. In particular, if, for each i ∈ I , Xi is a compactoid in Ei then i∈I Xi is a compactoid in i∈I Ei . Proof. Assume X is a compactoid. Then by 3.8.4(iv) each πi (X) is a compactoid. Conversely, let πi (X) be a compactoid for each i ∈ I . Let U be a zero neighbourhood in i∈I Ei . To prove that X ⊂ U + aco G for some finite set G we may assume that U has the form i∈I Ui , with Ui := Ei for all i ∈ I \ J , where J is a finite subset of I and, for i ∈ J , Ui is a zero neighbourhood in Ei . For each i ∈ J there is a finite set Gi ⊂ Ei such that πi (X) ⊂ Ui + aco Gi . Put Gi := {0} if i ∈ I \ J . Then X ⊂ i∈I πi (X) ⊂ U + aco G, where G is the finite set i∈I Gi . Corollary 3.8.15 A subset X of the projective limit of a projective system (En , πn )n∈N of locally convex spaces is a compactoid if and only if each πn (X) is a compactoid. Theorem 3.8.14 can also be applied to obtain the following characterization of the compactoid subsets of any locally convex space. Theorem 3.8.16 Let E be a locally convex space, let P be a collection of seminorms generating the topology, let X ⊂ E. Then X is a compactoid if and only if, for each p ∈ P, πp (X) is a compactoid in Ep . Proof. If X is a compactoid then, for each p ∈ P, πp (X) is a compactoid in Ep by 3.8.4(iv). Conversely, suppose πp (X) is a compactoid in Ep for each p ∈ P. First we prove that, for every continuous seminorm q on E, πq (X) is a compactoid in Eq . In fact, there exist p1 , . . . , pn ∈ P and C > 0 such that q ≤ C max(p1 , . . . , pn ). Then, for the subspace D := {(πp1 (x), . . . , πpn (x)) : x ∈ E} of Ep1 × · · · × Epn , the map D−→Eq (πp1 (x), . . . , πpn (x)) → πq (x) is a well-defined continuous linear map. So it suffices to see that the set

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{(πp1 (x), . . . , πpn (x)) : x ∈ X} is a compactoid in D. This follows because this set is contained in πp1 (X) × · · · × πpn (X) which, by assumption and 3.8.14, is a compactoid in Ep1 × · · · × Epn . Now compactoidity of πq (X) (q continuous seminorm) means that for every r > 0 there is a finite subset G of E such that πq (X) ⊂ πq ({x ∈ E : q(x) < r}) + aco πq (G) or, equivalently, X ⊂ {x ∈ E : q(x) < r} + aco G. Hence X is a compactoid. Next we describe compactoids in direct sums. Theorem 3.8.17 Let I be an index set and let, for each i ∈ I , a locally convex space Ei be given. For each i ∈ I , let πi : i∈I Ei −→Ei be the coordinate map. Then a set X ⊂ i∈I Ei is a compactoid if and only if πi (X) is a compactoid in Ei for each i and πi (X) ⊂ {0} for all but finitely many i. Proof. Let X be a compactoid. Then by 3.8.4(iv) each πi (X) is a compactoid. By boundedness of X and 3.6.4(ii) πi (X) ⊂ {0} for all but finitely many i. The proof of the converse can be done similarly as in (ii) of 3.6.4, but now using 3.8.4(ii) and (vi). Remark 3.8.18 There do not exist simple, general, descriptions of compactoids in inductive limits in the spirit of the previous theorems. But in Sections 11.3 and 11.5 we will discuss various interesting special cases. Compactoids in quotients of Fr´echet spaces will be described in 3.8.33 (see also 3.8.34). We now will prove some permanence properties of compactoids (3.8.23, 3.8.29 and 3.8.38). From now on in this chapter λ ∈ K , λ := 1 if the valuation of K is discrete, |λ| > 1 otherwise.

Lemma 3.8.19 Let B be a closed absolutely convex subset of a locally convex space E. Let C ⊂ K be absolutely convex, let x ∈ E and let i → yi := zi + γi x (zi ∈ B, γi ∈ C) be a net in B + C x converging to 0. Then we have the following: (i) If α ∈ K, α x ∈ B then |γi | < |α| for large i. (ii) If C x ∩ B = {0} then limi γi x = 0, limi zi = 0. (iii) If C x ∩ B = {0} then yi ∈ λ B for large i.

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Proof. (i). If J := {i ∈ I : |γi | ≥ |α|} were cofinal then from α γi−1 ≤ 1 we obtain 0 = limi∈J α γi−1 yi = limi∈J (α γi−1 zi + α x), and α γi−1 zi ∈ B. Hence α x ∈ −B = B, a contradiction. (ii). We may suppose C x = {0}. Let ε > 0, choose an α ∈ C with 0 < |α| < ε. Then α x ∈ B so, by (i), |γi | < |α| < ε for large i implying limi γi x = 0 and limi zi = 0. (iii). Let C1 := {µ ∈ K : µ x ∈ B}. Then C1 is absolutely convex. By assumption C1 = {0}, we may suppose C1 = K. So C1 is a ball and r := diam C1 is strictly between 0 and ∞. If the valuation of K is dense, choose an α ∈ K with r < |α| < |λ| r. Then α ∈ C1 , so α x ∈ B and by (i) we have |γi | < |α| for large i implying γi x ∈ λ C1 x ⊂ λ B and yi = zi + γi x ∈ B + λ B ⊂ λ B. If the valuation of K is discrete then r ∈ |K|. Choose α ∈ K, |α| = r ρ −1 , where ρ is a uniformizing element. Then α ∈ C1 , so by (i) we have |γi | < |α| for large i. So, for these i, |γi | ≤ r, γi ∈ C1 , so yi ∈ B. Corollary 3.8.20 Let B, E be as above. Let x1 , . . . , xn ∈ E and let C1 , . . . , Cn be absolutely convex subsets of K. Then (B + C1 x1 + · · · + Cn xn )e is closed. Proof. First consider the case n = 1. If C1 x1 ∩ B = {0} it follows from 3.8.19(ii) that B + C1 x1 , hence (B + C1 x1 )e is closed. Now suppose C1 x1 ∩ B = {0}; we prove that B + C1 x1 ⊂ λ (B + C1 x1 ), and by 3.3.11(iv) we are done. To this end, let i → yi := zi + γi x1 (zi ∈ B, γi ∈ C1 ) be a net in B + C1 x1 converging to y ∈ E. By 3.8.19(iii) yi − yj ∈ λ B for some j and all i ≥ j . Now yj + λ B is closed, so y = limi yi ∈ yj + λ B = zj + γj x1 + λ B ⊂ B + C1 x1 + λ B ⊂ λ (B + C1 x1 ). To prove the induction step n − 1 → n consider V := (B + C1 x1 + · · · + Cn−1 xn−1 )e . By the induction hypothesis V is closed, by the first part of this proof (V + Cn xn )e is closed. Then (by 3.1.6(v)) (B + C1 x1 + · · · + Cn xn )e = ((B + C1 x1 + · · · + Cn−1 xn−1 )e + Cn xn )e = (V + Cn xn )e is closed.

Remark 3.8.21 In 3.8.31(ii) we will see that B + C1 x1 + · · · + Cn xn itself needs not be closed. Corollary 3.8.22 Let B, E be as in 3.8.19. Suppose x1 , . . . , xn ∈ [B], let i → yi be a net in B + aco {x1 , . . . , xn } converging to 0. Then yi ∈ λ B for large i. Proof. Choose λ1 , . . . , λn ∈ K such that |λi | > 1 for each i and |λ1 · · · λn |3 ≤ |λ| if the valuation of K is dense, λi := 1 for each i if the valuation of K is discrete. By 3.8.20, for all i, yi ∈ (B + aco {x1 , . . . , xn })e , hence yi ∈ λn (B +

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aco {x1 , . . . , xn }). By 3.8.19(iii), for large i, yi ∈ λ2n (B + aco {x1 , . . . , xn−1 }), which is contained by 3.8.20 in λ3n (B + aco {x1 , . . . , xn−1 }). Continuation of this process leads to yi ∈ (λ1 · · · λn )3 B ⊂ λ B for large i. We are now ready to prove a first permanence theorem for compactoids. Theorem 3.8.23 Let X be a compactoid in a locally convex space (E, τ ). Let τ1 ≤ τ be a locally convex topology on E that is closely related to τ . Then τ = τ1 on X. Proof. Without loss, assume X is absolutely convex. Let i → yi be a net in X, τ1 τ yi −→ 0. It suffices to prove yi −→ 0. Let U be a zero neighbourhood in E. We may assume that U is absolutely convex and τ1 -closed. By compactoidity there are x1 , . . . , xn ∈ E such that X ⊂ λ−1 U + aco {x1 , . . . , xn }. By 3.8.22 we have yi ∈ λ (λ−1 U ) = U for large i. We will apply this result to obtain the important Theorem 5.2.12. More permanence properties, for spherically complete base fields only, will be proved in Chapter 6. For general K, to be able to prove anything we concentrate on metrizable compactoids. Notation Let X1 , X2 , . . . be subsets of a locally convex space E. We write limn Xn = 0 if for each zero neighbourhood U there is an m ∈ N such that Xn ⊂ U for n ≥ m. First, a metrizability theorem. Theorem 3.8.24 Let (E, τ ) be a Hausdorff locally convex space, let e1 , e2 , . . . be a sequence in E tending to 0. Then A := aco {e1 , e2 , . . .} is a metrizable compactoid. In fact, τ |A is induced by an invariant ultrametric. Proof. Compactoidity follows immediately from 3.8.4(iii). Now we prove metrizability. For technical reasons we construct a metric d on B := λ A, as follows. Let λ1 , λ2 , . . . ∈ K, 1 = |λ1 | > |λ2 | > · · · , limn λn = 0. For n ∈ N put Vn := λn B + aco {λ en+1 , λ en+2 , . . .}. Then B = V1 ⊃ V2 ⊃ · · · . By boundedness of B and the fact that limn λ en = 0 we have limn Vn = 0. In particular, n∈N Vn = {0} and we can define for x, y ∈ B 0 if x = y d(x, y) := 2−n if n ∈ N, x − y ∈ Vn \ Vn+1 .

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Then d is an invariant ultrametric on B, whose topology is stronger than τ |B. τ τ Now let i → yi be a net in A for which yi −→ 0. Then λ yi −→ 0 and λ yi ∈ B. Let n ∈ N. Since Vn is absorbing in [B] and B ⊂ Vn + aco {λ e1 , . . . , λ en } we have by 3.8.22 that, for large i, λ yi ∈ λ Vn i.e., yi ∈ Vn or d(yi , 0) ≤ 2−n showing that limi d(yi , 0) = 0. We see that τ |A is induced by d. The converse is more important. Theorem 3.8.25 Let A be an absolutely convex metrizable compactoid in a locally convex space E. Then there exist e1 , e2 , . . . ∈ λ A such that limn en = 0 and A ⊂ aco {e1 , e2 , . . .}. Proof. By metrizability of λ A there is a sequence V1 ⊃ V2 ⊃ · · · of zero neighbourhoods in E, which we may assume to be absolutely convex, for which limn Vn ∩ λ A = 0. Choose λ1 , λ2 , . . . ∈ K, |λi | > 1 for all i, ∞ i= |λi | ≤ |λ| if the valuation of K is dense, λi := 1 for all i otherwise. There is a finite set G1 ⊂ λ1 A such that A ⊂ V1 + aco G1 (3.8.9). Then A ⊂ (V1 ∩ λ1 A) + aco G1 . There is a finite set G2 ⊂ λ2 (V1 ∩ λ1 A) such that V1 ∩ λ1 A ⊂ V2 + aco G2 . Then A ⊂ (V2 ∩ λ1 λ2 A) + aco (G1 ∪ G2 ), etc. Inductively we arrive at finite sets G1 , G2 , . . . , where G1 ⊂ λ1 A, Gn+1 ⊂ (λ1 · · · λn+1 ) A ∩ λn+1 Vn and A ⊂ (Vn ∩ (λ1 · · · λn ) A) + aco (G1 ∪ · · · ∪ Gn ) for each n. Now Vn ∩ (λ1 · · · λn ) A ⊂ Vn ∩ λ A, and so limn Vn ∩ (λ1 · · · λn ) A = 0, and we have A ⊂ aco (G1 ∪ G2 ∪ · · · ). Since Gn+1 ⊂ λn+1 Vn ⊂ λ Vn we have limn Gn = 0, and the conclusion follows after observing that Gn ⊂ (λ1 · · · λn ) A ⊂ λ A for each n. Remarks 3.8.26 (a) If in the above the condition of absolute convexity of A is dropped the conclusion is no longer true. See 5.4.4. (b) Let the valuation of K be dense. The conclusion of the above theorem is false for λ = 1, as is shown by taking A := BK− . (c) In 8.1.4 we will apply 3.8.24 and 3.8.25 to describe the compactoids in c0 . Corollary 3.8.27 Let E, F be Hausdorff locally convex spaces, let T ∈ L(E, F ) and let A ⊂ E be an absolutely convex compactoid. If A is metrizable then so is T (A). Proof. By 3.8.25 A ⊂ aco {e1 , e2 , . . .} with limn en = 0. Then T (A) ⊂ aco {T (e1 ), T (e2 ), . . .} and limn T (en ) = 0, so by 3.8.24 T (A) is metrizable.

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Corollary 3.8.28 Let D be a dense subspace of a metrizable locally convex space E, Then, for each compactoid A ⊂ E there is a compactoid B in D with A ⊂ B. Proof. By 3.8.25 A ⊂ aco X where X := {x1 , x2 , . . .}, limn xn = 0. We shall prove that there is a precompact Y ⊂ D such that X ⊂ Y . (Then B := aco Y meets the requirements.) For each n ∈ N, let Bn be the collection of balls in E of radius 1/n, with respect to the ultrametric defining its topology, that have a non-empty intersection with X. By precompactness of X each Bn is finite and any member of Bn+1 is contained in a member of Bn . Let n ∈ N. By choosing a point in C ∩ D for each C ∈ Bn we obtain a finite set Yn ⊂ D. Now let Y := n∈N Yn . It is easily seen that Y is precompact and that X ⊂ Y . Now we are ready for a second permanence theorem. Theorem 3.8.29 Let A be a complete absolutely convex metrizable compactoid in a Hausdorff locally convex space E. (i) If F is a Hausdorff locally convex space and T ∈ L(E, F ) then (T (A))e is complete. (ii) If B ⊂ E is absolutely convex and closed (complete) then (A + B)e is closed (complete). Proof. By passing to completions and by 3.4.14 it suffices to prove that (T (A) + B)e is closed for any closed absolutely convex B ⊂ F . Let x ∈ T (A) + B; we prove that x ∈ λ (T (A) + B), and by 3.3.11(iv) we are done. Choose λ1 , λ2 , . . . ∈ K, |λi | > 1 for each i, |λ1 |2 ≤ |λ|, ∞ i=2 |λi | ≤ |λ1 | if the valuation of K is dense, λi := 1 for each i otherwise. By the previous theorem there exist e1 , e2 , . . . ∈ λ1 A with limn en = 0 such that A ⊂ aco {e1 , e2 , . . .}. For n ∈ N, set An := aco {en , en+1 , . . .}. Then limn An = 0 and limn T (An ) = 0. We have T (A) + B ⊂ aco {T (e1 )} + T (A2 ) + B which, by 3.8.20, is contained in λ2 aco {T (e1 )} + λ2 (T (A2 ) + B), so there is an x1 ∈ aco {e1 } such that x = λ2 T (x1 ) + λ2 y2 , where y2 ∈ T (A2 ) + B ⊂ aco {T (e2 )} + T (A3 ) + B ⊂ λ3 aco {T (e2 )} + λ3 (T (A3 ) + B). Hence x = λ2 T (x1 ) + λ3 λ2 T (x2 ) + λ3 λ2 y3 , where x2 ∈ aco {e2 } and y3 ∈ T (A3 ) + B, etc. Inductively we arrive at a sequence u1 , u2 , . . . , where ui ∈ λ2 . . . λi+1 aco {ei } ⊂ λ21 A ⊂ λ A for each i such that, for each n, x = T (u1 + · · · + un ) + zn+1 ,

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where zn+1 = (λ2 · · · λn+1 ) yn+1 ∈ λ1 (T (An+1 ) + B). From limn en = 0 we u exists and lies in obtain limn un = 0. By completeness of λ A, u := ∞ n=1 n λ A. Then z := limn zn must also exist. It lies in λ1 n∈N (T (An ) + B) which is λ1 B since B is closed and limn T (An ) = 0. We see that x = T (u) + z ∈ λ T (A) + λ1 B ⊂ λ (T (A) + B). Remarks 3.8.30 (a) Later (5.5.8) we will see the relevance of the metrizability condition in (i). (b) For locally compact K we have compactness of A (3.8.3), which implies that (i) T (A) is compact, hence complete, (ii) A + B is closed. Observe that absolute convexity (and even metrizability) of A can be dropped. For non-locally compact K this goes wrong. In fact, let λ1 , λ2 , . . . ∈ BK such that r := inf{|λn − λm | : n = m} > 0, choose µ1 , µ2 , . . . ∈ BK with |µ1 | > |µ2 | > · · · , limn µn = 0. Set X := {(λn , µn ) : n ∈ N} ⊂ K 2 . Then X is a complete compactoid. Now let T : K 2 → K be the projection (α, β) → β. Then T (X) = {µ1 , µ2 , . . .}. We see that 0 ∈ T (X) but 0 ∈ T (X), so T (X) is not closed. We will prove in 6.2.3 and 6.2.4 that when K is spherically complete the symbol ( )e in 3.8.29 can be eliminated without harm if A is c-compact (no metrizability condition is needed). However, for non-spherically complete K we are not so fortunate, as we show in the next example. Example 3.8.31 Let K be not spherically complete. Then we have the following: (i) There is a closed, absolutely convex, edged and compactoid subset A of c0 and a T ∈ L(c0 ) such that T (A) is not closed. (ii) There is a closed absolutely convex, edged and compactoid subset B of c0 and a y ∈ c0 such that aco {y} + B is not closed. Proof. We first show that (i) follows from (ii). So let B be as in (ii) and consider the space E := K × c0 . The map T : E → E given by T ((α, x)) := (0, α y + x)

(α ∈ K, x ∈ c0 )

is linear and continuous and maps the complete absolutely convex compactoid A := {(α, x) : α ∈ BK , x ∈ B} onto (0, aco {y} + B). Now (i) follows after observing that E c0 . Next we prove (ii). Let t ∈ (0, 1) and choose a t-orthogonal sequence e1 , e2 , . . . in c0 of non-zero vectors such that limn en = 0. Put B := aco {e1 , e2 , . . .}. As K is not spherically complete there exist λ1 , λ2 , . . . ∈ BK and µ1 , µ2 , . . . ∈ K with B(λ1 , |µ1 |) ⊃ B(λ2 , |µ2 |) ⊃

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· · · and n∈N B(λn , |µn |) = ∅. Then inf n∈N |µn | > 0 and for m > n we have λm ∈ B(λn , |µn |), whence |λn − λm | ≤ |µn | (m ≥ n). (3.15) ∞ −1 The formulas y := n=1 µ−1 n en , z := n=1 λn µn en define elements y, z ∈ c0 . We will complete the proof by showing that z ∈ aco {y} + B but z ∈ m −1 −1 Ky + B. Putting ym := m n=1 µn en and zm := n=1 λn µn en we have, using (3.15), ∞

zm − λm ym =

m

(λn − λm ) µ−1 n en ∈ aco {e1 , . . . , em } ⊂ B,

n=1

so that zm − λm (ym − y) ∈ λm y + B ⊂ aco {y} + B. Now limm (ym − y) = 0, limm zm = z and we obtain z ∈ aco {y} + B. Now suppose z ∈ Ky + B. Then z − α y ∈ B = aco {e1 , e2 , . . .} for some ∞ −1 α ∈ K. Since z − α y has the expansion n=1 (λn − α) µn en we must −1 have |λn − α| µn ≤ 1, i.e., |λn − α| ≤ |µn | for each n, implying that α ∈ n∈N B(λn , |µn |), which is a contradiction. Remark 3.8.32 Now we are able to provide an example of two absolutely convex sets A, B for which Ae + B e is strictly contained in (A + B)e (see 3.1.7(a)). In fact, let A := aco {y} and B be as in the proof of (ii) above. It is easily seen that A and B are edged. By 3.8.29(ii) (A + B)e is closed, but A + B is not. We use the previous theory to obtain the two following “pull back principles”. Theorem 3.8.33 (Pull Back Principle 1) Let E, F be Fr´echet spaces, let T ∈ L(E, F ) be a surjection. Then, if Y is a compactoid in F , there is a compactoid X ⊂ E such that T (X) = Y . Proof. By 3.5.10 we may assume that F = E/D, where D is a closed subspace and that T is the canonical quotient map. By 3.8.25 there are y1 , y2 , . . . ∈ E/D, tending to 0, such that λ Y ⊂ aco {y1 , y2 , . . .}. From 3.5.8 we find x1 , x2 , . . . ∈ E, tending to 0, with T (xn ) = yn for each n. Then A := aco {x1 , x2 , . . .} is a complete metrizable compactoid (3.8.24), so by 3.8.29(i) (T (A))e is complete, and we have λ (T (A)) ⊃ (T (A))e ⊃ aco {y1 , y2 , . . .} ⊃ λ Y , hence Y ⊂ T (A). Now take X := A ∩ T −1 (Y ). Remark 3.8.34 In 3.8.33 completeness is essential, i.e., “Fr´echet” cannot be replaced by “metrizable”. In fact, let R := {z ∈ c0 : z = 1}. The map π :

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c00 (R) → c0 given by π((λz )z∈R ) :=

λz z

((λz )z∈R ∈ c00 (R))

z∈R

makes c0 into a quotient of c00 (R). Indeed, let y ∈ c0 , y = 0, let µ ∈ K × with y = |µ|. Define x = (λz )z∈R ∈ c00 (R) by µ if z = µ−1 y λz := 0 otherwise. Then x = |µ| = y and π (x) = y. For each S ⊂ R we view the natural map c00 (S)−→c00 (R) as an inclusion. We now construct a compactoid Y in c0 such that π (X) = Y for no compactoid X in c00 (R), as follows: 1. Let ρ ∈ K, 0 < |ρ| < 1, let Y := {(λ1 , λ2 , . . .) ∈ c0 : |λn | ≤ |ρ|n for all n}. Clearly Y is a compactoid in c0 . Also, the map (µ1 , µ2 , . . .) → (µ1 ρ, µ2 ρ 2 , . . .) is a bijection ∞ → [Y ]. So [Y ] has uncountable dimension. 2. Let D be a subspace of c00 (R), D of countable type. Then D has countable dimension. In fact, D has a 12 -orthogonal base e1 , e2 , . . . (2.3.7). For each n, let en = (λn,z )z∈R and Sn := {z ∈ R : λn,z = 0}. Then Sn is finite, so S = n∈N Sn is countable and en ∈ c00 (S) for each n. As c00 (S) is closed in c00 (R) we have D ⊂ c00 (S). 3. Now let X ⊂ c00 (R) with π (X) = Y . Then π([X]) = [Y ], so [X] must have uncountable dimension. On the other hand, if X were a compactoid then [X] would be of countable type by 3.8.25, hence countably dimensional according to 2. We conclude that there is no compactoid X in c00 (R) such that π(X) = Y . Theorem 3.8.35 (Pull Back Principle 2) Let E, F be Hausdorff locally convex spaces, let A ⊂ E be absolutely convex, complete, metrizable and compactoid. Then, for each T ∈ L(E, F ) and each sequence y1 , y2 , . . . in T (A) converging to 0 there is a sequence x1 , x2 , . . . in λ A converging to 0 such that T (xn ) = yn for each n. Proof. Let µ ∈ K, 1 < |µ|3 ≤ |λ| if the valuation of K is dense, µ := 1 otherwise. There are absolutely convex zero neighbourhoods V1 ⊃ V2 ⊃ . . . in E such that limn Vn ∩ A = 0. By 3.8.9 there is a finite set G1 ⊂ µ A such that A ⊂ (V1 ∩ µ A) + aco G1 , hence T (A) ⊂ T (V1 ∩ µ A) + aco T (G1 ). Now V1 ∩ µ A is closed in µ A hence complete, so by 3.8.29(i), (T (V1 ∩ µ A))e is complete, so by 3.8.22 we have yn ∈ µ (T (V1 ∩ µ A))e for n ≥ n1 . By the same token, replacing in the above V1 by V2 we obtain an n2 > n1 such that

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yn ∈ µ (T (V2 ∩ µ A))e for n ≥ n2 and so on. Inductively we obtain n1 < n2 < · · · such that for each k and all n ≥ nk we have yn ∈ µ (T (Vk ∩ µ A))e ⊂ µ2 T (Vk ∩ µ A) ⊂ T (λ Vk ∩ λ A). Now let n ∈ N. If n ≤ n1 , choose xn ∈ A with T (xn ) = yn . Otherwise there is precisely one k with nk < n ≤ nk+1 . Then choose xn ∈ λVk ∩ λ A such that T (xn ) = yn . To see that limn xn = 0, let k ∈ N, n ≥ nk . Then xn ∈ λ (Vk ∩ A), so λ−1 xn ∈ Vk ∩ A and it follows that limn λ−1 xn = 0. We now prove that the conclusion of 3.8.35 fails for λ = 1. Example 3.8.36 Let K have a dense valuation. Then there exist a closed absolutely convex compactoid subset A of c0 , a T ∈ L(c0 ) and a sequence y1 , y2 , . . . in T (A) converging to 0 such that there is no sequence x1 , x2 , . . . in A converging to 0 with T (xn ) = yn for each n. Hence the restriction map T |A : A−→T (A) is not open. Proof. Choose α1 , α2 , . . . ∈ K with 0 < |αn | < 1 for each n and ∞ n=1 |αn | > 0, and choose ρ ∈ K, 0 < |ρ| < 1. The formula ∞ ∞ λn en ) := (λn − αn λn+1 ρ −1 ) en , T( n=1

n=1

where e1 , e2 , . . . is the standard base of c0 and λn ∈ K, limn λn = 0, defines a map T ∈ L(c0 ). Setting yn := ρ n en we have limn yn = 0, so that A := aco {y1 , y2 , . . .} is a closed compactoid in c0 . We first show that y1 , y2 , . . . are in T (A). In fact, we have yn = T (zn ), with zn := yn + αn−1 yn−1 + αn−2 αn−1 yn−2 + · · · + α1 α2 · · · αn−1 y1 ∈ A. Also, observe that Ker T = Kx, where x := (ρ, ρ 2 α1−1 , ρ 3 α1−1 α2−1 , . . .) ∈ c0 . Now let x1 , x2 , . . . be any sequence in A with T (xn ) = yn for each n. Then xn = zn − µn x for some µn ∈ K. Thus, µn x ∈ A implying |µn | ≤ ∞ i=1 |αi |. The first coefficient of xn (n ≥ 2) in the expansion with respect to y1 , y2 , . . . equals α1 α2 · · · αn−1 − µn , which does not tend to 0 since |µn | < |α1 · · · αn−1 |. It follows that x1 , x2 , . . . cannot tend to 0. Remark 3.8.37 For non-spherically complete K, also the set A of 3.8.31(i) does the job. As a corollary of 3.8.35 we obtain another permanence property. Notice the difference with 3.8.23. Theorem 3.8.38 Let A be a complete absolutely convex metrizable compactoid in a Hausdorff locally convex space (E, τ ). Let τ1 be a Hausdorff locally convex topology on E, weaker than τ . Then τ = τ1 on A.

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Proof. By 3.8.27 A is metrizable with respect to τ1 |A. Let x1 , x2 , . . . ∈ A, τ1 τ xn −→ 0. By the Pull Back Principle 2, xn −→ 0, and we are done. Corollary 3.8.39 Let (E, τ ) be a Fr´echet space. Let τ1 be a Hausdorff locally convex topology on E, τ1 ≤ τ . Then τ = τ1 on compactoids of E. Proof. Let X be a compactoid in E. Apply the previous theorem to A := aco X. Remark 3.8.40 We will prove in 6.2.1 that when K is spherically complete the conclusion of 3.8.38 holds when A is compactoid and complete (no metrizability condition is needed). However, we will show in 5.5.8 that when K is not spherically complete this theorem is false if metrizability is dropped.

3.9 Compactoidity vs orthogonality In this section we extend the concept of orthogonality to systems in locally convex spaces and characterize compactoids as those bounded sets in which each “orthogonal” sequence tends to zero. For more characterizations of compactoidity, see Section 3.10 and 8.4.9. We consider the normed case first. Theorem 3.9.1 Let X be a compactoid in a normed space E, let t ∈ (0, 1]. Then each t-orthogonal sequence in X tends to 0. Proof. Suppose the conclusion were not true. Then there exist a t ∈ (0, 1] and a t-orthogonal sequence e1 , e2 , . . . in X such that r := inf n∈N en > 0. By compactoidity there is a finite-dimensional space D ⊂ E such that X ⊂ B(0, 12 t r) + D. So, for each n ∈ N we can choose a dn ∈ D such that 1 t r < t r ≤ t en . 2 By the Full Perturbation Lemma 2.3.16(i) the sequence d1 , d2 , . . . is torthogonal, hence linearly independent, conflicting finite-dimensionality. en − dn ≤

The proof of the converse (3.9.6) is more involved. First a lemma for discretely valued K. Lemma 3.9.2 Let X be a non-empty bounded subset of a normed space E over a discretely valued field K. Suppose that each orthogonal sequence in X tends to 0. Then, on X, the norm . takes a maximum value. Proof. Let s := sup{x : x ∈ X}. If s is not attained there exist x1 , x2 , . . . ∈ X with 0 < x1 < x2 < · · · , limn xn = s. Let ρ ∈ K be a uniformizing

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159

element. Without loss, assume x1 > |ρ| s. If m, n ∈ N, n > m then it follows easily that |ρ| <

xm < 1, xn

implying that xm and xn are in different cosets of K × , so that x1 , x2 , . . . is orthogonal (2.2.8). Then, by assumption, limn xn = 0, a contradiction. Lemma 3.9.3 Let A be a bounded absolutely convex subset of c0 and suppose that each orthogonal sequence in A tends to 0. Then there exists an orthogonal sequence e1 , e2 , . . . in λ A such that limn en = 0 and A ⊂ aco {e1 , e2 , . . .}. In particular, A is a compactoid. Proof. If the valuation of K is dense, choose µ ∈ K such that 1 < |µ|2 < |λ| and α1 , α2 , . . . ∈ K with |αn | > 1 for each n and ∞ n=1 |αn | ≤ |µ|. If the valuation of K is discrete, put µ := 1, αn := 1 for each n. Set A1 := A. There is an x1 ∈ A1 such that x1 ≥ α1−1 diam A1 (for the discretely valued case, use the previous lemma). By 2.3.19 there is an orthogonal projection P of c0 onto Kx1 . Let x ∈ A1 . Then P (x) = α x1 for some α ∈ K. From α x1 = P (x) ≤ x ≤ diam A1 ≤ |α1 | x1 we conclude that |α| ≤ |α1 |. Thus, P (A1 ) ⊂ aco {α1 x1 } ⊂ α1 A1 . Then (I − P )(A1 ) ⊂ A1 + α1 A1 ⊂ α1 A1 . Set A2 := (I − P )(A1 ). We have A1 ⊂ A2 ⊕ ⊥ aco {α1 x1 }, where the symbol ⊕⊥ indicates that every vector of A2 is orthogonal to every vector in aco {α1 x1 }. Clearly we can further decompose A2 in a similar way obtaining A2 ⊂ A3 ⊕ ⊥ aco {α2 x2 }, where x2 ∈ A2 is such that x2 ≥ α2−1 diam A2 and A3 ⊂ α2 A2 , and so on. Inductively we arrive at an orthogonal sequence x1 , x2 , . . . and absolutely con vex sets A1 , A2 , . . . such that An+1 ⊂ αn An , xn ∈ An , xn ≥ αn−1 diam An and A ⊂ An+1 ⊕ ⊥ aco {α1 x1 , . . . , αn xn }

(3.16)

for each n. For each n ≥ 2 we have xn ∈ An ⊂ α1 · · · αn−1 A1 ⊂ µ A, so the sequence x1 , x2 , . . ., being orthogonal, tends to 0. Then from diam An ≤ |αn | xn ≤ |µ| xn it follows that limn diam An = 0. Hence from (3.16) we obtain A ⊂

160

Locally convex spaces

aco {α1 x1 , α2 x2 , . . .}. Now αn xn ∈ αn µ A ⊂ µ2 A ⊂ λ A. So the lemma is proved after putting en := αn xn . Theorem 3.9.4 Let A be a bounded absolutely convex set in a normed space E. Then A is a compactoid if and only if, for each t ∈ (0, 1], each t-orthogonal sequence in A tends to 0. Proof. The first half follows from 3.9.1. Conversely, suppose each t-orthogonal sequence in A tends to 0 for each t ∈ (0, 1]. By 3.8.12 it suffices to show that each countable subset of A is a compactoid, i.e., we may assume that [A] is of countable type and infinite-dimensional, so its completion [A]∧ is isomorphic to c0 (2.3.9). Thus, with an eye on 2.2.11 and 2.2.12, we may assume E = c0 . Now the proof is completed by using the previous lemma. Next we remove the condition of absolute convexity. Lemma 3.9.5 Let X be a bounded subset of an infinite-dimensional normed space (E, . ) such that aco X is a zero neighbourhood. Then there exist a t ∈ (0, 1] and a t-orthogonal sequence e1 , e2 , . . . in X with inf n∈N en > 0. Proof. The Minkowski function pA of A := aco X is a norm, equivalent to . . Without loss, assume . = pA . Then (3.1.9) {x ∈ E : x < 1} ⊂ A ⊂ {x ∈ E : x ≤ 1}, and, if the valuation of K is discrete, A = {x ∈ E : x ≤ 1}, E = |K| . ∞ Choose t, t1 , t2 , . . . ∈ (0, 1), n=1 tn = t if the valuation of K is dense; t = tn = 1 for all n if the valuation of K is discrete. Inductively we shall construct a sequence e1 , e2 , . . . in X such that dist(en , [em : m < n]) ≥ tn for each n. (Then by 2.2.16 we are done.) First choose e1 ∈ X with e1 ≥ t1 . Suppose we have constructed e1 , . . . , en−1 in the above fashion. To define en , let s ∈ (tn , 1) if the valuation of K is dense, s = 1 otherwise. There is a non-zero vector x ∈ E that is s-orthogonal to Dn−1 := [e1 , . . . , en−1 ] (apply 2.3.13 if the valuation of K is dense and apply 2.3.21 and 2.3.25 if the valuation of K is discrete). By a suitable scalar multiplication we can arrange that x ∈ A, x ≥ tn s −1 . Then dist(x, Dn−1 ) ≥ s x ≥ tn . It follows that there must be also an element en ∈ X with dist(en , Dn−1 ) ≥ tn (otherwise, the closed absolutely convex set {x ∈ E : dist(x, Dn−1 ) < tn } would contain X, hence aco X).

3.9 Compactoidity vs orthogonality

161

Theorem 3.9.6 Let X be a bounded subset of a normed space E. Then X is a compactoid if and only if, for each t ∈ (0, 1], each t-orthogonal sequence in X tends to 0. Proof. Again 3.9.1 takes care of one half. To prove the “if” part, suppose X is not a compactoid. Then by 3.8.12 there is a countable subset Y ⊂ X that is not a compactoid. Then [Y ]∧ is of countable type and we may assume E = [Y ]∧ . By 3.9.4 there is a t1 ∈ (0, 1] and a t1 -orthogonal sequence y1 , y2 , . . . in aco Y with inf n∈N yn > 0. Let D := [y1 , y2 , . . .] and let P : E−→D be a continuous projection onto D (2.3.13). Since aco {y1 , y2 , . . .} is open in D (apply 3.3.11(ii)) and aco {y1 , y2 , . . .} ⊂ P (aco Y ) ⊂ aco P (Y ), the set aco P (Y ) is open in D, so by applying 3.9.5 there is a sequence e1 , e2 , . . . in Y and an s ∈ (0, 1] such that P (e1 ), P (e2 ), . . . is s-orthogonal and r := inf n∈N P (en ) > 0. Let M := supn∈N en . Then for λ1 , . . . , λn ∈ K, we have P

n i=1

λi ei ≥

n

λi P (ei ) ≥ s max |λi | P (ei ) 1≤i≤n

i=1

≥ s r max |λi | ≥ s r M −1 max λi ei , i≤i≤n

1≤i≤n

and it follows that e1 , e2 , . . . is t-orthogonal where t := s r M −1 P −1 . Finally, we obtain inf n∈N en ≥ inf n∈N P −1 P (en ) ≥ r P −1 > 0. Thus, we have found a t-orthogonal sequence in X not tending to 0, a contradiction. For absolutely convex compactoid sets in normed spaces we have the following refinement of 3.8.25. Theorem 3.9.7 Let E be an infinite-dimensional normed space, let A ⊂ E be absolutely convex and compactoid. Then for each t ∈ (0, 1) there is a t-orthogonal sequence e1 , e2 , . . . in λ A such that limn en = 0 and A ⊂ aco {e1 , e2 , . . .}. Proof. We may assume E Banach and E = [A], so E is of countable type. Let t ∈ (0, 1). Choose µ ∈ K such that t < |µ| < 1 if the valuation of K is dense, and put µ := 1 if the valuation of K is discrete. Take t ∈ (0, 1) with t |µ| ≥ t. 1 for all By 2.3.7 E has a t -orthogonal base x1 , x2 , . . . such that 1 ≤ xn ≤ |µ| ∞ n (recall that norms are solid). Then T : c0 −→E, (λ1 , λ2 , . . .) → n=1 λn xn is a surjective linear homeomorphism satisfying t y∞ ≤ T (y) ≤

1 y∞ for all y ∈ c0 . |µ|

162

Locally convex spaces

The above inequalities imply that T maps orthogonal sequences in c0 into torthogonal sequences in E. By compactoidity and 3.9.1 we obtain that every orthogonal sequence in T −1 (A) tends to 0. Now the conclusion follows from 3.9.3 (note that T −1 (A) is absolutely convex and bounded in c0 ). Remark 3.9.8 If dim E = n < ∞ then, in the same spirit, we can find e1 , . . . , en ∈ λ A that are t-orthogonal such that A ⊂ aco {e1 , . . . , en }. Finally, we will discuss the extensions of 3.9.6 and 3.9.7 to the locally convex setting. To this end we extend the notion of orthogonality to the locally convex case. We shall need this concept only for sequences. Definition 3.9.9 Let p be a seminorm on a vector space E, let t ∈ (0, 1]. A sequence e1 , e2 , . . . of non-zero vectors in E is called t-orthogonal with respect to p if for each n ∈ N and λ1 , . . . , λn ∈ K we have n λi ei ≥ t max p(λi ei ). (3.17) p i=1

1≤i≤n

If t = 1 then e1 , e2 , . . . is called orthogonal with respect to p and (3.17) can be written as n p λi ei = max p(λi ei ). i=1

1≤i≤n

Definition 3.9.10 A sequence e1 , e2 , . . . in a locally convex space E is called “orthogonal” sequence if there exists a collection P of continuous seminorms generating the topology such that e1 , e2 , . . . is orthogonal with respect to p for each p ∈ P. Remark 3.9.11 It is easily seen that a sequence e1 , e2 , . . . is “orthogonal” if and only if the collection of all continuous seminorms p for which e1 , e2 , . . . is orthogonal with respect to p forms a base (see 3.3.10) of continuous seminorms. An “orthogonal” sequence in a Hausdorff space is linearly independent. The next theorem enables us to speak about “orthogonal” sequences without specifying a subspace. Theorem 3.9.12 A sequence in a locally convex space E is “orthogonal” in E if and only if it is “orthogonal” in its algebraic linear hull. Proof. The “only if” is obvious. To prove the “if” let e1 , e2 , . . . be “orthogonal” in D := [e1 , e2 , . . .], let P ∗ be a base of continuous seminorms on D for which e1 , e2 , . . . are orthogonal. We may assume that C q ∈ P ∗ for all

3.9 Compactoidity vs orthogonality

163

C ∈ K × , q ∈ P ∗ . Let P := {p : p continuous seminorm on E, p|D ∈ P ∗ }. By the above remark it is enough to prove that P is a base of continuous seminorms on E. For that, let p be a continuous seminorm on E. There is a q ∈ P ∗ such that p ≤ q on D. By 3.4.3 there is also a continuous seminorm p on E such that q = p on D. Now take for x ∈ E ˜ q(x) := inf max(p(x − d1 ), p (x − d2 ), q(d1 ), q(d2 )). d1 ,d2 ∈D

Then q˜ is a seminorm on E extending q and p ≤ q˜ ≤ p ∨ p , so q˜ ∈ P.

An “orthogonal” sequence is, in a natural sense, a base of its closed linear hull. Theorem 3.9.13 Let e1 , e2 , . . . be an “orthogonal” sequence in a Hausdorff locally convex space E. Then every x in the closed linear hull of {e1 , e2 , . . .} can be written uniquely as x = ∞ n=1 λn en , with λn ∈ K. Proof. Let D := [e1 , e2 , . . .]. First, we prove that every x ∈ D can be written in the required way. By 3.9.12 we may assume that E is complete and that E = D. For each n, gn : [e1 , e2 , . . .]−→K, given by gn (em ) = δnm is a continuous linear functional which extends uniquely to an fn ∈ E . We will see that for each x ∈ E, x = ∞ n=1 hn (x), where hn : E−→Ken is defined by hn (x) := fn (x) en . Let P be as in 3.9.10. “Orthogonality” implies that if p ∈ P then p = maxn∈N (p ◦ hn ) on [e1 , e2 , . . .]. Hence {hn |D : n ∈ N} is equicontinuous, then so is {hn : n ∈ N}. Since limn hn (d) = 0 for all d ∈ [e1 , e2 , . . .] we also have limn→∞ hn (x) = 0 for all x ∈ E. By completeness and equicontinuity the formula T (x) := ∞ n=1 hn (x) defines a continuous linear map T : E−→E. But T is the identity on [e1 , e2 , . . .], hence on E and we are done. Next we show uniqueness. Suppose x = ∞ n=1 λn en and also x = ∞ µ e (λ , µ ∈ K). Take m ∈ N. By 3.4.7 there is a seminorm pm ∈ P n n n=1 n n such that pm (em ) = 0. By using “orthogonality” we have 0 = pm (x − x) ≥ |λm − µm | pm (em ), from which we obtain λm = µm . Thus, the scalar coefficients in the series expansion of x are unique. Remark 3.9.14 Note that the coefficient functionals D−→K, x → λn are continuous, so e1 , e2 , . . . is a Schauder base in the sense of 9.1.1. We leave it to the reader to verify that “orthogonality” of 3.9.10 coincides, for normed spaces, with “orthogonality” of 2.2.11. This observation entails the following. Theorem 3.9.15 Let e1 , e2 , . . . be a sequence in a locally convex space E. Then e1 , e2 , . . . is “orthogonal” if and only if there is a base P of continuous

164

Locally convex spaces

seminorms on E and a map p → tp of P into (0, 1] such that, for each p ∈ P, e1 , e2 , . . . is tp -orthogonal with respect to p. Now we arrive at the final aim of this section. Theorem 3.9.16 Let X be a bounded subset of a locally convex space E. Then X is a compactoid if and only if each “orthogonal” sequence in X tends to 0. Proof. Suppose X is a compactoid, let e1 , e2 , . . . be an “orthogonal” sequence in X. Let p be a continuous seminorm on E for which e1 , e2 , . . . is orthogonal with respect to p. We prove that limn p(en ) = 0. In fact, πp (X) is a compactoid in Ep (3.8.16) and πp (e1 ), πp (e2 ), . . . is orthogonal with respect to p, so by 3.9.6 we have limn p(πp (en )) = 0, i.e., limn p(en ) = 0. Conversely, suppose each “orthogonal” sequence in X tends to 0. We derive a contradiction from the assumption that X is not a compactoid. By 3.8.16 there is a continuous seminorm p on E such that πp (X) is not a compactoid in Ep . By 3.9.6 there exists a sequence e1 , e2 , . . . in X such that, for some tp ∈ (0, 1], πp (e1 ), πp (e2 ), . . . is tp -orthogonal in Ep but p(πp (en )) 0, i.e., e1 , e2 , . . . is tp -orthogonal with respect to p and p(en ) 0. Without loss, assume p(en ) ≥ r > 0 for all n. Now let P be the collection of all continuous seminorms on E that are ≥ p. Then P is a base of continuous seminorms. Let q ∈ P. By boundedness of X we have M := supn∈N q(en ) < ∞. For n ∈ N, λ1 , . . . , λn ∈ K we have n n λi ei ≥ p λi ei ≥ tp max |λi | p(ei ) q i=1

1≤i≤n

i=1

≥ tp r max |λi | ≥ tp r M −1 max |λi | q(ei ) 1≤i≤n 1≤i≤n n λi ei . ≥ tp r M −1 q

(3.18)

i=1

We see that e1 , e2 , . . . is tp r M −1 -orthogonal with respect to q. By 3.9.15 e1 , e2 , . . . is “orthogonal”, so the sequence must tend to 0 conflicting p(en ) ≥ r. Remarks 3.9.17 (a) For an interesting class of spaces the locally convex version of 3.9.7 holds (9.2.9(i)). A counterexample to the general case can be found in 9.9.4. (b) For several applications of 3.9.16, see e.g. Section 5.6 and 6.1.13, 8.4.5, 9.2.10.

3.10 Characterization of compactoids by t-frames

165

3.10 Characterization of compactoids in normed spaces by means of t-frames In this section we will introduce – in normed spaces only – the concept of a t-frame, which is more general than t-orthogonality (i.e., every t-orthogonal set is a t-frame, but not conversely) and we prove (3.10.7) a characterization of compactoids in terms of t-frames in the spirit of 3.9.6. Though interesting on its own right, this section can be disregarded by the reader until Section 8.6 where we shall need the main result. One of the advantages of t-frames over t-orthogonal sets will be that for a maximal t-frame X (0 < t < 1) in a normed space E we have [X] = E (3.10.5), a property not shared by t-orthogonal sets X. From now on in this section E = (E, . ) is a normed space. We start off by defining the volume function. Definition 3.10.1 For n ∈ N and x1 , x2 , . . . , xn ∈ E we set Vol(x1 , . . . , xn ) := x1 dist (x2 , [x1 ]) dist (x3 , [x1 , x2 ]) · · · dist (xn , [x1 , . . . , xn−1 ]) n dist (xi , [xj : j < i]). = i=1

Clearly Vol(x1 , . . . , xn ) = 0 if and only if x1 , . . . , xn are linearly indepen dent. We also have Vol(x1 , . . . , xn ) ≤ ni=1 xi . Equality holds if and only if {x1 , . . . , xn } is orthogonal or one of the xi is 0. We will prove that Vol is a symmetric function. We first consider the case where n = 2. Lemma 3.10.2 For all x, y ∈ E we have x dist (y, [x]) = y dist (x, [y]). Proof. We may assume that x, y are linearly independent. By symmetry it suffices to show that, for each r > 1, r x dist (y, [x]) ≥ y dist (x, [y]).

(3.19)

Now (3.19) is obvious if s := dist (y, [x]) = y, so assume 0 < s < y. To prove (3.19) we also may assume r s < y. There is a λ ∈ K × such that

166

Locally convex spaces

y − λ x ≤ r s. Then y = λ x and r x s ≥ x y − λ x = |λ| x λ−1 y − x ≥ λ x dist (x, [y]) = y dist (x, [y]),

and we are done.

Theorem 3.10.3 Let x1 , . . . , xn ∈ E, let σ be a permutation of {1, . . . , n}. Then Vol(x1 , . . . xn ) = Vol(xσ (1) , . . . , xσ (n) ). Proof. It suffices to show that for i ∈ {1, . . . , n − 1}, Vol(x1 , . . . , xi−1 , xi+1 , xi , xi+2 , . . . , xn ) = Vol(x1 , . . . , xn ), which boils down to showing that dist (xi+1 , [x1 , . . . , xi−1 ]) dist (xi , [x1 , . . . , xi−1 , xi+1 ]) = dist (xi , [x1 , . . . , xi−1 ]) dist (xi+1 , [x1 , . . . , xi ]), in other words, that, with D := [x1 , . . . , xi−1 ], dist (xi+1 , D) dist (xi , [xi+1 ] + D) = dist (xi , D) dist (xi+1 , [xi ] + D). But this is just 3.10.2 replacing E by E/D and x, y by π (xi+1 ), π (xi ) respectively, where π : E−→E/D is the quotient map. The following crucial concept, somewhat weaker than t-orthogonality, will suit our purposes. Definition 3.10.4 Let X ⊂ E \ {0}, let t ∈ (0, 1]. X is called a t-frame if for all n ∈ N and x1 , . . . , xn ∈ X Vol(x1 , . . . , xn ) ≥ t n−1 x1 · · · xn .

(3.20)

Similarly, a sequence x1 , x2 , . . . in E \ {0} is called a t-frame sequence if (3.20) holds for each n ∈ N. Clearly t-frames are linearly independent. Every t-orthogonal set (sequence) is a t-frame (sequence), but the converse does not hold. In fact, let E := c0 with n canonical base e1 , e2 , . . .. Let ρ ∈ K, 0 < |ρ| < 1 and let a := ∞ n=1 ρ en . Then it is easily seen that a, e1 , e2 , . . . is a |ρ|-frame but t-orthogonal for no t ∈ (0, 1]. Subsets of t-frames are t-frames. By Zorn’s Lemma every t-frame is contained in a maximal one. It follows from 3.10.3 that a sequence x1 , x2 , . . . in

3.10 Characterization of compactoids by t-frames

167

E \ {0} is a t-frame sequence if and only if {x1 , x2 , . . .} is a t-frame. Also, a set (sequence) in E \ {0} is a 1-frame (sequence) if and only if it is orthogonal. Now we prove the results announced in the beginning of this section. Notice the clause t < 1 below! Theorem 3.10.5 Let X be a maximal t-frame in E, where 0 < t < 1. Then [X] = E. Proof. It suffices to prove that for each y ∈ E \ [X] we have dist (y, [X]) < t y. (In fact, choose y1 ∈ [X] such that y − y1 < t y. Repeating the argument for y − y1 instead of y we obtain a y2 ∈ [X] such that y − y1 − y2 < t y − y1 < t 2 y. Going on this way we find dist (y, [X]) = 0.) Now by maximality {y} ∪ X is no longer a t-frame, so there exist n ∈ N and x1 , . . . , xn ∈ X \ {0} such that Vol(x1 , . . . , xn , y) < t n y x1 · · · xn . On the other hand, we have Vol(x1 , . . . , xn , y) = dist (y, [x1 , . . . , xn ]) Vol(x1 , . . . , xn ) ≥ dist (y, [X]) t n−1 x1 · · · xn , and we see that dist (y, [X]) < t y.

A second nice property of t-frames is that one can extend the characterization of compactoids in terms of t-orthogonal sequences (3.9.6) to t-frames, see 3.10.7. Lemma 3.10.6 Let X ⊂ E be a compactoid. Then, for each sequence x1 , x2 , . . . in X, lim n Vol(x1 , . . . , xn ) = 0. n

√ Proof. Let ε > 0; we prove that n Vol(x1 , . . . , xn ) < 2 ε for large n. There is an M > 0 such that xn ≤ M for all n. Let E1 := [x1 , x2 , . . .]. By compactoidity there is a finite-dimensional space D ⊂ E1 such that xn ∈ D + B(0, ε)

(3.21)

for all n. Then D is contained in [x1 , . . . , xm ] for some m, so we may assume that D = [x1 , . . . , xm ]. Now (3.21) implies dist (xn , D) ≤ ε

(3.22)

168

Locally convex spaces

for all n. For n > m we have, using (3.22), Vol(x1 , . . . , xn ) = dist (xn , [x1 , . . . , xn−1 ]) · · · dist (xm+1 , D) Vol(x1 , . . . , xm ) n ≤ dist (xi , D) Vol(x1 , . . . , xm ) ≤ ε n−m−1 M m , i=m+1

so that

√ n n Vol(x1 , . . . , xn ) ≤ ( M m ε−m−1 ) ε < 2 ε

for large n.

Corollary 3.10.7 A bounded subset X of E is a compactoid if and only if, for each t ∈ (0, 1], each t-frame sequence in X tends to 0. Proof. By 3.9.6 we only need to prove the “only if” part. So, suppose X is a compactoid having for some t ∈ (0, 1], a t-frame sequence x1 , x2 , . . . not tending to 0. Then, by taking a suitable subsequence, we may assume that, for some r > 0, xn ≥ r for all n. Thus, Vol(x1 , . . . , xn ) ≥ t n−1 r n ≥ (tr)n for all n, which conflicts with 3.10.6.

3.11 Notes The concept of convexity was proposed by Monna in 1958, [158]. A first systematic treatment of non-Archimedean locally convex spaces was carried out by van Tiel in 1965, [227]. In 3.5.11 we mentioned some extensions of the Closed Graph Theorem, the Uniform Boundedness Principle and the Banach–Steinhaus Theorem given in Chapter 2 for Banach spaces. It was Prolla who, as far as we know, proved these extensions for the first time [186], even for spaces over valued division rings. Compactoidity was introduced, under the name precompactness, by Gruson and van der Put in [100]. Some other variants of “compact like” sets have been considered in the non-Archimedean literature, see [198] for an impression. The Ascoli Theorem 3.8.2 was originally given by De Grande-De Kimpe in [70]. Extended versions for various spaces of continuous functions can be found in [125], [149] and [176]. Theorem 3.8.9 has been proved for normed spaces by van Rooij, [193], and for locally convex spaces by Katsaras, [111]. Our proof is somewhat simplified.

3.11 Notes

169

The Pull Back Principle 3.8.33 is due to De Grande-De Kimpe, [71]. Also, the proof of 3.8.25 is a slight modification of the one of 2.3 of [71]. Combining the ideas of both proofs we even have, see [76], that if A is an absolutely convex metrizable compactoid in a locally convex space E, then there exist e1 , e2 , . . . ∈ λ A such that limn en = 0 and A ⊂ {x ∈ E : x = ∞ n=1 λn en , |λn | ≤ 1 for all n} (λ ∈ K as in Section 3.8). Absolutely convex subsets of (K-)vector spaces have a richer algebraic structure than their Archimedean counterparts; in fact, they are modules over the commutative ring BK . This observation gives rise to the study of locally convex BK -modules, initiated in [202] and continued in [146], [147], [168] and [169]. Among other things, this theory provides new proofs of compactoidity results, like the Pull Back Principle 3.8.35, see [202]. Note also that module theory is used in [210]. The permanence properties of 3.8.29 and 3.8.38 are taken from [199] and [203] respectively. In the last paper one also can find 3.8.36 and more stability properties for compactoid sets. From one of them, [203], 9.6(ii), we derive the following. Let K be not spherically complete, let A be an absolutely convex subset of a Banach space E. If for each Banach space F and each T ∈ L(E, F ) the image T (A) is closed then A is finite-dimensional (compare with 3.8.31). The concept of orthogonality in the locally convex setting (3.9.10) is due to De Grande-De Kimpe, [65]. For normed spaces, the characterization 3.9.4 of compactoidity for absolutely convex bounded sets, in terms of orthogonality, was proved by van Rooij in [193]; the assumption of absolute convexity was removed (3.9.6) in [200]. The final general form 3.9.16 was obtained by De Grande-De Kimpe, Ka¸kol, Perez-Garcia and Schikhof in [79]. The concept of a t-frame in a normed space E (3.10.4) as well as the characterization of compactoid subsets of E in terms of t-frames (3.10.7) were given in [96], where it was also proved that E is of countable type if and only if every t-frame in E is countable (t ∈ (0, 1)). One may investigate the possibility of a theory of locally convex spaces over trivially valued fields. Indeed, proceeding in the spirit of Sections 3.1–3.3 for such scalar fields one arrives at the conclusion that absolutely convex sets are subspaces and hence that a locally convex space should have a base of zero neighbourhoods consisting of subspaces (compare with 3.3.16). However, such spaces have been studied in detail in [144], Sections 10–13, which is the main reason for excluding the trivial valuation in this book.

4 The Hahn–Banach Theorem

Duality is an important part of the theory of real or complex locally convex spaces. Among other things, it includes the Hahn–Banach Theorem in its analytic and geometric form, one of the “pillars” of classical Functional Analysis. In this chapter we will discuss the Hahn–Banach Theorem in the nonArchimedean case. A first approach is of course to consider the “classical” one. We will see in 4.1.1 that it works in the non-Archimedean setting as well, provided K is spherically complete. If K is not the situation becomes more exciting and varied. Not only we will present non-trivial (Banach) spaces with zero dual (4.1.12) but – which is more important – we will select classes of locally convex spaces for which a satisfactory duality theory can be build. In fact, relaxing in 4.1.1 below the condition f ≤ p by f ≤ (1 + ε) p for some ε > 0 the conclusion holds for any K, provided the associated normed space Ep is of countable type (4.2.4). This leads to an investigation in Section 4.2 of the class of spaces of countable type (4.2.1) in its own right. Other than the above “(1 + ε)-Hahn–Banach Property” we prove hereditary properties (4.2.13) and a representation theorem (4.2.14). In the same spirit, we also discuss the slightly stronger notion of being “strictly of countable type” (4.2.3). In Section 4.3 we find out which of the examples discussed in the previous chapters are (strictly) of countable type. As a by-product we obtain a space that is strictly of countable type whereas some (dense) subspace is not (4.3.6(ii)). In Section 4.4 we further relax the “(1 + ε)-Hahn–Banach Property” by only asking in 4.2.4 the subspace D to be finite-dimensional. This leads to the large family of the so-called “polar spaces” (4.4.1, 4.4.5), including almost all examples previously treated (4.4.8). It is this class that is basic in the duality theory of this book; from our point of view non-polar spaces are being considered as “pathological”.

170

4.1 A first Hahn–Banach Theorem

171

4.1 A first Hahn–Banach Theorem: spherically complete scalar fields The proof of the analytic form of the Hahn–Banach Theorem for real scalars relies on the fact that a collection of closed bounded intervals with the finite intersection property has a non-empty intersection. The natural translation of this property in the non-Archimedean case is spherical completeness. Theorem 4.1.1 (Analytic form of the Hahn–Banach Theorem) Let E be a vector space over a spherically complete field K, let p be a seminorm on E. ∗ Then, for every subspace D of E and every f ∈ D with |f | ≤ p on D, there ∗ is an extension f ∈ E of f such that f ≤ p on E. Proof. By a standard application of Zorn’s Lemma we may assume that there is an x ∈ E, x ∈ D such that E = D + Kx. A linear extension f of f is determined as soon as λ0 := f (x) is fixed. Also, the requirement f ≤ p on E means that λ0 has to satisfy f (d + µ x) = |f (d) + µ λ0 | ≤ p(d + µ x) for all d ∈ D, µ ∈ K, which is equivalent to |λ0 − f (d)| ≤ p(d − x) for all d ∈ D. So we are done if we find a λ0 ∈ d∈D Bd , where Bd := {λ ∈ K : |λ − f (d)| ≤ p(d − x)}. To this end, observe that, by assumption, for every d1 , d2 ∈ D, |f (d1 ) − f (d2 )| ≤ p(d1 − d2 ) ≤ max(p(d1 − x), p(d2 − x)), from which we derive that Bd1 ∩ Bd2 = ∅. Now the existence of the required λ0 follows from the spherical completeness of K. As an immediate consequence of 4.1.1 we have the following two conclusions. Corollary 4.1.2 Let E be a normed space over a spherically complete field K, let D be a subspace of E. Then every f ∈ D can be extended to an f ∈ E such that f = f . Corollary 4.1.3 Let E be a locally convex space over a spherically complete field K, let D be a subspace of E. Then every f ∈ D can be extended to an f ∈ E. Proof. By 3.4.2 there is a continuous seminorm q on D such that |f | ≤ q on D. This q extends to a continuous seminorm p on E (3.4.3). Now use 4.1.1 to

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find an extension f ∈ E ∗ of f with f ≤ p on E. This last inequality clearly implies that f ∈ E . In 2.5.15 we proved that ∞ is not of countable type. For spherically complete K we have the following extension. Corollary 4.1.4 Let K be spherically complete, let E be an infinitedimensional normed space. Then E is not of countable type. Proof. We may assume E is Banach. By 2.3.9 there is a linear homeomorphism of c0 into E, and by 4.1.2 this leads to a continuous linear surjection E −→c0 , or equivalently (2.5.11) to a continuous linear surjection E −→ ∞ . Hence, since ∞ is not of countable type, neither is E . Remark 4.1.5 If K is not spherically complete the above corollary fails, as we will see in 4.1.12 and 5.5.5. Next we will derive a geometric form of the Hahn–Banach Theorem (4.1.8). First of all we point out the following separation theorem for seminorms which holds for any K. It is the locally convex version of 3.1.11. Theorem 4.1.6 Let A be a closed absolutely convex subset of a locally convex space E, let x ∈ E, x ∈ A. Then we have the following: (i) There is a continuous seminorm p on E with p(E) ⊂ |K| such that p(A) < 1 and p(x) = 1. (ii) If, in addition, A is edged then p can be chosen such that p(A) ≤ 1 and p(x) > 1. Proof. Just follow the same proof as in 3.1.11 observing that, by 3.3.11(ii), the absolutely convex subset U of E considered in that proof, which was maximal with respect to the properties A ⊂ U and x ∈ U , can be chosen to be open. Hence, the Minkowski function of U is a continuous seminorm on E (3.3.15). Then the seminorm p found in the proof of 3.1.11, which meets our requirements, is also continuous. For spherically complete scalar fields we can go one step further by choosing in 4.1.6 for p a seminorm of the form |f |, where f ∈ E . Lemma 4.1.7 Let K be spherically complete. Let A be a closed absolutely convex subset of a locally convex space E and let x ∈ E, x ∈ A. Then we have the following:

4.1 A first Hahn–Banach Theorem

173

(i) There is an f ∈ E with |f (A)| < 1 and f (x) = 1. (ii) If, in addition, A is edged then f can be chosen such that |f (A)| ≤ 1 and |f (x)| > 1. Proof. (i). By 4.1.6,(i) there is a continuous seminorm p on E such that p(A) < 1 and p(x) = 1. For the continuous linear functional g : Kx → K defined by g(x) := 1 we have |g(λ x)| = |λ| = p(λ x) for all λ ∈ K. By the Hahn–Banach Theorem 4.1.1 there is an f ∈ E extending g such that |f (y)| ≤ p(y) < 1 for all y ∈ A. Further, f (x) = g(x) = 1. So this f satisfies the desired conditions. (ii). First let us assume that the valuation of K is discrete. By (i) there is a g ∈ E with |g(A)| < 1 and g(x) = 1. Let y0 ∈ A be such that |g(y0 )| = max{|g(y)| : y ∈ A} < 1. If g(y0 ) = 0 then (ii) follows immediately. If g(y0 ) = 0 then (ii) holds for f := g(y0 )−1 g. Now let us assume that K is densely valued. Since x ∈ A = Ae there is a λ ∈ K with |λ| > 1 such that x ∈ λ A. Applying again (i) we obtain a g ∈ E with |g(λ A)| < 1 and g(x) = 1. Then take f := λ g. As an application we have the following. Theorem 4.1.8 (Geometric form of the Hahn–Banach Theorem) Let K be spherically complete. Let E be a locally convex space and let C1 , C2 be convex subsets of E such that C1 ∩ C2 = ∅ and C1 − C2 is closed. Then there is an f ∈ E with f (C1 ) ∩ f (C2 ) = ∅. Proof. From 4.1.7(i) it follows easily that if C is a closed convex subset of E and x ∈ E, x ∈ C then there is an f ∈ E with f (x) ∈ f (C). Now apply this result for x := 0 and C := C1 − C2 . Remark 4.1.9 The condition “C1 − C2 is closed” required in 4.1.8 is satisfied e.g. if one of the sets is open (3.3.11(ii)) or if one of the sets is closed and the other one is c-compact (6.2.4, see 6.1.1 for the concept of c-compactness). When K is not spherically complete the previous Hahn–Banach theory fails dramatically. In fact, we will show that in this case there exist infinite-dimensional Banach spaces E having trivial dual. Clearly, for such E, we can see that 4.1.3, 4.1.7 and 4.1.8 are false. We need two lemmas, valid for any K. Lemma 4.1.10 ∞ /c0 is spherically complete. Proof. Let π : ∞ → ∞ /c0 be the canonical map. Observe that for every x = (λ1 , λ2 , . . .) ∈ ∞ we have π(x) = lim supn |λn |. To prove spherical completeness let B(y1 , r1 ) ⊃ B(y2 , r2 ) ⊃ · · · be balls in ∞ /c0 such that r1 > r2 > · · · . By induction we can select x1 , x2 , . . . ∈ ∞ with π(xn ) = yn for all n and xn+1 − xn ≤ rn−1 for all n ≥ 2. For each n, let xn := (λn,1 , λn,2 , . . .)

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(λn,i ∈ K, i ∈ N). Define x := (λ1,1 , λ2,2 , . . .). Then x ∈ ∞ and, for each n ≥ 2, π (x) − yn = π (x − xn ) = lim sup λi,i − λn,i ≤ lim sup xi − xn ≤ rn−1 . i

Thus, π(x) ∈

n≥2

i

B(yn , rn−1 ) =

B(yn+1 , rn ) =

n∈N

B(yn , rn ).

n∈N

Lemma 4.1.11 If D is a closed subspace of a spherically complete normed space E then the quotient space E/D is also spherically complete. Proof. Let π : E → E/D be the canonical map and let B(y1 , r1 ) ⊃ B(y2 , r2 ) ⊃ · · · be a nested sequence of balls in E/D such that r1 > r2 > · · · . We can construct a sequence x1 , x2 , . . . in E with π (xn ) = yn for all n and xn+1 − xn ≤ rn−1 for all n ≥ 2. As E is spherically complete there is an x ∈ E such that x − xn ≤ rn−1 for all n ≥ 2. Then π(x) ∈ n≥2 B(yn , rn−1 ) = n∈N B(yn , rn ). So E/D is spherically complete. Theorem 4.1.12 Let K be not spherically complete. Then ( ∞ /c0 ) = {0}. Proof. Write E := ∞ /c0 . Assume there is an f ∈ E , f = 0. From 4.1.10 and 4.1.11 we have that the one-dimensional space E/Ker f is spherically complete. By 2.1.12 we obtain that K is also spherically complete, a contradiction. For later use we make this result into a characterization of spherical completeness of K. Corollary 4.1.13 The following properties on K are equivalent: (α) K is spherically complete. (β) The functional (λ1 , λ2 , . . .) → ∞ n=1 λn on c0 can be extended to an f ∈ ( ∞ ) . (γ ) ( ∞ /c0 ) = {0}. Proof. (α) =⇒ (β) and (γ ) =⇒ (α) follow from 4.1.3 and 4.1.12 respectively. (β) =⇒ (γ ). Let f ∈ ( ∞ ) be as in (β). Define T : ∞ −→ ∞ by T ((λ1 , λ2 , . . .)) := (0, λ1 , λ2 , . . .)

((λ1 , λ2 , . . .) ∈ ∞ ).

Then T is a continuous linear map for which f ◦ T − f = 0 on c0 . If (γ ) would be not true then we would have f ◦ T = f . For every x = (λ1 , λ2 , . . .) ∈ ∞ ,

4.2 A second Hahn–Banach Theorem

175

let y := (λ1 , λ1 + λ2 , λ1 + λ2 + λ3 , . . .) ∈ ∞ . Then x = y − T (y), so that f (x) = f (y − T (y)) = 0. Hence f = 0, which is a contradiction.

4.2 A second Hahn–Banach Theorem: spaces of countable type In this section we prove that for a wide class of locally convex spaces the Hahn–Banach Theorem can be “saved”. We first extend the notion “of countable type” (see 2.3.1) to the locally convex setting. Definition 4.2.1 A seminorm p on a vector space E is called of countable type if the normed space Ep is of countable type. A locally convex topology τ on E is said to be of countable type if every continuous seminorm p is of countable type. Then we say that the locally convex space (E, τ ) is of countable type. From compactoid sets we can produce spaces of countable type in the following way. Theorem 4.2.2 Let X be a compactoid in a locally convex space E. Then [X] is of countable type. Proof. Since compactoidity does not depend on the embedding space (3.8.9) we may assume E = [X]. Let p a continuous seminorm on E. By 3.8.16 πp (X) is a compactoid in the normed space Ep , from which it follows easily that Ep = [πp (X)] is of countable type. Hence so is E. If p, q are seminorms of countable type on a vector space E then r := p ∨ q is again of countable type. (Consider the natural isometry Er −→Ep × Eq .) Also, if p is a seminorm of countable type on E and q is a seminorm on E with q ≤ C p, for some C > 0, then q is of countable type (see Section 2.3). The above implies that, for a locally convex topology τ (or for a locally convex space (E, τ )), to be of countable type it suffices that τ is generated by a collection of seminorms of countable type. So, in particular, we have indeed extended the notion “of countable type” from the normed to the locally convex case. In 2.3.9 we have seen that c0 is the “model” of Banach spaces of countable type. We will show (4.2.14) that c0 is also “generic” for the Hausdorff locally convex spaces of countable type. The spaces of countable type form the non-Archimedean counterpart of the so-called transseparable spaces of the classical theory (see e.g. [171], Section 2.5); the reader should not confuse this with the following.

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Definition 4.2.3 A locally convex space E is said to be strictly of countable type if there is a countable subset of E whose linear hull is dense in E. It is easily seen that if E is strictly of countable type then E is of countable type and that the converse is true if E is metrizable. But will see in 4.3.6(i) that this converse does not always hold. More examples of spaces that are and are not (strictly) of countable type we will meet along this section. Now we can obtain our first goal. Theorem 4.2.4 (Analytic form of the Hahn–Banach Theorem) Let E be a vector space and let p be a seminorm of countable type on E. Then, for every ∗ subspace D of E, for every ε > 0 and for every f ∈ D with |f | ≤ p on D, ∗ there is an extension f ∈ E of f such that f ≤ (1 + ε) p on E. Proof. First, let us assume that p is a norm of countable type and equip E with the topology induced by p. Then f ∈ D . To find the desired extension f we also may suppose that E is Banach and D is closed. By 2.3.13 there is a continuous surjective projection Q : E → D with p(Q(x)) ≤ (1 + ε) p(x) for all x ∈ E. Then f := f ◦ Q meets the requirements. Now let p be any seminorm of countable type on E and let πp : E−→Ep be the natural map. Then fp : πp (D)−→K, πp (d) → f (d) is a well-defined linear functional on the subspace πp (D) of Ep such that fp ≤ p on πp (D). Since Ep is a normed space of countable type we apply the first part of the can ∗ proof to find an extension f p ∈ Ep of fp with f p ≤ (1 + ε) p on Ep . Then f := f p ◦ πp ∈ E ∗ satisfies the desired conditions. Without proof we state the following immediate consequences of 4.2.4, similar to 4.1.2 and 4.1.3 respectively. Corollary 4.2.5 Let E be a normed space of countable type, let D be a subspace of E. Then, for every ε > 0 and every f ∈ D , there is an extension f ∈ E of f with f ≤ (1 + ε) f . Corollary 4.2.6 Let E be a locally convex space of countable type, let D be a subspace of E. Then every f ∈ D can be extended to an f ∈ E . Remark 4.2.7 As an application we deduce that C∨p , the spherical completion of Cp (see Chapter 1), is not of countable type as a Banach space over Cp , and hence it is not separable (we announced this in the introduction of Section 2.3). In fact, C∨p is a spherically complete Banach space over the non-spherically complete field Cp . Hence, with the same proof as in 4.1.12, we obtain that (C∨p ) = {0} as a Cp -Banach space. In particular, continuous linear functionals

4.2 A second Hahn–Banach Theorem

177

defined in one-dimensional subspaces do not have extensions to (C∨p ) . By the above corollary, C∨p cannot be of countable type. The reader may wonder whether the conclusion of 4.2.5 holds for ε = 0. The answer is no. Example 4.2.8 Let K be not spherically complete. Then there exist a subspace D of c0 and an f ∈ D that cannot be extended to f ∈ c0 with f = f . Proof. By non-spherical completeness there exist λ1 , λ2 , . . . ∈ K and r0 , r1 , r2 , . . . ∈ (0, ∞) with r0 > r1 > r2 > · · · such that |λn+1 − λn | ≤ rn for all n and n∈N BK (λn , rn ) = ∅. The valuation of K being dense, we can choose e1 , e2 , . . . in c0 , multiples of the canonical base, such that rn < en < rn−1 for all n. Next define dn := en − en+1 (n ∈ N). Since dn − en < en for all n, by the Perturbation Lemma (2.2.9), we have that d1 , d2 , . . . are orthogonal. Let D := [d1 , d2 , . . .]. There is a unique f ∈ D ∗ with f (dn ) = λn − λn+1 for all n. By orthogonality of the dn , for each d = m n=1 µn dn ∈ D (m ∈ N, µ1 , . . . , µm ∈ K), we have m |f (d)| = µn (λn − λn+1 ) ≤ max |µn | rn 1≤n≤m n=1

≤ max |µn | en = max |µn | dn = d, 1≤n≤m

1≤n≤m

so f ∈ D and f ≤ 1. Suppose there is an extension f ∈ c0 of f with f = f ≤ 1. Set λ := λ1 − f (e1 ). We prove that |λ − λn | ≤ rn for each n, which contradicts the choice of the λn . In fact, for each n we have −en+1 = −e1 + d1 + · · · + dn and hence −f (en+1 ) = λ − λn+1 . Therefore, |λ − λn | ≤ max(|λ − λn+1 | , |λn+1 − λn |) ≤ max(f en+1 , rn ) ≤ max(en+1 , rn ) = rn . The above example plays in c0 , an infinite-dimensional space with an orthogonal base. One may wonder whether there exist similar examples in a finitedimensional setting. In a finite-dimensional normed space with an orthogonal base every subspace is orthocomplemented (2.3.21), so the conclusion of 4.2.5 holds with ε = 0. But we do have the following weird case. Example 4.2.9 Let K be not spherically complete. Then there exist a twodimensional space E, a subspace D of E and an f ∈ D that cannot be extended to f ∈ E with f = f .

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The Hahn–Banach Theorem

Proof. Let E := Kν2 (see 2.3.26). If f ∈ E , f = 0 and if there were a onedimensional subspace D with f |D = f then Ker f is ν-orthogonal to D, an impossibility. Our next goal is to give a geometric form of the Hahn–Banach Theorem for spaces of countable type. First we show that for these spaces 4.1.7(ii) also holds. Lemma 4.2.10 Let E be a space of countable type. Let A be a closed absolutely convex and edged subset of E. Then, for every x ∈ E, x ∈ A, there is an f ∈ E with |f (A)| ≤ 1 and |f (x)| > 1. Proof. Thanks to 4.1.7(ii) we may assume that K is not spherically complete and hence that the valuation of K is dense. By 4.1.6(ii) there is a continuous seminorm p on E such that p(A) ≤ 1 and p(x) > 1. Since the valuation of K is dense there exist λ, µ ∈ K with |µ| > 1 and 1 < |λ| < |µ|−1 p(x). The linear functional g : Kx → K defined by g(x) := λ satisfies |g| ≤ |µ|−1 p on Kx. By 4.2.4 there is an f ∈ E extending g such that |f (A)| ≤ p(A) ≤ 1. Further, |f (x)| = |g(x)| = |λ| > 1. So this f satisfies the desired properties. The following theorem will not come as a surprise to the reader. By using 4.2.10 its proof follows as in 4.1.8. Theorem 4.2.11 (Geometric form of the Hahn–Banach Theorem) Let E be a space of countable type. Let C1 , C2 be convex subsets of E such that C1 ∩ C2 = ∅ and C1 − C2 is closed and edged. Then there is an f ∈ E with f (C1 ) ∩ f (C2 ) = ∅. Next we show that in the above theorem edgedness cannot be dropped when K is not spherically complete, proving also that the conclusion of 4.1.7 does not hold. Example 4.2.12 Let K be not spherically complete. Then for every normed space E of dimension > 1 there exists a two-dimensional closed absolutely convex subset A of E and a point x outside it but such that f (x) ∈ f (A) for every f ∈ E . Proof. Let D be a two-dimensional subspace of E. By 2.1.14(iii) assume without loss D = Kν2 . Take x ∈ D with ν(x) = 1 and A := {y ∈ D : ν(y) < 1}. Suppose there is an f ∈ D with f (x) ∈ f (A). We may assume that |f (A)| < 1 and f (x) = 1. From |f (A)| < 1 we obtain f ≤ 1 and from f (x) = 1 we have f = 1. But, for z ∈ Ker f , z = 0 one verifies ν(x + z) ≥ |f (x + z)| = |f (x)| = 1 = ν(x), so x is ν-orthogonal to z, an impossibility.

4.2 A second Hahn–Banach Theorem

179

The hereditary properties of spaces of countable type are contained in the next result. Theorem 4.2.13 (i) Subspaces, continuous linear images, in particular quotients, of spaces of countable type are of countable type. (ii) If D is a dense subspace of a locally convex space E then D is of countable type if and only if E is of countable type. In particular, the completion of a Hausdorff space of countable type is of countable type. (iii) Products and projective limits of spaces of countable type are of countable type. (iv) Let E be a locally convex space, let E1 ⊂ E2 ⊂ · · · be subspaces of count able type of E such that n∈N En is dense in E. Then E is of countable type. (v) Countable locally convex direct sums and inductive limits of spaces of countable type are of countable type. Proof. Let us denote the class of all spaces of countable type by (S0 ). (i). Let E ∈ (S0 ) and let D be a subspace of E. Let p be a continuous seminorm on D. There is a continuous seminorm q on E whose restriction to D is p (3.4.3). Let i : D−→E be the canonical inclusion. In the commutative diagram EO

πq

j

i

D

/ Eq O

πp

/ Dp

the map j is a linear isometry. Since Eq ∈ (S0 ) we have Dp ∈ (S0 ) (2.3.14). It follows that D ∈ (S0 ). Similarly, let E ∈ (S0 ) and let T : E−→F be a continuous linear map of E to a locally convex space F . If p is a continuous seminorm on T (E) then q := p ◦ T is a continuous seminorm on E, so there is a countable subset X of E whose linear hull is q-dense in E. It follows that T (X) is a countable subset of T (E) whose linear hull is p-dense in T (E), hence T (E) ∈ (S0 ). (ii). By (i) we only need to prove the “only if”. Let E be a locally convex space and let D ∈ (S0 ) be a dense subspace of E. For each continuous seminorm q on E, with restriction p on D, we have that Dp ∈ (S0 ). Now the isometry j in the above diagram has dense image, so Eq ∈ (S0 ). It follows that E ∈ (S0 ).

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The Hahn–Banach Theorem

(iii). We first prove that E1 , E2 ∈ (S0 ) implies E1 × E2 ∈ (S0 ). It suffices to see that (E1 × E2 )p1 ×p2 ∈ (S0 ) for p1 (resp. p2 ) a continuous seminorm on E1 (resp. E2 ) and where (p1 × p2 )(x, y) := max(p1 (x), p2 (y)). But (E1 × E2 )p1 ×p2 is isometrically isomorphic to (E1 )p1 × (E2 )p2 ∈ (S0 ), so (E1 × E2 )p1 ×p2 ∈ (S0 ). It follows that (S0 ) is closed for finite products. For general products, let I be an indexing set and, for each i ∈ I , let Ei ∈ (S0 ). Let p be a continuous seminorm on i∈I Ei . Since the closed unit ball of p is open in the product topology it contains a subset of the form i∈I Ui , where Ui is open in Ei for each i and where Ui = Ei only for i ∈ {i1 , . . . , in } for some n ∈ N. Thus, p factors through nj=1 Eij : i∈I

Ei FF FF F p FFF F"

g

R

/

n

w ww wwp˜ w w w {w

j =1

Eij

where the map g in the diagram is the canonical projection. The seminorm p˜ on nj=1 Eij is continuous, so by what we have just proved there is a countable ˜ in nj=1 Eij . subset X of nj=1 Eij such that the linear hull of X is p-dense Then if Y is a countable subset of i∈I Ei such that g(Y ) = X, the linear hull of Y is p-dense in i∈I Ei . Thus, p is of countable type. Hence (S0 ) is closed for products. It follows from (i) that (S0 ) is also closed for projective limits. (iv). By (ii) we may assume that n∈N En = E. Let p be a continuous seminorm on E. For each n ∈ N, choose a countable set Xn ⊂ En such that the linear hull of Xn is p-dense in En . Then X := n∈N Xn is countable and its linear hull is p-dense in E, so p is of countable type. Therefore, E ∈ (S0 ). (v). First we prove that (S0 ) is closed for countable locally convex direct sums. For that, let E := n∈N En be the locally convex direct sum of a sequence of spaces E1 , E2 , . . . ∈ (S0 ). For m ∈ N, set Fm := m n=1 En considered as a m subspace of E. Them Fm ∼ n=1 En , so by (iii) Fm ∈ (S0 ). Since E is the union of the sequence F1 , F2 , . . . it follows from (iv) that E ∈ (S0 ). Now that the inductive limit of a system of spaces ∈ (S0 ) is in (S0 ) follows from the above and the assertion of (i) for quotients. As a consequence we obtain the following characterization. Theorem 4.2.14 Let E be a Hausdorff locally convex space. Then E is of countable type if and only if it is isomorphic to a subspace of c0I for some set I . Proof. By 4.2.13(i) and (iii) each subspace of any c0I is of countable type. Conversely, if E is of countable type then, for each continuous seminorm p on

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181

E, the Banach space Ep∧ is of countable type, so it is isomorphic to a subspace of c0 (2.3.9). Further, if P is a collection of seminorms generating the topology of E then E is linearly and homeomorphically embedded into p∈P Ep∧ (proof of 3.4.10). Thus, E is isomorphic to a subspace of c0P . Corollary 4.2.15 A locally convex space is metrizable and of countable type if and only if it is isomorphic to a subspace of c0N . Proof. First let us assume that E is metrizable and of countable type. Since its topology is induced by a countable collection of continuous seminorms (3.5.2) we can follow the same arguments as before to conclude that E is isomorphic to a subspace of c0N . Conversely, since c0N is metrizable (3.5.5) and of countable type then so is any of its subspaces. Spaces that are strictly of countable type have the following hereditary properties. Theorem 4.2.16 (i) Continuous linear images, in particular quotients of spaces strictly of countable type are strictly of countable type. (ii) If D is a dense subspace of a locally convex space E and D is strictly of countable type then so is E. In particular, the completion of a Hausdorff space strictly of countable type is strictly of countable type. (iii) Countable products of spaces strictly of countable type are strictly of countable type. (iv) Let E be a locally convex space, let E1 ⊂ E2 ⊂ · · · be subspaces strictly of countable type of E such that n∈N En is dense in E. Then E is strictly of countable type. (v) Countable locally convex direct sums and inductive limits of spaces strictly of countable type are strictly of countable type. Proof. The proofs of (i) and (ii) are straightforward. To prove (iii), let E1 , E2 , . . . be spaces strictly of countable type. For each n, let Sn be a countable set such that [Sn ] is dense in En . Then n

!"#$ {(0, 0, . . . , 0, sn , 0, . . .) : sn ∈ Sn , n ∈ N} is countable and its linear hull is dense in n∈N En . Now we prove (iv). By (ii) we may assume that n∈N En = E. For each n there is a countable set Sn ⊂ En such that [Sn ] is dense in En . Hence n∈N Sn is countable and its linear hull is dense in E. Finally, (v) follows from (i) and (iv).

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Remarks 4.2.17 (a) Uncountable locally convex direct sums of spaces of countable type need not be of countable type, even when the spaces are strictly of countable type. In fact, let E be a vector space endowed with the strongest locally convex topology and let I be a set whose cardinality equals dim E. Then E is isomorphic to K (I ) , i.e., to i∈I Ei , where Ei = K for all i (3.4.22(iv)). Clearly each Ei is strictly of countable type. However, if I is uncountable then E is not of countable type because, by 2.5.8, the supremum norm on c00 (I ) is not of countable type. (b) In 4.3.6 we will see that dense subspaces and uncountable products of spaces strictly of countable type may fail to be strictly of countable type. Problem Is the class of spaces that are strictly of countable type stable under taking closed subspaces or under taking projective limits?

4.3 Examples of spaces (strictly) of countable type In this section we present some examples of locally convex spaces (strictly) of countable type, in addition to the normed ones given in Section 2.5. First observe that the concept of (topological) base given in 2.3.10 trivially extends to locally convex spaces. Certainly spaces with such a base are (strictly) of countable type. Also, the next spaces are of countable type. Definition 4.3.1 A locally convex space E is said to be of finite type if for every continuous seminorm p on E the normed space Ep is finite-dimensional. Clearly K I is of finite type for any index set I , and the same holds for any of its subspaces. Another example of space of finite type is furnished by the weak topology σ (E, E ), defined on a locally convex space E by the family of seminorms {|f | : f ∈ E }. Weak topologies and spaces of finite type will be studied thoroughly in Chapter 5. In 4.3.5 we will see that there are sets I for which K I is not strictly of countable type. Next we discuss when the spaces of continuous functions considered in Subsection 3.7.1 are (strictly) of countable type. From here until 4.3.14 X will be a non-empty zero-dimensional Hausdorff topological space. Notice that (C(X), τs ) is of finite type, so it is of countable type.

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Theorem 4.3.2 The following are equivalent: (α) (BC(X), β0 ) is of countable type. (β) (C(X), τc ) is of countable type. (γ ) Every compact subset of X is ultrametrizable. Proof. (α) =⇒ (β). Since τc ≤ β0 on BC(X) it follows from (α) that (BC(X), τc ) is of countable type. By 4.2.13(ii) we obtain (β) as soon as we see that BC(X) is τc -dense in C(X). To this end, let Y ⊂ X be compact, let f ∈ C(X). By 2.5.23 there is a g ∈ BC(X) with g|Y = f |Y . Hence pY (f − g) = 0, and we are done. (β) =⇒ (γ ). Let Y be a compact subset of X, let πY : C(X)−→C(X)pY be the natural map. The map C(X)pY −→C(Y ), πY (f ) → f |Y is a well-defined linear isometry, which by 2.5.23 is also surjective. Since by (β) the normed space C(X)pY is of countable type, so is C(Y ). It follows from 2.5.24 that Y is ultrametrizable. (γ ) =⇒ (α). Let φ ∈ W0 , let 0 < ε < 1. (Recall that W0 is the collection of all bounded functions φ : X−→K that vanish at infinity, see 3.7.14.) There is a compact subset Y of X with |φ| < ε off Y . By 2.5.23 there is a linear isometry T : C(Y )−→BC(X) with T (g)|Y = g for all g ∈ C(Y ). (The isometry is with respect to the supremum norms . ∞ .) By (γ ) the compact set Y is ultrametrizable, so C(Y ) is of countable type (2.5.24). Let {g1 , g2 , . . .} be a countable set whose linear hull is dense in C(Y ). Call F := {T (gn ) : n ∈ N}. Clearly F is countable. To finish the proof we verify that for every f ∈ BC(X) there is an h ∈ [F ] with pφ (f − h) ≤ ε. We may assume f ∞ ≤ 1 and φ∞ := sup{|φ(x)| : x ∈ X} ≤ 1. There is a g ∈ [g1 , g2 , . . .] with |f (x) − g(x)| ≤ ε for all x ∈ Y , in particular g∞ ≤ 1. Put h := T (g) ∈ [F]. Then |f − h| ≤ ε on Y , from which we obtain |f (x) − h(x)| |φ(x)| ≤ |f (x) − h(x)| φ∞ ≤ ε for all x ∈ Y . Further, h∞ = g∞ ≤ 1 and, since |φ| < ε off Y , we also have |f (x) − h(x)| |φ(x)| ≤ ε (f ∞ ∨ h∞ ) ≤ ε for all x ∈ X \ Y . It follows that pφ (f − h) ≤ ε, so this h satisfies the required conditions.

To describe when (C(X), τs ) and (C(X), τc ) are strictly of countable type we use the following. Lemma 4.3.3 There exists a continuous injection : K N −→BK . Proof. We first construct continuous injections φ : K−→BK and ϕ : BKN −→BK as follows.

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Let ρ ∈ K, 0 < |ρ| < 1, let R be a full set of representatives of BK /BK (0, ρ), R ⊂ BK , 0 ∈ R. According to 12.3 of [195], each λ ∈ K has a unique representation λ = n∈Z µn ρ n , where µn ∈ R, µn = 0 for large negative n. Set φ(λ) := µ0 + µ1 ρ + µ−1 ρ 2 + µ2 ρ 3 + µ−2 ρ 4 + · · · One can check easily that φ is continuous and injective. To define ϕ, let x := (λ1 , λ2 , . . .) ∈ BKN , where λ1 = µ10 + µ11 ρ + µ12 ρ 2 + · · · λ2 = µ20 + µ21 ρ + µ22 ρ 2 + · · · .. . are the respective expansions in the above sense. Now set, with the diagonal procedure, ϕ(x) := µ10 + µ11 ρ + µ20 ρ 2 + µ30 ρ 3 + · · · Straightforward verification shows continuity and injectivity of ϕ. Now φ induces naturally a continuous injection ψ : K N −→BKN and := ϕ ◦ ψ : K N −→BK is the required map. Theorem 4.3.4 The following are equivalent: (α) (C(X), τc ) is strictly of countable type. (β) (C(X), τs ) is strictly of countable type. (γ ) There exists a continuous injection X−→K. Proof. The implication (α) =⇒ (β) is clear because τs ≤ τc on C(X). (β) =⇒ (γ ). Let f1 , f2 , . . . ∈ C(X) be such that [f1 , f2 , . . .] in τs -dense in C(X). Define a map : X−→K N by (x) := (f1 (x), f2 (x), . . .)

(x ∈ X).

Obviously is continuous and, since X is Hausdorff, is injective. Then ◦ (with as in 4.3.3) is a continuous injection X−→BK . (γ ) =⇒ (α). Let ω : X−→K be a continuous injection, let ω−1 : ω(X)−→X be its inverse. We claim that D := [Xn ◦ ω : n ∈ N ∪ {0}] is τc -dense in C(X). In fact, let f ∈ C(X), Y ⊂ X be compact, ε > 0. We find a g ∈ D for which pY (f − g) ≤ ε as follows. The set ω(Y ) is compact and the function f ◦ ω−1 is continuous on ω(Y ). So by the (1.2.17) there is a Weierstrass Theorem polynomial function P such that (f ◦ ω−1 )(λ) − P (λ) ≤ ε for all λ ∈ ω(Y ),

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i.e., |f (y) − (P ◦ ω)(y)| ≤ ε for all y ∈ Y . So we are done after taking g := P ◦ ω. We see that, for (C(X), τc ) and (C(X), τs ), the property of being strictly of countable type depends on K! We have a surprising, interesting consequence. Corollary 4.3.5 Let I be an index set. Then K I is strictly of countable type if and only if I ≤ K. Proof. Take the discrete topology on I . Then K I ∼ (C(I ), τc ) = (C(I ), τs ), so by the above theorem K I is strictly of countable type if and only if there is an injection I −→K (which automatically is continuous), which comes down to saying that I ≤ K. Now we can give two examples already announced before (see comments following 4.2.3 and in 4.2.17(b)). Examples 4.3.6 (i) There exist index sets I such that the space K I (which is of countable type, 4.2.14) is not strictly of countable type. (ii) There exist index sets I such that K I is strictly of countable type, but has a dense subspace that is not strictly of countable type. Proof. If we take I with I > K then K I is not strictly of countable type by 4.3.5, which yields (i). For (ii), choose I with I = K. Then (4.3.5) K I is strictly of countable type. But the dense subspace D := K (I ) (with the induced product topology) is not. In fact, if f1 , f2 , . . . ∈ D then every element of [f1 , f2 , . . .] is zero off the countable set n∈N {i ∈ I : fn (i) = 0}, so that [f1 , f2 , . . .] = D. Now we study when (BC(X), β0 ) is strictly of countable type. Theorem 4.3.7 Let (BC(X), β0 ) be strictly of countable type. Then there exists a continuous injection X−→K. Proof. Since τc ≤ β0 on BC(X) we have that (BC(X), τc ) is strictly of countable type. As in (α) =⇒ (β) of 4.3.2 we can prove that BC(X) is τc -dense in C(X). By 4.2.16(ii) we obtain that (C(X), τc ) is strictly of countable type, and the conclusion follows from 4.3.4. A partial converse of the above theorem is the following. Theorem 4.3.8 Let X be locally compact and σ -compact. Suppose there exists a continuous injection X−→K. Then (BC(X), β0 ) is strictly of countable type.

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Proof. X has a fundamental sequence U1 ⊂ U2 ⊂ · · · of compact open sets. Let ω : X−→K be a continuous injection, let ω−1 : ω(X)−→X be its inverse. We will show that D := [(Xn ◦ ω) ξUm : n ∈ N ∪ {0}, m ∈ N] is β0 -dense in BC(X). To this end, let f ∈ BC(X), φ ∈ W0 , ε > 0. Let us produce a g ∈ D such that pφ (f − g) ≤ ε, that is, |f (x) − g(x)| |φ(x)| ≤ ε for all x ∈ X.

(4.1)

We may assume f ∞ ≤ 1, φ∞ := sup{|φ(x)| : x ∈ X} ≤ 1. Since φ vanishes at infinity there is an m such that {x ∈ X : |φ(x)| ≥ ε} ⊂ Um . The set ω(Um ) is compact and the function f ◦ ω−1 is continuous on ω(Um ). So by the Weierstrass Theorem (1.2.17) there is a polynomial function P such that f ◦ ω−1 − P ≤ ε on ω(Um ), i.e., |f − P ◦ ω| ≤ ε on Um . Set g := (P ◦ ω) ξUm ∈ D. We prove that (4.1) holds for this choice of g. If x ∈ Um then |f (x) − g(x)| |φ(x)| = |f (x) − (P ◦ ω)(x)| |φ(x)| ≤ ε φ∞ ≤ ε. If x ∈ Um then g(x) = 0 and |φ(x)| < ε, so |f (x) − g(x)| |φ(x)| ≤ ε f ∞ ≤ ε. Remarks 4.3.9 With respect to the conditions on X in 4.3.8 it might be worth noting the following: (a) The existence of the continuous injection X−→K does not follow from the first two assumptions on X. In fact, take X := ZIp with I > K. (b) For locally compact X the assertion “X is σ -compact and there exists a continuous injection X−→K” is equivalent to the more natural looking “X is ultrametrizable and separable”. In fact, assume the first statement. Then apply 2.5.35(β) =⇒ (γ ) (the continuous injection X−→K guarantees that each compact subset of X is ultrametrizable). For the converse, 2.5.35(γ ) =⇒ (α) yields σ -compactness. To construct a continuous injection X−→K, let (Bi )i∈I be a partition of X into compact balls (1.1.3). By separability I is countable. Each Bi can be homeomorphically embedded into K (1.2.21) and it is not hard to see that we can arrange, by applying suitable translations, that the images of the Bi are mutually disjoint. By gluing together the embeddings we arrive at a continuous injection X−→K. Next we will give another converse of 4.3.7. The following lemma will be useful for this purpose.

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Lemma 4.3.10 Let R ⊂ BK be a full set of representatives of BK /BK− , let µ1 , . . . , µn ∈ R be mutually distinct points of R and α1 , . . . , αn ∈ K. Then there exists a polynomial function P with P (µi ) = αi for all i ∈ {1, . . . , n} and sup{|P (λ)| : λ ∈ BK } ≤ max1≤i≤n |αi |. Proof. Define the (classical Lagrange basic) polynomials L1 (λ) := .. . Ln (λ) :=

(λ − µ2 ) (λ − µ3 ) · · · (λ − µn ) (µ1 − µ2 ) (µ1 − µ3 ) · · · (µ1 − µn ) (λ − µ1 ) (λ − µ2 ) · · · (λ − µn−1 ) , (µn − µ1 ) (µn − µ2 ) · · · (µn − µn−1 )

and set P (λ) := ni=1 αi Li (λ) (λ ∈ K). It is obvious that P (µi ) = αi for all i ∈ {1, . . . , n}. Also, for λ ∈ BK , |Li (λ)| =

|(λ − µ1 ) · · · (λ − µi−1 ) (λ − µi+1 ) · · · (λ − µn )| |(µi − µ1 ) · · · (µi − µi−1 ) (µi − µi+1 ) · · · (µi − µn )|

= |(λ − µ1 ) · · · (λ − µi−1 ) (λ − µi+1 ) · · · (λ − µn )| ≤ 1, so we have |P (λ)| ≤ max1≤i≤n |αi |.

Theorem 4.3.11 Let X be a discrete space such that X ≤ k, where k is the residue class field of K. Then (BC(X), β0 ) is strictly of countable type. Proof. Let R ⊂ BK be as in 4.3.10. By assumption on cardinalities there exists a (continuous) injection ω : X−→R, let ω−1 : ω(X)−→X be its inverse. We claim that D := [Xn ◦ ω : n ∈ N ∪ {0}] is β0 -dense in BC(X). To see this, let f ∈ BC(X), φ ∈ W0 , ε > 0. We have to find an h ∈ D such that |f (x) − h(x)| |φ(x)| ≤ ε for all x ∈ X.

(4.2)

We may assume f ∞ ≤ 1. Since φ vanishes at infinity the set Y:= {x ∈ X : |φ(x)| ≥ ε} is contained in a compact subset of X and, by discreteness, Y must be finite. By 4.3.10 there exists a polynomial function P such that P = f ◦ ω −1 on ω(Y ) and |P (λ)| ≤ max{|f (y)| : y ∈ Y } ≤ f ∞ for all λ ∈ BK . To finish we prove that the element h := P ◦ ω, which is in D, satisfies (4.2). Let x ∈ X. If x ∈ Y it is clear because f (x) = h(x). Now let

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x ∈ Y . Then |φ(x)| < ε, so we have |f (x) − h(x)| |φ(x)| = |f (x) − (P ◦ ω)(x)| |φ(x)| ≤ ε max(|f (x)| , |P (ω(x))| ≤ ε f ∞ ≤ ε. We do not know of general necessary and sufficient conditions in order that (BC(X), β0 ) is strictly of countable type. For (Cc (X), τi ) (see 3.7.19, 3.7.21) we find out when it is of countable type. Theorem 4.3.12 Let X be locally compact. Then the following are equivalent: (α) (β) (γ ) (δ)

(Cc (X), τi ) is of countable type. (Cc (X), τi ) is strictly of countable type. is σ -compact and each compact subset of X is ultrametrizable. X is ultrametrizable and separable.

Proof. 2.5.35 takes care of (γ ) ⇐⇒ (δ). (α) =⇒ (γ ). The supremum norm on Cc (X) is τi -continuous, so there exist f1 , f2 , . . . ∈ Cc (X) such that [f1 , f2 , . . .] is uniformly dense in Cc (X). For each n, let Yn := supp fn . Suppose there exists a y ∈ X \ n∈N Yn . Choose an f ∈ Cc (X) with f (y) = 0. There are λ1 , . . . , λm ∈ K such that f (x) − m λn fn (x) < |f (y)| for all x ∈ X. Putting x = y we obtain n=1 |f (y)| < |f (y)|, a contradiction. Hence X = n∈N Yn and X is σ -compact. Now let U ⊂ X be compact. To show ultrametrizability we may assume that U is open compact. The natural embedding (C(U ), . ∞ )−→(Cc (X), τi ) is easily seen to be a homeomorphism. So (C(U ), . ∞ ) is of countable type and by 2.5.24 U is ultrametrizable. (γ ) =⇒ (β). Let U1 ⊂ U2 ⊂ · · · be a fundamental sequence of open compact sets. Then (Cc (X), τi ) = lim(C(Uj ), . ∞ ) −→

(see 3.7.54). By assumption and 2.5.24, for each j , (C(Uj ), . ∞ ) is strictly of countable type, hence by 4.2.16(v) so is the inductive limit. (β) =⇒ (α) is trivial. Corollary 4.3.13 Let X be locally compact. Then (α) − (δ) of 4.3.12 are equivalent to (ε) (BCWc (X), τWc ) is (strictly) of countable type. If (α) − (ε) hold then (Cc (X), τi ) = (BCWc (X), τWc ). Proof. If (BCWc (X), τWc ) is of countable type then (Cc (X), τi ) being a subspace by 3.7.22, is of countable type (4.2.13(i)), and we have (α).

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Conversely, let (α)–(δ) hold. Then by ultrametrizability and 3.7.23 we obtain (Cc (X), τi ) = (BCWc (X), τWc ), and the result follows. In the next theorem we show that the spaces of analytic functions considered in Subsection 3.7.2 are strictly of countable type. Theorem 4.3.14 (i) For each r ∈ (0, ∞], A(r) is strictly of countable type. (ii) For each r ∈ [0, ∞), A† (r) is strictly of countable type. Proof. We use the descriptions of A(r) and A† (r) in terms of sequence spaces given in Subsection 3.7.2. First, A(r) is isomorphic to the space E := {(λ0 , λ1 , . . .) ∈ K N∪{0} : limn |λn | s n = 0 for each 0 < s < r}, equipped with the locally convex topology induced by the norms given by (λ0 , λ1 , . . .)s := max |λn | s n , n∈N∪{0}

with 0 < s < r. It is straightforward to see that each x = (λ0 , λ1 , . . .) ∈ E can be written uniquely as x = ∞ n=0 λn en+1 , where e1 = (1, 0, 0, . . .), e2 = (0, 1, 0, . . .), . . . are the unit vectors. Thus, E = [e1 , e2 , . . .] is strictly of countable type and so is A(r). Now A† (r) is isomorphic to lim Esm , where ∞ > s1 > s2 > . . . , limm sm = −→ r, and where for each m ∈ N, Esm := {(λ0 , λ1 , . . .) ∈ K N∪{0} : limn |λn | smn = 0}, equipped with the corresponding norm . sm defined as above by taking s := sm . Each Esm is a normed space strictly of countable type and by 4.2.16(v) its inductive limit is also strictly of countable type, hence so is A† (r). For the space O(A) of 3.7.29 we have the following. Theorem 4.3.15 Let K be algebraically closed and separable. Then, for any A ⊂ K, O(A) is of countable type. If, in addition, A has a fundamental sequence of complete bounded sets then O(A) is strictly of countable type. Proof. For each bounded complete set B ⊂ A, the restriction map (O(A), . B )−→(H (B), . B ), f → f |B is an isometry. Since by 2.5.72 the Banach space (H (B), . B ) is of countable type, also so is (O(A), . B ) and hence so is O(A). If, in addition, A has a fundamental sequence of complete bounded sets, then O(A) is strictly of countable type, because in this case it is a Fr´echet space (3.7.31). Next we discuss the spaces of differentiable functions introduced in Subsection 3.7.3.

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From here until the end of this section X is a non-empty subset of K without isolated points. Clearly the spaces (C n (X), τsn ) (n ∈ {0, 1, . . .}) and (C ∞ (X), τs∞ ) are of finite (hence of countable) type (3.7.43). Theorem 4.3.16 The spaces (C n (X), τcn ) (n ∈ {0, 1, . . .}) and (C ∞ (X), τc∞ ) are of countable type. Proof. By 3.4.31 (C ∞ (X), τc∞ ) is isomorphic to the projective limit of the projective sequence ((C n (X), τcn ))n∈N∪{0} , so by 4.2.13(iii) it suffices to see that for each n ∈ {0, 1, . . .} the space (C n (X), τcn ) is of countable type. To this end, notice that the map (C n (X), τcn )−→(C(X), τc ) × (C(X2 ), τc ) × . . . × (C(X n+1 ), τc ), f → (f, 1 f, . . . , n f ) is an embedding. Since each Xi (i ∈ {1, . . . , n + 1}) is ultrametrizable it follows from 4.3.2 that (C(Xi ), τc ) is of countable type, and by 4.2.13(i) and (iii) so is (C n (X), τcn ). Remark 4.3.17 By using the Weierstrass Theorem for differentiable functions given in [17], we even have that the spaces (C n (X), τcn ) (n ∈ {0, 1, . . .}) and (C ∞ (X), τc∞ ) are strictly of countable type, therefore so are (C n (X), τsn ) (n ∈ {0, 1, . . .}) and (C ∞ (X), τs∞ ) . To describe when the space BC ∞ (X) (3.7.46) is of countable type we need a lemma which also will be used later on. Lemma 4.3.18 Each f ∈ BC ∞ (X) extends uniquely to an f ∈ BC ∞ (X), where X is the closure of X in K. The map f → f is a linear homeomorphism of BC ∞ (X) onto BC ∞ (X). Proof. From Rule III of 2.5.78 we obtain, for each f ∈ BC ∞ (X), n f (z1 ) − n f (z2 ) ≤ z1 − z2 ∞ n+1 f ∞ for all n ∈ {0, 1, . . .}, z1 , z2 ∈ X n+1 . It follows that n f is uniformly continuous on X n+1 and can uniquely be extended to a continuous (bounded) function n+1 hn on X . By continuity hn = n h0 for all n. Hence h0 ∈ BC ∞ (X) and we can take f := h0 . In the next theorem notice the difference with 2.5.84 and 2.5.85. Theorem 4.3.19 BC ∞ (X) is (strictly) of countable type if and only if X is precompact. Proof. Suppose X is precompact, i.e., X is compact. By 2.5.84, for each n ∈ {0, 1, . . .}, BC n (X) is of countable type, hence so is BC ∞ (X), and by 4.3.18

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so is BC ∞ (X). Now BC ∞ (X), being Fr´echet (3.7.47), is automatically also strictly of countable type. Conversely, suppose X is not precompact and let us prove that BC ∞ (X) is not of countable type. From non-precompactness of X and with the same reasoning as in 2.5.90, we find an infinite set I , a subspace D of BC ∞ (X) and a linear map D−→ ∞ (I ) that is a surjective homeomorphism with we endow D with any of the norms . n . Then D, with the topology induced by BC ∞ (X), is isomorphic to ∞ (I ). As by 2.5.15 ∞ (I ) is not of countable type, neither is BC ∞ (X) (apply 4.2.13(i)). The next result gives the differentiable version of 4.3.12 and 4.3.13. Theorem 4.3.20 Let X be locally compact, let W := C ∞ (X) and let n ∈ {0, 1, . . .}. Then the following are equivalent: (α) (β) (γ ) (δ) (ε) (ζ )

(Ccn (X), τWn ) is of countable type. (Cc∞ (X), τW∞ ) is of countable type. (Ccn (X), τWn ) is strictly of countable type. (Cc∞ (X), τW∞ ) is strictly of countable type. X is σ -compact. X is separable.

Proof. Lemma 2.5.35 takes care of (ε) ⇐⇒ (ζ ). Also, it is clear that (γ ) =⇒ (α), (δ) =⇒ (β). For (α) =⇒ (ε) (resp. (β) =⇒ (ε)) we can use the same reasoning as in the first part of (α) =⇒ (γ ) of 4.3.12 replacing Cc (X) by Ccn (X) (resp. by Cc∞ (X)) and τi by τWn (resp. by τW∞ ). (ε) =⇒ (γ ), (δ). Let X be σ -compact with fundamental sequence of compact open sets U1 ⊂ U2 ⊂ · · · . By 2.5.84, for each j the Banach space (C n (Uj ), . n ) is strictly of countable type. Then, since (Ccn (X), τWn ) = lim(C n (Uj ), . n ) −→

(see 3.7.54), we can apply 4.2.16(v) to conclude that the inductive limit is strictly of countable type, and we have (γ ). Now we obtain (δ). For each j , (C ∞ (Uj ), τW∞j ) (with Wj := C ∞ (Uj )) is a Fr´echet space which, as we saw in 3.7.54, is isomorphic to the projective limit of the projective sequence of Banach spaces (C m (Uj ), . m )m∈N∪{0} . By 4.2.13(iii) we derive that (C ∞ (Uj ), τW∞j ) is of countable type and by metrizability it is strictly of countable type. Also, it is easily seen that for each j the natural embedding ϕj : (C ∞ (Uj ), τW∞j )−→(Cc∞ (X), τW∞ ) is continuous, and that Cc∞ (X) = j ∈N ϕj (C ∞ (Uj )). It follows from 4.2.16(i) and (iv) that (Cc (X)∞ , τW∞ ) is strictly of countable type, and we arrive at (δ).

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To finish this section we consider the spaces of differentiable functions introduced in 3.7.55, 3.7.57 and 3.7.64 respectively. Theorem 4.3.21 Let balls in X be compact. Then we have the following: (i) Let W be the collection of all polynomial functions X−→K. Then n ∞ (X) (n ∈ {0, 1, . . .}) and BCW (X) are strictly of countable type. BCW n ∞ (ii) C0 (X) (n ∈ {0, 1, . . .}) and C0 (X) are strictly of countable type. (iii) S(X) is strictly of countable type. Proof. Firstly notice that the above spaces are metrizable, so it suffices to show that they are of countable type. n (i). We only have to see that for each n ∈ {0, 1, . . .} the space BCW (X) is of ∞ countable type. (In fact, since by 3.4.31 BCW (X) is isomorphic to the projective n limit of the projective sequence (BCW (X))n∈N∪{0} , we can apply 4.2.13(iii) to ∞ obtain that BCW (X) is of countable type.) So let n ∈ {0, 1, . . .}. For each j ∈ N, let Bj := {x ∈ X : |x| ≤ j }, Ej := n (X) : supp f ⊂ Bj }. We prove that every Ej is of countable type {f ∈ BCW n and that j ∈N Ej is dense in BCW (X) (then by 4.2.13(iv) we are done). For the first assertion, let j ∈ N and observe that by 3.7.34 there is a constant C > 0 n (f Xm ) ≤ C pBn j (f Xm ) for all f ∈ Ej and all m ∈ {0, 1, . . .}. such that pX n (f Xm ) ≤ Now pBn j (Xm ) ≤ j for all m (apply rule X of 2.5.78), so that pX n C j pBj (f ). Thus, the topology on Ej is induced by the single norm pBn j , hence Ej is isomorphic to (a subspace of) C n (Bj ) which is of countable type by compactness of Bj and 2.5.84. To prove density it is enough to verify n that, for f ∈ BCW (X), limj f ξBj = f in the topology τWn , i.e., that for each n ((f − f ξBj ) Xm ) = 0. But, since f Xm is in C0n (X), m ∈ {0, 1, . . .}, limj pX the proof of 3.7.61 does the job. (ii). With the same arguments as in (i), we only have to see that C0n (X) (n ∈ {0, 1, . . .}) is of countable type. To this end, the map C0n (X)−→C0 (X) × C0 (X 2 ) × . . . × C0 (X n+1 ), f → (f, 1 f, . . . , n f ) is a linear isometry. Since each Xi (i ∈ {1, . . . , n + 1}) is locally compact, σ -compact and ultrametrizable it follows from 2.5.36 that C0 (X i ) is of countable type, and by 4.2.13(i), (iii) so is C0n (X). (iii). Is a direct consequence of (i).

4.4 A third Hahn–Banach Theorem: polar spaces In this section we consider a class of locally convex spaces (“polar spaces”) which is wider than the class of spaces of countable type, and for which we prove a weaker version of the Hahn–Banach Theorem.

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Definition 4.4.1 A seminorm p on a vector space E is called polar if p = sup{|f | : f ∈ E ∗ , |f | ≤ p}. A locally convex space E is said to be polar if its topology is induced by a collection of polar seminorms. Such a topology is called polar. Remark 4.4.2 Observe that for a seminorm p on a vector space E and an f ∈ E ∗ the property |f | ≤ p appearing above is, by solidity assumption, equivalent to |f | ≤ 1 on {x ∈ E : p(x) ≤ 1}. If follows from 4.1.1 that if K is spherically complete then every seminorm is polar. Analogously, from 4.2.4 we deduce that seminorms of countable type are polar. Therefore, we can conclude the following. Theorem 4.4.3 (i) If K is spherically complete then every locally convex space is polar. (ii) Every locally convex space of countable type is polar. The space ∞ is polar but, by 2.5.15, it is not of countable type. More examples of polar and non-polar spaces will be given in 4.4.8. The supremum of a pointwise bounded family of polar seminorms is again polar. From this we easily obtain that a normed space is polar if and only if its topology is generated by a polar norm. We also deduce that for a polar space the polar continuous seminorms form a base. The set of polar seminorms is closed under multiplication by elements of |K|. Moreover, if E, F are vector spaces, p is a polar seminorm on F and T : E−→F is a linear map then p ◦ T is a polar seminorm on E. The following algebraic characterization of polar seminorms will be useful in the sequel. Lemma 4.4.4 Let p be a seminorm on a vector space E, let A := {x ∈ E : p(x) ≤ 1}. Then p is polar if and only if for every x ∈ E, x ∈ A there is an f ∈ E ∗ such that |f (A)| ≤ 1 and |f (x)| > 1. Proof. The “only if” is clear. To prove the “if”, let x ∈ E with p(x) > 0 and let λ ∈ K with |λ| < p(x). By assumption and 4.4.2 there is an f ∈ E ∗ such that |f | ≤ p and |f (x)| > |λ|. Since this is true for all λ ∈ K with |λ| < p(x) we derive that p(x) ≤ sup{|f (x)| : f ∈ E ∗ , |f | ≤ p}, so p is polar. Polar seminorms can also be characterized in terms of the following Hahn– Banach type properties. Theorem 4.4.5 (Analytic form of the Hahn–Banach Theorem) Let p be a seminorm on a vector space E. Then the following are equivalent:

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(α) p is polar. (β) For every one-dimensional subspace D of E, every ε > 0 and every f ∈ ∗ ∗ D with |f | ≤ p on D, there is an extension f ∈ E of f such that f ≤ (1 + ε) p on E. (γ ) For every finite-dimensional subspace D of E, every ε > 0 and every ∗ ∗ D with |f | ≤ p on D, there is an extension f ∈ E of f such that f ∈ f ≤ (1 + ε) p on E. Proof. (α) ⇒ (β). Let D := Kd (d ∈ E, d = 0) be a one-dimensional subspace of E. Fix ε > 0 and f ∈ D ∗ with |f | ≤ p on D. We may assume p(d) > 0 (otherwise, f = 0 and f := 0 satisfies (β)). By (α) there is a g ∈ E ∗ with |g| ≤ p such that (1 + ε)−1 p(d) < |g(d)|. Then f : E → K, x → g(d)−1 g(x) f (d) meets the requirements of (β). (β) =⇒ (γ ). Let D be a finite-dimensional subspace of E and let ε > 0. It suffices to see that there is a surjective projection Q : E → D with p ◦ Q ≤ (1 + ε) p. Then, for every f ∈ D∗ with |f | ≤ p on D, f := f ◦ Q ∈ E ∗ extends f and f ≤ (1 + ε) p on E, so we have (γ ). We are going to prove the existence of such a projection Q by induction on the dimension of D. First let us assume D := K d for some d ∈ E, p(d) > 0 (if p(d) = 0 then any projection of E onto D will do). Choose ε > 0 and µ ∈ K such that (1 + ε )2 < 1 + ε and (1 + ε )−1 p(d) < |µ| < p(d). Then f : D → K given by f (d) := µ defines a linear functional on D with |f | ≤ p on D. By (β) there is an f ∈ E ∗ extending f such that f ≤ (1 + ε ) p on E. Then Q : E−→D, x → µ−1 f (x) d is a projection of E onto D with p ◦ Q ≤ (1 + ε) p. Now let n ∈ N, n ≥ 2 and assume that for every m-dimensional subspace D of E, m < n and every ε > 0 there is a surjective projection Q : E → D with p ◦ Q ≤ (1 + ε) p. We prove that the same property holds for any ndimensional subspace D of E. Let ε > 0 and let ε > 0 be such that (1 + ε )2 < (1 + ε). Choose an n − 1-dimensional subspace M of D. By the induction hypothesis there is a projection R of E onto M with p ◦ R ≤ (1 + ε ) p. There is a non-zero d ∈ D such that R(d) = 0. Also, by the validity of the property for one-dimensional subspaces, there is a projection S of E onto Kd with p ◦ S ≤ (1 + ε ) p. As R ◦ S = 0 the map S ◦ (I − R) is a projection of E onto Kd with p ◦ (S ◦ (I − R)) ≤ (1 + ε) p. Since R ◦ (S ◦ (I − R)) = 0 and (S ◦ (I − R)) ◦ R = 0 it follows that Q := R + S ◦ (I − R) is a projection of E onto M + Kd (that is, onto D) for which p ◦ Q ≤ (1 + ε) p, and the conclusion follows. (γ ) =⇒ (α). Let x ∈ E with p(x) > 1. According to 4.4.4 polarity of p follows as soon as we find an f ∈ E ∗ with |f | ≤ p and |f (x)| > 1. Since

4.4 A third Hahn–Banach Theorem

195

p(x) ∈ |K| there exist λ, µ ∈ K such that |µ| > 1 and 1 < |λ| < |µ|−1 p(x). The linear functional g : Kx → K defined by g(x) := λ satisfies |g| ≤ |µ|−1 p on Kx. By (γ ) there is an extension f ∈ E ∗ of g with |f | ≤ p on E. Further, |f (x)| = |g(x)| = |λ| > 1, so this f satisfies the conditions. In the proof of the previous theorem the following corollary is implicitly shown. Corollary 4.4.6 A seminorm p on a vector space E is polar if and only if for every one(finite)-dimensional subspace D of E and every ε > 0 there is a surjective projection Q : E → D with p ◦ Q ≤ (1 + ε) p. Remark 4.4.7 Taking ε = 0 in the above leads to a falsity as we have already seen in 4.2.9. The same happens if we allow D to be infinite-dimensional. In fact, we have seen in 4.1.13 that if K is not spherically complete then the continuous functional (λ1 , λ2 , . . .) → ∞ n=1 λn on c0 cannot be extended to an element of ( ∞ ) . Before discussing further properties of polar spaces we show the abundance of spaces that are polar. Examples 4.4.8 (a) The normed spaces considered in Subsections 2.5.1−2.5.7 are polar. The locally convex spaces introduced in Section 3.7 are also polar with the possible exception of (Cc (X), τi ) when X is not locally compact (see 3.7.21 and 3.7.22). (b) Any vector space E equipped with the strongest locally convex topology is polar. (Let (ei )i∈I be an algebraic base of E. Then the norms i∈I λi ei → maxi∈I |λi | Ci , where Ci runs through |K| are polar and generate the topology, see 3.4.22.) On the other hand, according to 4.2.13(v) and 4.2.17(a), E is of countable type if and only if I is countable (c) An example of a non-polar normed space is ∞ /c0 for K not spherically complete, since ( ∞ /c0 ) = {0} (4.1.12). Now we can see why not every normed space can be embedded into some ∞ (I ) (see 2.5.14 and compare with 2.5.6). Theorem 4.4.9 For a normed space E the following are equivalent: (α) E can be linearly and isometrically embedded into some ∞ (I ). (β) The norm of E is polar.

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Proof. (α) =⇒ (β) is clear. (β) =⇒ (α). By polarity of the norm, the map x ∈ E → x ∈ BC(BE ), where x (f ) := f (x), f ∈ BE is an isometrical linear embedding. As BC(BE ) ⊂ ∞ (BE ), the conclusion holds for I := BE . Next we study the relations between polar seminorms (spaces) and polar sets. As in the classical case, these sets are defined in the following way. Definition 4.4.10 Let E be a locally convex space and let X ⊂ E. Set X◦ := {f ∈ E : |f (x)| ≤ 1 for all x ∈ X} (the polar of X), X ◦◦ := {x ∈ E : |f (x)| ≤ 1 for all f ∈ X◦ } (the bipolar of X). X is called (a) polar (set) if X = X ◦◦ . In other words, if for every x ∈ E, x ∈ X there is an f ∈ E such that |f (X)| ≤ 1 and |f (x)| > 1 (compare with 4.1.7(ii) and 4.2.10). Note For subspaces 4.4.10 formally conflicts 4.4.1, but we trust the reader to understand which “polarity” is meant in forthcoming cases. The next corollary is a direct application of 4.4.4. Corollary 4.4.11 Let E be a locally convex space, let p be a continuous seminorm on E. Then p is polar if and only if {x ∈ E : p(x) ≤ 1} is a polar subset of E. Remark 4.4.12 Note that each polar set is edged. Further, it follows from 4.4.11 that if U is an edged absolutely convex zero neighbourhood in a locally convex space then U is a polar set if and only if the associated Minkowski function pU is a polar seminorm. Observe that a set X ⊂ E is polar if and only if there is a family F in E such that X = f ∈F {x ∈ E : |f (x)| ≤ 1}. As a direct consequence of 4.4.11 we also obtain the following description of polar sets. Corollary 4.4.13 Let E be a locally convex space. A subset X of E is polar if and only if there is a family P of polar continuous seminorms on E such that X = p∈P {x ∈ E : p(x) ≤ 1}. Intersections of polar sets are polar. We derive some characterizations of polar spaces. Theorem 4.4.14 Let E be a locally convex space. Then the following are equivalent: (α) E is polar. (β) For every continuous seminorm q on E there is a polar continuous seminorm p on E such that q ≤ p. (γ ) The topology of E has a base of polar zero neighbourhoods.

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197

Proof. (α) =⇒ (β). Let P be a family of polar continuous seminorms defining the topology of E and let q be a continuous seminorm on E. There exist pi1 , . . . , pin ∈ P and r ∈ |K| such that q ≤ r maxi≤j ≤n pij . Then p := r maxi≤j ≤n pij is a polar continuous seminorm on E with q ≤ p. (β) =⇒ (γ ). Let U be an absolutely convex zero neighbourhood in E and let pU be its associated Minkowski function. Let us prove that U contains a polar zero neighbourhood. We may assume that U is edged. By (β) there is a polar continuous seminorm p on E with pU ≤ p. Then V := {x ∈ E : p(x) ≤ 1} ⊂ {x ∈ E : pU (x) ≤ 1} = U . So V is a zero neighbourhood in E, V ⊂ U , and by 4.4.11 V is polar. (γ ) =⇒ (α). If B is a base of polar zero neigbourhoods for the topology of E then {pU : U ∈ B} is a family of polar seminorms on E defining its topology. (For the polarity of the pU see 4.4.12.) Thus, E is polar. Remark 4.4.15 In 5.2.16 we will discuss a geometric form of the Hahn– Banach Theorem for polar spaces. Now we give the hereditary properties of polar spaces. Theorem 4.4.16 (i) Subspaces of polar spaces are polar. (ii) If D is a dense subspace of a locally convex space E then D is polar if and only if E is polar. In particular, the completion of a Hausdorff polar space is polar. (iii) If E is a polar space and D is a finite-dimensional subspace of E then E/D is also polar. (iv) Products and projective limits of polar spaces are polar. (v) Locally convex direct sums of polar spaces are polar. Proof. We only pay attention to (iii) and (v). The rest are straightforward. (iii). If suffices to see that, for every polar seminorm p on E, the quotient seminorm pπ of p on E/D is also polar. To this end we use 4.4.5. Let M be a finite-dimensional subspace of E/D. Take ε > 0 and g ∈ M ∗ with |g| ≤ pπ on M. We will find an extension g ∈ (E/D)∗ of g with |g| ≤ (1 + ε) pπ on E/D, and by 4.4.5 we arrive at polarity of pπ . For that, let π : E−→E/D be the canonical map. Then N := π −1 (M) is a finite-dimensional subspace of E containing D and for which π (N) = M. Also, f := g ◦ (π |N ) is a linear functional on N such that |f | ≤ p on N . Since p is polar we can apply 4.4.5 to obtain an f ∈ E ∗ extending f such that f ≤ (1 + ε) p on E. As f (d) = f (d) = 0 for all d ∈ D the map g : E/D−→K, π (x) → f (x) (x ∈ E) is a well-defined linear functional on E/D. It follows easily that g satisfies the required conditions.

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(v). Let E be the locally convex direct sum of the polar spaces Ei (i ∈ I ), where I is an indexing set. For each i, let φi : Ei −→E be the canonical injection. Let q be a continuous seminorm on E. We will find a polar continuous seminorm p on E such that q ≤ p, and then polarity of E follows from 4.4.14. To this end, note that for each i, the seminorm q ◦ φi is continuous, so there is a polar continuous seminorm pi on Ei with q ◦ φi ≤ pi (4.4.14). Each x ∈ E has a unique representation x = i∈I φi (xi ), where xi ∈ Ei for all i and where {i ∈ I : xi = 0} is finite. Set p(x) := maxi∈I pi (xi ). Then p is a seminorm on E and it is continuous since p ◦ φi = pi for each i. Further, for every x ∈ E, we have q(x) = q φi (xi ) ≤ max(q ◦ φi )(xi ) ≤ max pi (xi ) = p(x), i∈I

i∈I

i∈I

so that q ≤ p. It remains to prove that p is polar, or equivalently, that {y ∈ E : p(y) ≤ 1} is a polar subset of E (4.4.11). For that, let x = i∈I φi (xi ) ∈ E with p(x) > 1. We have p(x) = pj (xj ) for some j ∈ I . By polarity of pj there is an fj ∈ Ej such that fj ≤ pj and fj (xj ) > 1. Define f ∈ E by the formula f ( i∈I φi (yi )) := fj (yj ). Then |f | ≤ 1 on {y ∈ E : p(y) ≤ 1}, |f (x)| > 1, and we are done. Remark 4.4.17 4.4.8(c) shows that quotients of polar spaces by infinitedimensional subspaces may not be polar. This also follows from the fact that, by 2.5.6, every (non-necessarily polar) Banach space is a quotient of some c0 (I ), which is polar (recall that our norms are solid!). Inductive limits of polar spaces may also fail to be polar, as we will see in 11.2.11. Curiously the natural converse of 4.4.16(iii) is not known. Problem Let E a locally convex space over a non-spherically complete field K . Let x ∈ E \ {0}. If E/K x is polar, must E be polar?

4.5 Notes The analytic form of the Hahn–Banach Theorem 4.1.1 was proved by Monna in 1946, [156] for a seminorm p whose values have 0 as the only accumulation point, which implies that the valuation of K is discrete. In [39], [56] and [167] the work of Monna was extended, by showing that the result holds for any discretely valued field K and any seminorm p. Finally, Ingleton showed in 1952, [102], that the mentioned theorem holds for any spherically complete field K and any seminorm p.

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Spaces of countable type and polar spaces were introduced by Schikhof in [196], in which the Hahn–Banach Theorem 4.2.4 was obtained (the normed version appeared already in [193]). The Hahn–Banach Theorem 4.4.5 was proved by Perez-Garcia in [175]. In [102] Ingleton gave the following converse of 4.1.2. Suppose that, for every normed space E over K, every subspace D of E and every f ∈ D , there is an extension f ∈ E of f with f = f . Then K is spherically complete. We quote a stronger converse of 4.1.2, which was given by van Rooij in [191] (see also 4.1.13). Theorem 4.5.1 ([191], 5.1) Suppose there exists an infinite-dimensional normed space E with the following property. For every subspace D of E and every f ∈ D , there is an extension f ∈ E of f with f = f . Then K is spherically complete. But a similar converse of 4.2.5 is not known. Problem Let K be not spherically complete. Let E be a normed space such that, for every subspace D of E, for every ε > 0 and for every f ∈ D , there is an extension f ∈ E of f with f ≤ (1 + ε) f . Does it follow that E is of countable type? The central role played by the Hahn–Banach Theorem in Functional Analysis on one hand and the deviation from the classical case on the other, make it worthwhile to discuss various Hahn–Banach-type properties in locally convex spaces, as follows: Let us say that for a seminorm p on a vector space E and a subspace D of E

r r

the Hahn–Banach property holds if, for every f ∈ D ∗ with |f | ≤ p on D, there is an extension f ∈ E ∗ of f such that f ≤ p on E; the almost Hahn–Banach property holds if, for every ε > 0 and every ∗ ∗ D with |f | ≤ p on D, there is an extension f ∈ E of f such that f ∈ f ≤ (1 + ε) p on E.

Also, we will say that a subspace D of a locally convex space E has the weak Hahn–Banach property if every f ∈ D can be extended to an f ∈ E . First we consider the Hahn–Banach property. Theorem 4.5.2 For a Hausdorff locally convex space E the following are equivalent: (α) The Hahn–Banach property holds for every continuous semimorm and for every subspace.

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(β) The Hahn–Banach property holds for every continuous semimorm and for every one-dimensional subspace. (γ ) Either K is spherically complete or dim E ≤ 1. Proof. We only prove (β) =⇒ (γ ). (The first part of (γ ) =⇒ (α) follows from 4.1.1, and the rest is obvious.) Suppose (γ ) is not true, that is, K is not spherically complete and dim E > 1. Select a two-dimensional subspace D of E and a norm . on D making it isometrically isomorphic to Kν2 of 2.3.26. Then . can be extended to a continuous seminorm p on E (3.4.3). Suppose the Hahn–Banach property holds for this p and for every one-dimensional subspace of D. Then it holds also for the norm ν and for every one-dimensional subspace of Kν2 . In particular, the map f : Ke1 −→K, λ e1 → λ has an extension f ∈ (Kν2 ) with norm equals ν(e1 )−1 , where e1 = (1, 0) ∈ Kν2 . Thus, the kernel of f is an orthogonal complement of Ke1 in Kν2 , so this two-dimensional space has an orthogonal base, an impossibility. Hence (β) fails. Theorem 4.5.3 For a locally convex space E the following are equivalent: (α) The Hahn–Banach property holds for every member of some base P of continuous seminorms and all subspaces. (β) The Hahn–Banach property holds for every member of some base P of continuous seminorms and all subspaces of countable type. (γ ) Either K is spherically complete or E is of finite type (see 4.3.1). Proof. Clearly (α) =⇒ (β). (β) =⇒ (γ ). Let E be not of finite type. Then there is a p ∈ P such that Ep is infinite-dimensional. We will show that the Hahn–Banach property holds in Ep for its canonical norm p and every subspace D of countable type of Ep . (Then by taking any infinite-dimensional subspace of countable type D of Ep we obtain that the Hahn–Banach property also holds in D for the restriction of p to it and all its subspaces, hence by 4.5.1 K must be spherically complete, and we are done.) So, let f ∈ D ∗ with |f | ≤ p on D and let X be a countable subset of E such that [πp (X)] is p-dense in D. For each x ∈ [X], set g(x) := f (πp (x)). Then g ∈ [X] and |g| ≤ p on [X]. By (β) there is an extension g ∈ E of g with |g| ≤ p on E. One now checks easily that the formula f (πp (x)) := g(x) defines an f ∈ Ep∗ for which f ≤ p, and that f extends f . (γ ) =⇒ (α). By 4.1.1 we may assume that E is of finite type. Let P be the collection of those continuous seminorms p on E for which the (finitedimensional) normed space (Ep , p) has an orthogonal base. Then P is a base of continuous seminorms. Now let p ∈ P, let D be a subspace of E, and let f ∈ D ∗ with |f | ≤ p on D. Then f = g ◦ πp , where g ∈ πp (D) and |g| ≤ p on πp (D). By 2.3.21 there is a surjective projection Q : Ep −→πp (D) with

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≤ p(y) for all y ∈ Ep . Now f := g ◦ Q ◦ πp ∈ E ∗ extends f and p(Q(y)) f ≤ p. Remark 4.5.4 (α), (β), (γ ) of the previous theorem are not equivalent to: (δ) The Hahn–Banach property holds for every member of some base P of continuous seminorms and all finite-dimensional subspaces. In fact, let K be not spherically complete and let E be an infinite-dimensional vector space equipped with the strongest locally convex topology. Then E does not satisfy (γ ). To see that (δ) holds for E, choose for P the collection of all norms on E for which there exists an orthogonal base. Then P is a base of continuous seminorms and, since every finite-dimensional subspace has a p-orthogonal complement for every p ∈ P (2.5.3), we have (δ). Now we pass to the almost Hahn–Banach property. The next theorem was proved by Schikhof in [196]. Theorem 4.5.5 ([196], 4.2) For a locally convex space the following are equivalent: (α) The almost Hahn–Banach property holds for every continuous seminorm and for every subspace. (β) The almost Hahn–Banach property holds for every continuous seminorm and for every one-dimensional subspace (in other words, every continuous seminorm is polar, see 4.4.5). Locally convex spaces satisfying (α), (β) are called strongly polar in [196]. Theorem 4.2.4 states that spaces of countable type are strongly polar. Problem Let E be a strongly polar space over a non-spherically complete field. Does it follow that E is of countable type? We believe that a solution to this long standing problem (and to the related one posed after 4.5.1) would cause a breakthrough in p-adic analysis. We also have the following related problem. Problem Let E1 , E2 be strongly polar spaces. Is E1 × E2 strongly polar? It is also natural to consider the following four properties for a locally convex space E: (α) The almost Hahn–Banach property holds for every member of some base P of continuous seminorms and all subspaces. (β) The almost Hahn–Banach property holds for every member of some base P of continuous seminorms and all subspaces of countable type.

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The Hahn–Banach Theorem

(γ ) The almost Hahn–Banach property holds for every member of some base P of continuous seminorms and all finite-dimensional subspaces. (δ) The almost Hahn–Banach property holds for every member of some base P of continuous seminorms and all one-dimensional subspaces. Clearly (α) =⇒ (β) =⇒ (γ ). By 4.4.5 (γ ) and (δ) are equivalent to the polarity of E. Problem Let K be not spherically complete. Does (α) imply that E is strongly polar? However, (β) does not imply that E is strongly polar. In fact, let K be not spherically complete and let I be a set such that c0 (I ) has ∞ /c0 as a quotient (2.5.6). Since ( ∞ /c0 ) = {0} (4.1.12) we have that c0 (I ) is not strongly polar. We see that c0 (I ) satisfies (β) for P consisting of its canonical norm. To this end, let D be a subspace of countable type of c0 (I ). Observing that for each x = (λi )i∈I ∈ c0 (I ) its support {i ∈ I : λi = 0} is countable one arrives easily at a countable set J ⊂ I such that, for every element of D, its support is in J . Let M be the (closed) subspace of countable type of all elements of c0 (I ) with support in J . Now let f ∈ D , ε > 0 and let ε > 0 be such that (1 + ε )2 < 1 + ε. By 4.2.5 f extends to an f1 ∈ M with f1 ≤ (1 + ε ) f . Now let f := f1 ◦ P , where P is a continuous projection of c0 (I ) onto M with P ≤ 1 + ε (2.5.9(ii)). Then f ∈ c0 (I ) , extends f , and f ≤ (1 + ε) f . Finally we discuss the weak Hahn–Banach property. Consider the following statements: (α) The weak Hahn–Banach property holds for all subspaces. (β) The weak Hahn–Banach property holds for all subspaces of countable type. (γ ) The weak Hahn–Banach property holds for all finite-dimensional subspaces. (δ) The weak Hahn–Banach property holds for all one-dimensional subspaces. Clearly we have (α) =⇒ (β) =⇒ (γ ) ⇐⇒ (δ). When K is not spherically complete the converses of the above implications are not true. In fact, ∞ satisfies (γ ) but, by 4.1.13, it does not satisfy (β). Also, (β) holds for the space c0 (I ) considered before but (α) fails for this space. (Take a closed subspace D such that c0 (I )/D ∞ /c0 and an x ∈ c0 (I ), x ∈ D. Then D + Kx does not have the weak Hahn–Banach property.) Property (α) was studied by De Grande-De Kimpe and Perez-Garcia in [89]. For (β) see the notes to Chapter 5. Spaces satisfying (γ ), (δ) are called dualseparating. This just means that the continuous linear functionals separate the

4.5 Notes

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points of the space. (For more on dual-separating spaces, see Section 5.1.) One should not think that it is the same as polarity. In fact, the following example shows that there are non-polar spaces satisfying (α)! Example 4.5.6 Let K be not spherically complete. Then there exist non-polar spaces for which the weak Hahn–Banach property holds for all subspaces. Proof. Let E := ∞ /c0 equipped with the locally convex topology τ generated by the canonical norm . on E and the seminorms |f |, where f runs through E ∗ . Then E = E ∗ , so clearly all subspaces of E have the weak Hahn–Banach property. We now derive a contradiction from the assumption that E is polar. There is a polar τ -continuous norm . on E with . ≤ . (4.4.14). By τ -continuity . ≤ C max( . , |f1 | , . . . , |fn |) for some C > 0 and f1 , . . . , fn ∈ E ∗ . Let H := 1≤i≤n Ker fi . Then, for any g ∈ E ∗ with |g| ≤ . , we obtain |g(x)| ≤ x ≤ C x

(x ∈ H ).

Thus, g ∈ (H, . ) . But, since ( ∞ /c0 , . ) = {0} (4.1.12) and H is finitecodimensional, we have that H is . -dense, so that (H, . ) = {0}. It follows that g = 0 on H , conflicting the polarity of . . On the other hand, it was shown in [178] that a metrizable space satisfying (β) is polar (see also 5.8.4). In [193] a non-polar dual-separating Banach space was constructed. Although non-locally convex spaces are not the subject of this book, we will devote a few words in these notes to the problem of extension of continuous linear functionals in the non-locally convex setting. Examples are found among the so called A-normed spaces, i.e., pairs (E, . ) formed by a vector space E and an A-norm . on E (for this last concept see 2.1.2). The study of this kind of spaces was proposed by Monna in 1974, [161]. Clearly A-normed spaces are equipped with the topology defined by the metric associated to the “norm". It is easily seen that, among all seminorms that are ≤ . there is a largest one, . n.a . We point out that if E is an A-normed space then on the dual vector space E of all continuous linear functionals f : E−→K, the map f → f := sup{|f (x)| /x : x ∈ E, x = 0} satisfies the strong triangle inequality, hence is a norm in our terminology. We even have |f (x)| ≤ f xn.a for all x ∈ E. With this in mind it is not difficult to prove that, for instance, ( 2 ) is isometrically isomorphic to ∞ , where 2 is an in Section 2.4. For more examples of non-locally convex A-normed spaces

204

The Hahn–Banach Theorem

and their duals we refer to [59] and [193] where, among other things, it was proved that there are A-normed spaces with trivial dual, no matter whether K is spherically complete or not. One of the main problems concerning extensions of continuous linear functionals in A-normed spaces is the following. Problem Let E be an A-normed space such that the weak Hahn–Banach property holds for all subspaces. Does it follow that E is locally convex? Some partial affirmative answers to this problem were given in [151], [170] and [177]. Next, we pay attention to the following natural question. Do we have separation of convex sets by hyperplanes as in the classical case? To formulate a result in this line we have to define the non-Archimedean substitute of the classical concept of a side of a hyperplane. Recall that in real vector spaces two disjoint convex sets C1 and C2 are said to be separated by a hyperplane H := {x ∈ E : f (x) = r} (f ∈ E ∗ \ {0}, r ∈ R) if f (C1 ) < r and f (C2 ) > r. So C1 and C2 lie “at different sides” of H , the two sides of H being {x ∈ E : f (x) < r} and {x ∈ E : f (x) > r}. Both sides are maximal convex sets that are disjoint with H . Now we consider the non-Archimedean case. Let E be a vector space and let H be a hyperplane in E defined by the equation f (x) = λ, f ∈ E ∗ \ {0}, λ ∈ K. For each x ∈ E \ H , consider the non-empty set Hx := {y ∈ E : |f (y) − f (x)| < |f (x) − λ|}. Note that y ∈ Hx if and only if λ ∈ co {f (x), f (y)}, or equivalently, co {x, y} ∩ H = ∅. We claim that the sets Hx , x ∈ E \ H , are the maximal convex subsets of E that are disjoint with H . In fact, clearly each Hx is convex and Hx ∩ H = ∅. To show maximality, take y ∈ Hx and z ∈ E, z ∈ Hx . Then |f (y) − f (x)| < |f (x) − λ| ≤ |f (z) − f (x)| , so |f (y) − λ| = |f (x) − λ| and |f (y) − f (z)| = |f (z) − f (x)| . We conclude that |f (y) − λ| ≤ |f (y) − f (z)|, i.e., co {y, z} ∩ H = ∅, and maximality follows. Finally, we show that every non-empty convex subset C of E that is disjoint with H is contained in some Hx . For that, choose x ∈ C. For each y ∈ C, co {x, y} is contained in C, so co {x, y} ∩ H = ∅, i.e., y ∈ Hx . Thus, C ⊂ Hx .

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The formula x ∼ y if and only if Hx = Hy defines an equivalence relation on E \ H whose classes are called the sides of H . Contrary to the real case H has infinitely many sides. Now we can define the following: Definition 4.5.7 Given subsets X1 , X2 of a vector space E we say that X1 and X2 are separated by a hyperplane H in E if they are contained in different sides of H . We have the following refinement of the geometric forms of the Hahn–Banach Theorem given in 4.1.8 and 4.2.11 (for a polar version see 5.8.1). Theorem 4.5.8 (Separation of convex sets) Suppose the residue class field k of K has at least three elements. Let E be a locally convex space. Let C1 , C2 be convex subsets of E such that C1 ∩ C2 = ∅ and C1 − C2 is closed. Suppose that either: (i) K is spherically complete, or (ii) E is of countable type and C1 − C2 is edged. Then there is a closed hyperplane in E separating C1 and C2 . Proof. (i). We may assume that C1 and C2 are non-empty sets. By 4.1.8 there is an f ∈ E with f (C1 ) ∩ f (C2 ) = ∅. Now f (C1 ) and f (C2 ) are disjoints balls or singletons in K. Take λ1 ∈ f (C1 ), λ2 ∈ f (C2 ) and let r := |λ1 − λ2 | > 0. Then, for each i ∈ {1, 2} we have BK (λ1 , r) = BK (λ2 , r) ⊃ BK (λi , r − ) ⊃ f (Ci ). Since k has at least three elements there is a λ ∈ BK (λ1 , r) such that λ ∈ BK (λ1 , r − ) ∪ BK (λ2 , r − ). Hence |λ − λ1 | = |λ − λ2 | = |λ1 − λ2 | . Therefore, the closed hyperplane H := {x ∈ E : f (x) = λ} separates C1 and C2 . The proof of (ii) is identical to the one given in (i) but now using 4.2.11 instead of 4.1.8. Remark 4.5.9 The above assumption on k cannot be dropped. In fact, let E := K := Q2 and let C1 := BK− , C2 := BK (1, 1− ) (= {λ ∈ K : |λ| = 1}). Then (i) and (ii) are satisfied. Suppose C1 and C2 can be separated by a hyperplane in E. Then there is a λ ∈ K, λ ∈ C1 ∪ C2 and such that λ ∈ co (C1 ∪ C2 ). But C1 ∪ C2 = BK is convex, a contradiction.

206

The Hahn–Banach Theorem

The notion of side of a hyperplane was introduced by Monna in 1964 in [159] where, among other things, he proved 4.5.8(i). The geometric form of the Hahn–Banach Theorem and the separation of convex sets for spherically complete fields were also studied by Carpentier in [31] and by van Tiel in [227]. The Hahn–Banach Theorem plays a key role in the proof of the classical Krein–Milman Theorem, which states that a convex compact set in a Hausdorff locally convex space over R is the closed convex hull of the set of its extreme points. (Here, a point x of a convex set C is called an extreme point if C \ {x} is convex.) It was Monna in [161] who posed the problem on the existence of a nonArchimedean analogue, which is hard to produce (taking over the above definition of extreme point leads to nothing, compact convex sets are rare, the simple set BK− = {λ ∈ K : |λ| < 1} is, if the valuation of K is dense, not the (absolutely) convex hull of any compact set). For simplicity we work in the sequel with absolutely convex rather than convex sets (which are translates of absolutely convex ones). In [150] a first progress was made – using locally compact scalar fields only to ensure the existence of non-trivial convex compact sets – by defining a notion of extreme point generalizing the real case and by proving a Krein–Milman-like theorem, although those extreme sets are never singleton ones. In [172] a new definition of an extreme set was introduced by using supporting hyperplanes, so as to allow singletons to be extreme sets. Also, the theory was generalized to so-called c-compact sets (see Chapter 6) C in locally convex spaces over spherically complete fields. Again it was proved that such C are the closed convex hull of the set of its extreme points. In [30] the condition of spherical completeness was dropped and the theory was generalized to a certain class of compactoids (c -compact sets). In the above papers the set ext A of extreme points of an absolutely convex set A was defined canonically, yet such ext A are rarely compact. It is not unreasonable to look for compact sets X (preferably contained in ext A) such that A = aco X. It seems that for this one has to leave the canonical approach and select such X by brute force. As a case in point, consider the closed unit ball BE in an n-dimensional space E with an orthonormal base. The above theory yields ext BE = {x ∈ E : x = 1}. Clearly BE = aco ext BE , but also BE = aco {e1 , . . . , en }, where e1 , . . . , en is any orthonormal base. There seems to be no reason why one should prefer one orthonormal base over the other. To avoid having to deal with sets like BK− of above, we may restrict our attention to Krein–Milman (KM)-compactoids, i.e., complete absolutely convex sets A for which there exists a compact X ⊂ A such that A = aco X. We then have the following from [205].

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Let E be a polar Hausdorff space. A subset {xi : i ∈ I } of E is called a free simplex if the formula ϕ((λi )i∈I ) = λi xi i∈I

defines a homeomorphism ϕ of BKI onto aco {xi : i ∈ I }. Theorem 4.5.10 ([205], 6.6) Let A ⊂ E be a KM-compactoid. Then there exists a free simplex X such that A = aco X. If Y is a second free simplex such that A = aco Y then there exists a convex homeomorphism of A onto A sending X onto Y . (A map φ : A−→A is convex if φ(λ x + µ y) = λ φ(x) + µ φ(y) for all x, y ∈ A, λ, µ ∈ BK .) Free simplexes X are compact and minimal in the sense that if Y X, Y compact then aco Y aco X. Concluding we may say that, unlike the set of extreme points, free simplexes generating A are not unique, but they have minimality properties, and are “as unique as one can get” in the sense that any two of them are linked by an automorphism of A. If (E, τ ) is a locally convex space then, among all the topologies of countable type on E coarser than τ (such as the weak topology), there exists obviously a finest one: the topology induced by all τ -continuous seminorms that are of countable type. This topology is denoted by τc.t and is called the topology of countable type associated to τ . It was studied by De Grande-De Kimpe in [72] where, among other things, she presented the following description of τc.t . Theorem 4.5.11 Let (E, τ ) be a locally convex space and put ϒ0 := {(fn )n∈N ⊂ E : (fn )n∈N is equicontinuous and fn → 0 pointwise}. For each S := (fn )n∈N ∈ ϒ0 let pS be the seminorm on E defined by pS := max{|fn | : n ∈ N}.

(4.3)

Then τc.t is the topology generated by {pS : S ∈ ϒ0 }. Proof. Let τϒ0 be the topology on E generated by the seminorms pS , S ∈ ϒ0 . By equicontinuity it follows immediately that τϒ0 ≤ τ . Also, for each S = (fn )n∈N ∈ ϒ0 , the map EpS −→c0 , πpS (x) → (f1 (x), f2 (x), . . .) is a welldefined linear isometry, so pS is of countable type, hence so is τϒ0 . Thus, τϒ0 is a topology of countable type that is ≤ τ . Finally we show that it is the finest one among them. Let υ be a topology of countable type on E, υ ≤ τ , let p a υ-continuous seminorm. Ep is of countable type, so by 2.3.9 we may assume that Ep is a

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subspace of c0 equipped with the restriction of the supremum norm . ∞ . For each n ∈ N let en be the nth unit vector of ∞ and let gn := B(., en ) ∈ c0 , where B : c0 × ∞ −→K is the bilinear form considered in 2.5.11. Clearly gn → 0 pointwise, and for each z ∈ c0 we have z∞ = max |gn (z)| . n∈N

Thus, the formula fn := gn ◦ πp (n ∈ N) defines a sequence in (E, υ) ⊂ (E, τ ) which tends to 0 pointwise. Also, since for all x ∈ E, p(x) = p(πp (x)) = max gn (πp (x)) = max |fn (x)| , n∈N

n∈N

we conclude that (fn )n∈N is equicontinuous in (E, υ) (hence in (E, τ ) because υ ≤ τ ), so that p is τϒ0 -continuous. Therefore υ ≤ τϒ0 , and we are done. Next we apply 4.5.11 to describe the finest topology of countable type on a vector space. Corollary 4.5.12 Let E be a vector space and put ϒ0∗ := {(fn )n∈N ⊂ E ∗ : fn → 0 pointwise}. For each S := (fn )n∈N ∈ ϒ0∗ let pS be the seminorm on E defined as in (4.3). Then the finest topology of countable type on E is the one generated by the collection of seminorms {pS : S ∈ ϒ0∗ }. Proof. The finest topology of countable type is the topology of countable type associated to the strongest locally convex topology τ on E. Then the conclusion follows directly from 4.5.11 after observing that in this case (E, τ ) = E ∗ (3.4.22(ii)) and that every sequence (fn )n∈N in E ∗ converging pointwise to 0 is τ -equicontinuous. Analogously, if (E, τ ) is a locally convex space then, among all the polar topologies on E coarser than τ (the weak topology is again one of them), there exists obviously a finest one: the topology induced by all τ -continuous polar seminorms. This topology is denoted by τpol and is called the polar topology associated to τ . It was studied by De Grande-De Kimpe and Perez-Garcia in [90], where the following description of τpol was obtained. Theorem 4.5.13 ([90], 2.4) Let (E, τ ) be a locally convex space. Then we have the following: (i) Let ϒ := {S ⊂ E : S is equicontinuous} and for each S ∈ ϒ let pS be the polar seminorm on E defined by pS := sup{|f | : f ∈ ϒ}. Then τpol is generated by {pS : S ∈ ϒ}.

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(ii) Let P be a family of seminorms generating τ and for each p ∈ P define p˜ := sup{|f | : f ∈ E ∗ , |f | ≤ p}. Then {p˜ : p ∈ P} is a family of polar seminorms on E generating τpol . The polar version of 4.5.12 is obtained in an easy way. In fact, it is clear that the finest polar topology on a vector space E is the strongest locally convex topology (for its polarity see 4.4.8(b)). Note that if E is a locally convex space with E = {0} (e.g. E := ∞ /c0 when K is not spherically complete, 4.1.12) then τc.t and τpol are indiscrete topologies. In the same spirit one can construct the topology of finite type associated to a given locally convex topology. For details, see 5.3.5.

5 The weak topology

The weak topology on a locally convex space is defined as usual. Spaces for which their weak topology is Hausdorff are called dual-separating and are studied in Section 5.1. In Section 5.2 we prove that, under some restrictions, closed (edged) convex sets are weakly closed (5.2.1, 5.2.2), but we also provide, for non-spherically complete K, an example of a clopen bounded absolutely convex set in c0 that is not weakly closed (5.2.3). A slight modification of the classical Bipolar Theorem (5.2.7) is given, and also the important result that, on compactoids in a polar space, the weak topology and the initial topology coincide (5.2.12). In Section 5.3 we study the spaces of finite type, i.e., the spaces whose topology equals the weak topology. The most relevant result of Section 5.4 is the fact that, in polar spaces, each weakly bounded set is bounded (5.4.5). The theory presented in Sections 5.1–5.4 will be fundamental for duality theory later on. Weakly convergent sequences are the subject of Section 5.5. It turns out that, contrary to the classical case, in many spaces such sequences are convergent (5.5.2). In passing we also prove the important fact that, if K is not spherically complete, ( ∞ ) c0 ! (5.5.5). We continue, in the spirit of Section 5.5, by considering in Section 5.6 weakly (pre)compact sets (that are, in general, not convex); we derive conditions under which such sets are (pre)compact (5.6.1). We take over the classical notion of “admissible topology” in Section 5.7 and define the Mackey topology as the strongest polar admissible topology, if it exists (5.7.5). We show existence if K is spherically complete (5.7.8). For non-spherically complete K existence is an open problem.

5.1 Weak topologies and dual-separating spaces As in the classical case, for a locally convex space we can define the so-called weak topology as follows. 210

5.1 Weak topologies and dual-separating spaces

211

Definition 5.1.1 Let E be a locally convex space. The smallest topology on E for which all f ∈ E are continuous is called the weak topology on E and is denoted by σ (E, E ). The weak topology is locally convex. In fact, it is induced by the family of seminorms {|f | : f ∈ E }. It is of countable type, even of finite type (for more precision on this see 5.3.4). For every x ∈ E the weak topology has as a neighbourhood base at x the sets {y ∈ E : |fi (y) − fi (x)| < ε for all i ∈ {1, . . . , n}} (n ∈ N, f1 , . . . , fn ∈ E , ε > 0). Note that σ (E, E ) is smaller than the initial topology on E and that the duals of E and (E, σ (E, E )) are the same. We often use the terms “weakly closed”, “weakly bounded”, “weakly continuous”, . . . instead of “σ (E, E )-closed”, “σ (E, E )-bounded”, “σ (E, E )σ continuous”, . . . The weak closure of a subset X of E is denoted by X . Observe that a sequence x1 , x2 , . . . in E converges weakly to x ∈ E if and only if limn f (xn ) = f (x) in K for all f ∈ E . Analogously, x1 , x2 , . . . is weakly Cauchy if and only if, for every f ∈ E , f (x1 ), f (x2 ), . . . is a Cauchy (equivalently, convergent) sequence in K. Further, a set X ⊂ E is weakly bounded if and only if f (X) is bounded in K for all f ∈ E . Identifying c0 and ∞ according to 2.5.11 we often write σ (c0 , ∞ ) to indicate the weak topology on c0 . In contrast to the classical case, we have to point out that – even for Hausdorff locally convex spaces – the weak topology may not be Hausdorff. In fact, suppose K is not spherically complete and take E := ∞ /c0 . Then E = {0} (4.1.12) and so σ (E, E ) is the indiscrete topology, which clearly is not Hausdorff. With an eye on this we introduce the class of dual-separating spaces. We already said a few words about them in our discussion related to the weak Hahn– Banach property that was done in the notes to Chapter 4. This section is a good place to show more on dual-separating spaces as related to the weak topology. Definition 5.1.2 Let E be a vector space. We say that a subspace F of E ∗ separates the points of E if for every x ∈ E, x = 0 there is an f ∈ F with f (x) = 0. A locally convex space E is called dual-separating if E separates the points of E. Note that dual-separating spaces are Hausdorff. Also, it is straightforward to check that: Theorem 5.1.3 Every polar Hausdorff space is dual-separating.

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The weak topology

From this and 4.4.3 we may conclude the following. Corollary 5.1.4 Every Hausdorff locally convex space over a spherically complete field and every Hausdorff locally convex space of countable type is dualseparating. Also, the spaces treated in 4.4.8(a) and (b) are dual-separating. On the other hand, there are dual-separating spaces that are not polar, see 4.5.6. Lemma 5.1.5 Let E be a finite-dimensional vector space. If F is a subspace of E ∗ that separates the points of E then F = E ∗ . Proof. Let f1 , . . . , fn (n ≤ dim E) be an algebraic base of F. By assumption the linear map E−→K n , x → (f1 (x), . . . , fn (x)) is injective. Then n = dim E = dim E ∗ , so F = E ∗ . As in the notes to Chapter 4 we say that a subspace D of a locally convex space E has the weak Hahn–Banach property if every f ∈ D can be extended to an f ∈ E. Theorem 5.1.6 Let E be a Hausdorff locally convex space. Then the following are equivalent: (α) E is dual-separating. (β) (E, σ (E, E )) is Hausdorff. (γ ) Every one(finite)-dimensional subspace of E has the weak Hahn–Banach property. (δ) For every one(finite)-dimensional subspace D of E there is a continuous projection of E onto D. Proof. The equivalence (α) ⇐⇒ (β) follows directly from the definition of σ (E, E ) and 3.4.7. (α) =⇒ (γ ). Let D be a finite-dimensional subspace of E, let e1 , . . . , en be an algebraic base of D. Let RD : E −→D , f → f |D be the canonical restriction map. By (α) RD (E ) is a subspace of D that separates the points of D, and by 5.1.5 RD (E ) = D = D ∗ . Hence D has the weak Hahn–Banach property. (γ ) =⇒ (δ). Let D be a one (finite)-dimensional subspace of E. By (γ ) we can find f1 , . . . , fn ∈ E such that fi (ej ) = δi,j for all i, j ∈ {1, . . . , n}. Then notice that P : E → D, x → ni=1 fi (x) ei is a continuous projection of E onto D.

5.2 Weakly closed convex sets

213

(δ) =⇒ (α). Suppose (δ) is true for one-dimensional subspaces. Let x ∈ E, x = 0. Since Kx is dual-separating there is an f ∈ (Kx) such that f (x) = 0. If P : E → Kx is a continuous projection of E onto Kx then g := f ◦ P is in E and g(x) = 0. Hence E is dual-separating. We finish this section by discussing the hereditary properties of dual-separating spaces. Theorem 5.1.7 (i) Subspaces of dual-separating spaces are dual-separating. (ii) If E is a dual-separating space and D is a finite-dimensional subspace of E then E/D is also dual-separating. (iii) Products and projective limits of dual-separating spaces are dualseparating. (iv) Locally convex direct sums of dual-separating spaces are dual-separating. Proof. We only pay attention to (ii). The rest are straightforward. Let π : E−→E/D be the canonical map. Let y ∈ E/D, y = 0, let us find a g ∈ (E/D) with g(y) = 0. There is an x ∈ E, x ∈ D such that y = π (x). Since D ⊕alg Kx is a Hausdorff finite-dimensional space it is normable (3.4.24), hence the linear functional f : D ⊕alg Kx−→K, d + λ x → λ (d ∈ D, λ ∈ K) is continuous (2.1.14(ii)). By 5.1.6 there is an extension f ∈ E of f . As D ⊂ Ker f there exists a unique g ∈ (E/D) such that g ◦ π = f . Then g(y) = g(π (x)) = f (x) = 1 = 0, and we are done. Remark 5.1.8 Let K be not spherically complete. The fact that ( ∞ /c0 ) = {0} (4.1.12) shows that quotients of dual-separating spaces by infinite-dimensional subspaces may not be dual-separating. Completions of dual-separating spaces may also fail to be dual-separating, see the chapter notes. In 11.2.2 we will meet an inductive sequence of dual-separating spaces whose inductive limit is not even Hausdorff (see 11.2.3). Polarity is a notion that is somewhat stronger and more quantitative than dualseparating. In the framework of our theory polar spaces will play a more fundamental role than the dual separating ones.

5.2 Weakly closed convex sets A well-known consequence of the Hahn–Banach Theorem in classical Functional Analysis states that a convex subset (e.g. a subspace) of a real or complex locally convex space is closed if and only if it is weakly closed.

214

The weak topology

With similar proof as the classical one we prove now that in the nonArchimedean case closedness and weak closedness coincide for convex subsets of locally convex spaces over spherically complete fields. Notice that weakly closed always implies closed because the weak topology is smaller than the initial one. Theorem 5.2.1 Let E be a locally convex space over a spherically complete field K. Then every closed convex subset of E is weakly closed. Proof. By carrying out translations we may assume that the closed set A is absolutely convex. Let x ∈ E, x ∈ A. By 4.1.7(i) there is an f ∈ E with |f (A)| < 1 and f (x) = 1. The set {y ∈ E : |f (y)| ≥ 1} is weakly clopen, contains x, and is disjoint with A. Thus, A is weakly closed. For arbitrary scalar fields K and spaces of countable type we have the following restricted form of 5.2.1. Theorem 5.2.2 Let E be a locally convex space of countable type. Then every closed convex edged subset of E is weakly closed. Proof. It is a simple adaptation of the previous one by using now 4.2.10 instead of 4.1.7(i). The next example shows that, for non-spherically complete K, the conclusion of 5.2.2 is false when edgedness is dropped. Example 5.2.3 Let K be not spherically complete. Then there is an absolutely convex closed subset A of c0 that is not weakly closed. Proof. By 2.5.49 there exists on c0 an equivalent norm . for which c0 = |K| and such that no two non-zero vectors are . -orthogonal. So it is enough to prove the example for E := (c0 , . ) instead of c0 (note that E and c0 have the same weak topologies). Let us consider the absolutely convex closed set A := {x ∈ E : x < 1}. We claim that A is not weakly closed. For that we are going to see that the weak closure of A is {x ∈ E : x ≤ 1}. This last set is closed and edged, so it is weakly closed by 5.2.2. Let x ∈ E, x = 1, let ε > 0 and let f1 , . . . , fn ∈ E . We prove that {y ∈ E : |fi (y) − fi (x)| < ε for all i ∈ {1, . . . n}} meets A (hence x is the weak closure of A). In fact, {y ∈ E : fi (y) − fi (x) = 0 for all i ∈ {1, . . . n}} = x + 1≤i≤n Ker fi intersects A. Suppose not. Then x + y ≥ 1 = x for all y ∈ 1≤i≤n Ker fi , i.e., x is orthogo nal to 1≤i≤n Ker fi , a contradiction. The following is a direct consequence of 5.2.1 and 5.2.2.

5.2 Weakly closed convex sets

215

Corollary 5.2.4 Let E be a locally convex space. If either K is spherically complete or E is of countable type then every closed subspace of E is weakly closed. Next we show the necessity of the assumptions on K or E in the above corollary. Theorem 5.2.5 Let E be a locally convex space, let D be a subspace of E. Then the following are equivalent: (α) (E/D) = {0}. (β) For every f ∈ E that vanishes on D we have f = 0. (γ ) D is weakly dense in E. Proof. The proof of (α) ⇐⇒ (β) is straightforward. Also, (γ ) =⇒ (β) follows because the initial and the weak topologies on E have the same duals. To σ prove (β) =⇒ (γ ), suppose (β) holds and there is an x ∈ E, x ∈ D . By 4.2.10 applied to thelocally convex space of countable type (E, σ (E, E )) there is an σ f ∈ E with f D ≤ 1 and |f (x)| > 1. So f vanishes on D but f = 0, which contradicts (β). Corollary 5.2.6 Let K be not spherically complete. Then c0 is weakly dense in ∞ .

Proof. Apply 4.1.12 and 5.2.5.

The following is the p-adic counterpart of the classical Bipolar Theorem. Recall (4.4.10) that for X ⊂ E, we set X ◦ := {f ∈ E : |f (x)| ≤ 1 for all x ∈ X}, X◦◦ := {x ∈ E : |f (x)| ≤ 1 for all f ∈ X ◦ }. Theorem 5.2.7 (Bipolar Theorem) Let A be an absolutely convex subset of a locally convex space E. Then A

σ e

= A◦◦ .

Proof. A◦◦ is weakly closed, absolutely convex and edged, and A ⊂ A◦◦ , so e σ A ⊂ A◦◦ . σ

For the opposite inclusion, let x ∈ E, x ∈ (A )e . By 4.2.10 applied space of countable type (E, σ (E, E )) there is an f ∈ E with again toσ the e ≤ 1 (hence f ∈ A◦ ) and |f (x)| > 1. Thus, x ∈ A◦◦ . f A The next corollary follows.

216

The weak topology

Corollary 5.2.8 An absolutely convex edged subset of a locally convex space is weakly closed if and only if it is polar. Further, from 5.2.1, 5.2.2 and 5.2.8 we deduce the following. Corollary 5.2.9 Let E be a locally convex space. Suppose either K is spherically complete or E is of countable type. Then every closed absolutely convex edged subset (in particular, every closed subspace) of E is polar. Some facts for the strongest locally convex topology are contained in the next result. Theorem 5.2.10 Let E be a vector space equipped with the strongest locally convex topology. Then we have the following: (i) If K is spherically complete every absolutely convex (resp. edged) subset of E is weakly closed (resp. polar). (ii) Every subspace of E is a polar set. Proof. The proof of (i) follows from 5.2.1 (resp. 5.2.9) after noting that, by 3.4.22(i), every convex subset of E is closed. To prove (ii), let D be a subspace of E, let x ∈ E, x ∈ D. Then there is an f ∈ E ∗ such that f (D) = {0} and |f (x)| > 1. By 3.4.22(ii) f ∈ E , so f ∈ D ◦ and x ∈ D ◦◦ . Hence D = D ◦◦ , i.e., D is polar. Remark 5.2.11 If K is not spherically complete the conclusion of (i) is not true. In fact, let E := ∞ equipped with the strongest locally convex topology, let π : E−→E/c0 be the quotient map, let . be the quotient norm on E/c0 . By 4.1.12 we have that the (continuous) seminorm p on E defined by p(x) := π (x) is not polar. Then the closed unit ball of p is an edged subset of E which is not polar (4.4.11). From 5.2.8 we deduce that this ball is not weakly closed. The final aim of this section is to show that for (non-necessarily convex) compactoid sets in polar spaces, closedness and weak closedness coincide. As an application we will obtain a geometric form of the Hahn–Banach Theorem for polar spaces, which was already announced in 4.4.15. The next two results will be crucial for our purpose. Theorem 5.2.12 Let E be a polar space. Then, on compactoids, the weak topology and the initial topology coincide. Proof. By polarity and 4.4.14 the topology of E has a base of polar, hence weakly closed, zero neighbourhoods. Now apply 3.8.23.

5.2 Weakly closed convex sets

217

Theorem 5.2.13 Let E a polar space. If X ⊂ E is a compactoid then so are σ X◦◦ and X . σ

Proof. Only compactoidity of X ◦◦ has to be proved because X ⊂ X◦◦ . Let U be a zero neighbourhood in E, we construct a finite set G ⊂ E such that X ◦◦ ⊂ U + aco G. Let λ ∈ K, |λ| > 1. There is a polar zero neighbourhood V with λ V ⊂ U . Writing A := aco X we have by 5.2.7 σ e A = A◦◦ = X◦◦ . By compactoidity there is a finite set G1 ⊂ E such that A ⊂ V + aco G1 . By σ 3.3.11(iv) and 3.8.20 applied to the weak topology we have V + aco G1 ⊂ (V + aco G1 )e . Therefore, σ e σ e ⊂ V + aco G1 ⊂ (V + aco G1 )e ⊂ λ V X ◦◦ = A + aco λ G1 ⊂ U + aco G, where G := λ G1 .

We now arrive at the desired result. Corollary 5.2.14 Let E be a polar space, let X ⊂ E be a compactoid. Then we have the following: (i) X is closed if and only if X is weakly closed. (ii) X is closed, absolutely convex and edged if and only if X is polar. Proof. The proof of (i) follows from 5.2.12 and 5.2.13, and (ii) is a direct consequence of (i) and 5.2.8. Remark 5.2.15 Let K be not spherically complete. 5.2.6 shows that the conclusions of the above corollary fail when compactoidity of X is dropped. As an application we arrive at a geometric form of the Hahn–Banach Theorem for polar spaces. Theorem 5.2.16 (Geometric form of the Hahn–Banach Theorem for polar spaces) Let E be a polar space. Let C1 , C2 be compactoid convex subsets of E such that C1 ∩ C2 = ∅ and C1 − C2 is closed and edged. Then there is an f ∈ E with f (C1 ) ∩ f (C2 ) = ∅. Proof. From 5.2.14(ii) it follows easily that if C is a closed convex edged compactoid in E and x ∈ E, x ∈ C then there is an f ∈ E with f (x) ∈ f (C). Now apply this result for x := 0 and C := C1 − C2 which, by 3.8.4(vi), is a compactoid.

218

The weak topology

Remark 5.2.17 Let K be not spherically complete. By dropping edgedness the above theorem is false. In fact, an example was already given in 4.2.12. (Observe that the two-dimensional set considered there is bounded, hence compactoid by 3.8.4(viii).) Also, the conclusion fails for non-compactoid sets. For an example, consider the polar normed space ∞ . Let C1 := c0 and let C2 := {x} for some x ∈ ∞ , x ∈ c0 . Suppose there is an f ∈ ( ∞ ) with f (C1 ) ∩ f (C2 ) = ∅, that is, f (x) ∈ f (c0 ). Now f (c0 ) is a subspace of K, f (c0 ) = K, so f (c0 ) = {0}. Thus, f is a non-zero continuous linear functional on ∞ that vanishes on c0 , contradicting ( ∞ /c0 ) = {0} (4.1.12).

5.3 Weak topologies and spaces of finite type As we have already announced in the comments after 4.3.1, we will prove (5.3.4) that locally convex spaces having its weak topology coincide with locally convex spaces of finite type. We recall the definition. Definition 5.3.1 A seminorm p on a vector space E is called of finite type if the normed space Ep is finite-dimensional. A locally convex topology τ on E is said to be of finite type if every continuous seminorm p is of finite type. Then we say that the locally convex space (E, τ ) is of finite type. Note that every space of finite type is of countable type, hence polar. On the other hand, there are locally convex spaces of finite type that are not strictly of countable type (see 4.3.6(i)). If p, q are seminorms of finite type on a vector space E then r := p ∨ q is again of finite type. (Consider the natural isometry Er −→Ep × Eq .) Also, if p is a seminorm of finite type on E and q is a seminorm on E with q ≤ C p, for some C > 0, then q is of finite type. The above implies that, for a locally convex topology τ (or for a locally convex space (E, τ )), to be of finite type it suffices that τ is generated by a collection of seminorms of finite type. Remark 5.3.2 For non-spherically complete K we gave in 4.5.3 a description of spaces of finite type in terms of the Hahn–Banach property. The next theorem characterizes the weakly continuous seminorms among all continuous seminorms on a locally convex space. Theorem 5.3.3 Let E be a locally convex space, let p be a continuous seminorm on E. Then p is weakly continuous if and only if p is of finite type.

5.3 Weak topologies and spaces of finite type

219

Proof. First let us assume that p is weakly continuous. There exist C > 0 and f1 , . . . , fn ∈ E such that p ≤ C max1≤i≤n |fi |. For each i ∈ {1, . . . , n}, |fi | is clearly a seminorm of finite type on E. Hence p is also of finite type. Now let us assume that p is of finite type. Let {πp (e1 ), . . . , πp (en )} be an algebraic base of the finite-dimensional normed space Ep , let c := max1≤i≤n p(πp (ei )). Since all linear functionals on Ep are continuous (2.1.14(ii)) we have that, for i ∈ {1, . . . , n}, the maps gi : Ep −→K, ni=1 λi πp (ei ) → λi (λ1 , . . . , λn ∈ K) are in Ep and satisfy p(πp (x)) ≤ c max1≤i≤n gi (πp (x)) for all x ∈ E. Now fi := gi ◦ πp is in E (i ∈ {1, . . . , n}) and p(x) = p(πp (x)) ≤ c max |fi (x)| 1≤i≤n

Thus, p is weakly continuous.

(x ∈ E).

As a spin-off we characterize spaces of finite type. Corollary 5.3.4 A locally convex space E is of finite type if and only if the initial topology on E coincides with its weak topology. Remark 5.3.5 Let (E, τ ) be a locally convex space. Then the finest topology of finite type on E that is ≤ τ is its weak topology. In fact, for any topology of finite type υ such that σ (E, E ) ≤ υ ≤ τ we clearly have (E, υ) = (E, τ ) . Applying 5.3.4 we obtain that necessarily υ = σ (E, E ). It follows that the finest topology of finite type on a vector space E is the weak topology σ (E, E ∗ ) associated to the strongest locally convex topology on E (see 3.4.22). Clearly a normed space is of finite type if and only if it is finite-dimensional. Also, since every finite-dimensional normed space E is isomorphic to K n , with n = dim E (2.1.15) we have that the spaces K n are the “models” of normed spaces of finite type. We will show (5.3.9) that K is also a “generator” of all Hausdorff locally convex spaces of finite type. With simple adaptations of the proofs carried out in properties (i)−(iii) of 4.2.13 we obtain the following hereditary properties of spaces of finite type. Theorem 5.3.6 (i) Subspaces, continuous linear images, in particular quotients of spaces of finite type are of finite type. (ii) If D is a dense subspace of a locally convex space E then D is of finite type if and only if E is of finite type. In particular, the completion of a Hausdorff space of finite type is of finite type. (iii) Products and projective limits of spaces of finite type are of finite type. We also have the following.

220

The weak topology

Theorem 5.3.7 If every countably generated subspace of a locally convex space E is of finite type then so is E. Proof. Let p be a continuous seminorm on E, let X be a countably infinite set in Ep , let πp : E−→Ep be the natural map. Select a countable set Y ⊂ E with πp (Y ) = X. By assumption [Y ] is of finite type, so there is a non-trivial linear combination of elements of Y in Ker p. Applying πp we conclude that X is not linearly independent. Hence Ep is finite-dimensional. Remark 5.3.8 Infinite locally convex direct sums of spaces of finite type need not be of finite type. In fact, let E be a vector space equipped with the strongest locally convex topology and let I a set whose cardinality is dim E. Then E ∼ i∈I Ei , where Ei = K for all i ∈ I (3.4.22(iv)). Clearly each Ei is of finite type. But if I is infinite then E is not of finite type because the supremum norm on c00 (I ) is not of finite type. In other words, E is of finite type if and only if E is finite-dimensional (compare with 4.4.8(b)). By taking I := N in this example we see that inductive limits of spaces of finite type also may fail to be of finite type. We have the following consequences of 5.3.6, similar to 4.2.14 and 4.2.15 respectively. The proofs are left to the reader. Theorem 5.3.9 A Hausdorff locally convex space is of finite type if and only if it is isomorphic to a subspace of K I for some set I . Corollary 5.3.10 A locally convex space is metrizable and of finite type if and only if it is isomorphic to a subspace of K N . Next we shall characterize the complete spaces of finite type. First, we remark on the following fact, whose verification is elementary. For an index set I the spaces K I and K (I ) are duals of one another via the bilinear form B : K I × K (I ) −→K, (x, y) → i∈I λi µi (x = (λi )i∈I ∈ K I , y = (µi )i∈I ∈ K (I ) ). More precisely, the maps K I −→(K (I ) ) , x → B(x, .), K (I ) −→(K I ) , y → B(., y) are algebraic isomorphisms. In the following theorem we identify K I and K (I ) with their images under these isomorphisms. Theorem 5.3.11 For a Hausdorff locally convex space E the following are equivalent: (α) (β) (γ ) (δ)

E is complete and of finite type. E is isomorphic to a closed subspace of some power of K. E is isomorphic to some power of K. For every Hausdorff locally convex topology τ1 weaker than the initial one τ we have τ1 = τ .

5.3 Weak topologies and spaces of finite type

221

Proof. (α) =⇒ (β) follows from 5.3.9. (β) =⇒ (γ ). We have to see that for any index set I , every closed subspace D of K I is isomorphic to a power of K. For that, notice that K I is of countable type, so by 5.2.9 D is polar in K I . Hence there is a subspace D1 of K (I ) such that D = D1◦ = {x ∈ K I : B(x, y) = 0 for all y ∈ D1 }. Let D2 be an algebraic complement of D1 in K (I ) . Then K (I ) = D1 ⊕alg D2 and, by 3.4.22(ii), the associated projections are continuous. So the linear map D−→D2 , x → B(x, .)|D2 is a bijection. If (ej )j ∈J is an algebraic base of D2 then D2 ∼ K (J ) (3.4.22(iv)). Thus, the above bijection leads to the map D−→K J , x → (B(x, ej ))j ∈J , which is easily seen to be a linear surjective homeomorphism. (γ ) =⇒ (δ). By (γ ) we may assume that E = K I and that τ is the product topology on K I . Let τ1 be a Hausdorff locally convex topology on K I weaker than τ . Then (K I , τ1 ) is a subspace of F := K (I ) . As τ1 is Hausdorff and of countable type we have that (K I , τ1 ) is dual-separating (5.1.4), hence (K I , τ1 ) is σ (F, K I )-dense in F . But every subspace of F is σ (F, K I )-closed (5.2.10(ii)), so that (K I , τ1 ) = F , i.e., (K I , τ1 ) and K I have the same duals. Since τ and τ1 are of finite type it follows from 5.3.4 that they coincide with their weak topologies. Therefore, τ1 = τ . (δ) =⇒ (α). To prove completeness of E, suppose there is an x ∈ E ∧ , x ∈ E; we derive a contradiction. The space Kx is closed, so E ∧ /Kx is Hausdorff (3.4.8). The quotient map π : E ∧ −→E ∧ /Kx is injective on E. Now let (xi )i be a net in E converging to x in E ∧ . Then it does not converge in E while π(xi ) → 0. Hence π|E is not a homeomorphism, conflicting (δ). Next we prove that E is of finite type. By 5.3.6(i) and 5.3.7 it suffices to show that, for any countable X ⊂ E, D := [X] is of finite type. This will be done as soon as we construct a Hausdorff locally convex topology τ1 on E such that τ1 |D = σ (D, D ) and τ1 ≤ τ . (Then by (δ) we have τ |D = σ (D, D ) and by 5.3.4 D is of finite type.) Let τ1 be the locally convex topology generated by all τ -continuous seminorms p on E such that p|D is σ (D, D )-continuous. Then obviously τ1 ≤ τ and τ1 |D ≤ σ (D, D ). By 3.4.3 every σ (D, D )-continuous seminorm on D can be extended to a τ -continuous seminorm on E, so we have also τ1 |D ≥ σ (D, D ), i.e., τ1 |D = σ (D, D ). Finally, we see that τ1 is Hausdorff. For that, first observe that D is of countable type, so it is dual-separating (5.1.4), hence by 5.1.6 σ (D, D ) is Hausdorff. Thus, τ1 is Hausdorff if we find, for each x ∈ E, x ∈ D, a τ1 -continuous seminorm q such that q(x) = 0. Now D is closed, so E/D is Hausdorff (3.4.8). Let π : E−→E/D be the quotient map. Since π(x) = 0 there is a τ -continuous seminorm p on E such that pπ (π (x)) = 0 (pπ is the quotient seminorm of p, see 3.4.4). Then q := pπ ◦ π

222

The weak topology

satisfies the requirements. (Notice that q is automatically τ1 -continuous because q|D = 0.) Remarks 5.3.12 (a) For metrizable spaces E properties (α) and (δ) are equivalent to (β) and (γ ) below: (β) E is isomorphic to a closed subspace of K N . (γ ) E is finite-dimensional or isomorphic to K N . (b) An example of a quasicomplete but non-complete Hausdorff space of finite type will be given in 5.4.7. We know that the only normed spaces of finite type are the finite-dimensional ones. Next we study which spaces considered in Section 3.7 are of finite type. Let X be a non-empty zero-dimensional Hausdorff topological space. As we already said prior to 4.3.2, (C(X), τs ) is of finite type. For the other spaces of continuous functions of Subsection 3.7.1 we have the following. Theorem 5.3.13 (C(X), τc ) is of finite type if and only if every compact subset of X is finite. Proof. If every compact subset of X is finite we have τs = τc on C(X), so obviously the “if” is true. To prove the converse, let Y ⊂ X be compact. The restriction map R : (C(X), τc )−→C(Y ) is continuous (as usual, C(Y ) is equipped with the supremum norm). By the zero-dimensional Tietze–Urysohn Extension Lemma 2.5.23, R is surjective. It follows from 5.3.6(i) that C(Y ) is of finite type, hence finite-dimensional, so Y must be finite. Theorem 5.3.14 The following are equivalent: (α) (BC(X), β0 ) is of finite type. (β) X is finite. If, in addition, X is locally compact then properties (α), (β) are equivalent to: (γ ) (BCWc (X), τWc ) is of finite type. (δ) (Cc (X), τi ) is of finite type. Proof. The implications (β) =⇒ (α), (γ ) are clear. Also, (γ ) =⇒ (δ) follows from 3.7.22 and 5.3.6(i). Now we prove (α) =⇒ (β). (Similar reasoning works for (δ) =⇒ (β) when X is locally compact, using 3.7.22.) Assume X is infinite. Let x1 , x2 , . . . be mutually distinct elements of X. Let λ1 , λ2 , . . . ∈ K, |λ1 | >

5.3 Weak topologies and spaces of finite type

223

|λ2 | > · · · , limn λn = 0. Define φ : X−→K by λn if n ∈ N, x = xn φ(x) := 0 otherwise. Then {x ∈ X : |φ(x)| ≥ ε} is finite for each ε > 0, so φ is a bounded function that vanishes at infinity, i.e., φ ∈ W0 . Then the seminorm f −→ max |f (xn )| |λn | n∈N

(f ∈ BC(X))

is β0 -continuous. But its kernel is infinite-codimensional since the set of all (f (x1 ), f (x2 ), . . .) ∈ K N , where f ∈ BC(X), contains vectors of the form (1, . . .), (0, 1, . . .), (0, 0, 1, . . .), . . . . Hence (BC(X), β0 ) is not of finite type. Remark 5.3.15 The spaces of analytic functions A(r) (r ∈ (0, ∞]) and A† (r) (r ∈ [0, ∞)) introduced in Subsection 3.7.2 are not of finite type because they are infinite-dimensional and there exist continuous norms on them. Spaces of rigid analytic functions are of finite type only in uninteresting cases, as we will see now. Theorem 5.3.16 O(A) is of finite type if and only if A is discrete. Proof. Let O(A) be of finite type. By completeness of O(A) and 5.3.11 there are no strictly weaker Hausdorff locally convex topologies on O(A), so the canonical topology on O(A) equals the topology τs of pointwise convergence. Let f ∈ K A . For every finite subset G of A there exists a polynomial function pG for which pG = f on G. The pG |A form in a natural way a τs -Cauchy net in O(A) converging to f and it follows that O(A) = K A . Since every member of O(A) is continuous we must have that A is discrete. Conversely, if A is discrete, its complete bounded subsets B are finite, so the defining seminorms . B are of finite type and we get that O(A) is of finite type. Remark 5.3.17 Clearly the spaces (C n (X), τsn ) (n ∈ {0, 1, . . .}) and (C ∞ (X), τs∞ ) are of finite type. Let (E, τ ) be any of the other spaces of differentiable functions considered in Subsection 3.7.3 where, among other things, we saw that it is complete. Let us prove that it is not of finite type. Suppose it is; we derive a contradiction. By assumption and 5.3.11 there are no strictly weaker Hausdorff locally convex topologies on E. Thus, τ equals the topology of pointwise convergence on E. Let x ∈ X. There is a sequence of balls B1 ⊃ B2 ⊃ · · · about x with radius tending to 0 (the balls may be assumed to be compact in case X is

224

The weak topology

locally compact). Clearly (ξBi )i is pointwise Cauchy, hence τ -Cauchy. But by completeness of (E, τ ) there exists f ∈ E such that f = limi ξBi pointwise, which implies that f = ξ{x} , a contradiction because, as x is not isolated, ξ{x} is not even continuous.

5.4 Weakly bounded sets We know that compactoid sets are bounded (3.8.4(vii)). On the other hand, any ball in a normed space E is bounded but, by 3.8.6, it is a compactoid if and only if E is finite-dimensional. However, for spaces of finite type these two notions, compactoidity and boundedness, coincide. Theorem 5.4.1 In a locally convex space of finite type each bounded set is a compactoid. Proof. Let X be a bounded subset of E, let p be a continuous seminorm. We have to find a finite set G ⊂ E such that X ⊂ {x ∈ E : p(x) ≤ 1} + aco G. The set πp (X) is bounded in the normed space Ep , which is finitedimensional. By 3.8.4(viii) πp (X) is a compactoid in Ep , so there is a finite set G ⊂ E such that πp (X) ⊂ {z ∈ Ep : p(z) ≤ 1} + aco πp (G) = πp ({x ∈ E : p(x) ≤ 1}) + πp (aco G). Hence X ⊂ {x ∈ E : p(x) ≤ 1} + aco G + Ker p ⊂ {x ∈ E : p(x) ≤ 1} + aco G. Putting together 5.3.4 and 5.4.1 we derive the following. Corollary 5.4.2 Every weakly bounded subset of a locally convex space is weakly compactoid. Remark 5.4.3 In Chapter 8 we will study the class of locally convex spaces for which each bounded set is a compactoid, the so-called semi-Montel spaces. Now we have enough material to give the example announced in 3.8.26(a). Example 5.4.4 There exists a metrizable compactoid set whose closed absolutely convex hull is not metrizable. Proof. Let E := (c0 , σ ), where σ := σ (c0 , ∞ ). Then E is Hausdorff. Let A := {e1 , e2 , . . .}, where the ei are the unit vectors. Clearly A is bounded,

5.4 Weakly bounded sets

225

hence a compactoid in E by 5.4.2. Also, A is σ -discrete, hence metrizable in E. But acoσ A = Bc0 is not σ -metrizable. Indeed, suppose it is; we derive a contradiction. Since Bc0 is a compactoid in E, we can apply 3.8.25 to find a σ sequence x1 , x2 , . . . in λ Bc0 (λ ∈ K, |λ| > 1) such that xn −→ 0 and Bc0 ⊂ acoσ {x1 , x2 , . . .} ⊂ λ Bc0 .

(5.1)

Then X := {0, x1 , x2 , . . .}, equipped with the topology σ |X, is compact and, by 3.8.24, it is ultrametrizable. Thus, the Banach space C(X) is of countable type (2.5.24). Further, from (5.1) we obtain that the linear map T : c0 −→C(X), f → f |X satisfies f ≤ T (f )∞ ≤ |λ| f

(f ∈ c0 ).

Hence T is a homeomorphism, from which we deduce that c0 is of countable type, i.e., ∞ is of countable type (2.5.11), which contradicts 2.5.15. As A is metrizable in E but acoσ A is not, applying 3.8.24 we see that the conclusion of 3.8.25 fails for these E and A. The topology of a locally convex space E is stronger than its weak topology, so it is clear that boundedness implies weak boundedness for subsets of E. In the next theorem we show that the converse is true for polar spaces. Even more, we will prove later that for a metrizable space the validity of this converse characterizes polarity (5.4.11). Hence, for any non-polar normed space (e.g. ∞ /c0 over a non-spherically complete K, 4.4.8(c)) there exist weakly bounded sets that are not bounded, see also 5.4.12(a). Theorem 5.4.5 Every weakly bounded subset of a polar space is bounded. Proof. Let E be a polar space, let X be a weakly bounded subset of E. First, let us assume that E is a normed space whose topology is defined by a polar norm . . For every x ∈ E, hx : E −→K, f → f (x) is a continuous linear functional on E . Weak boundedness means that for each f ∈ E the set {f (x) : x ∈ X} is bounded in K, that is, {hx : x ∈ X} is a pointwise bounded set in the dual of the Banach space E . (E and its dual are equipped with the usual Lipschitz (solid) norms, see Section 2.1.) By the Uniform Boundedness Principle 2.1.20 this set is norm bounded in E , i.e., there is an M > 0 such that hx ≤ M for all x ∈ X. Polarity of . implies that, for all x ∈ X, x = sup{|f (x)| : f ∈ E , |f | ≤ . } = sup{|hx (f )| : f ∈ E , f ≤ 1} = hx ≤ M. Hence X is norm bounded.

226

The weak topology

Now let E be any polar space and let p be a polar continuous seminorm on E. It is easily seen that weak boundedness of X in E implies weak boundedness of πp (X) in the polar normed space Ep . So we can apply the first part of the proof to obtain that πp (X) is norm bounded in Ep , i.e., p is bounded on X. Thus, X is bounded. Corollary 5.4.6 Let E be a locally convex space. If either K is spherically complete or E is of countable type then every weakly bounded subset of E is bounded. Proof. In both cases E is polar (4.4.3). Then apply 5.4.5.

Now we can give the example promised in 5.3.12. Example 5.4.7 There exists a quasicomplete space of finite type that is not complete. Proof. Let E := (c0 , σ (c0 , c0∗ )). By 5.3.4 E is of finite type. Since c0 equipped with the strongest locally convex topology is polar (4.4.8(b)), it follows from 3.6.6(a) and 5.4.5 that every bounded subset X of E is finite-dimensional, so [X] is complete (3.4.24). Hence X is complete as soon as it is closed, from which we obtain that E is quasicomplete. On the other hand, the product topology on K N induces a Hausdorff locally convex topology on c0 that is ≤ σ (c0 , c0∗ ) and different from it. (The ≤ is clear. Further, the sequence e1 , e2 , . . . of unit vectors is not σ (c0 , c0∗ )-convergent in c0 but tends to 0 in K N .) By 5.3.11 we conclude that E cannot be complete. Clearly if E, F are locally convex spaces and T : E−→F is a continuous linear map then T is also continuous when we equip E and F with their weak topologies. For normed spaces the following converse is true. Theorem 5.4.8 Let E, F be normed spaces, F polar, let T : E−→F be a linear map. If T : (E, σ (E, E ))−→(F, σ (F, F )) is continuous then T is continuous. Proof. Since the weak and the norm topologies have the same duals we obtain that for every f ∈ F , f ◦ T ∈ E and hence f (T (BE )) is bounded in K. This means that T (BE ) is weakly bounded in F . By 5.4.5 we conclude that T (BE ) is bounded in F , that is, T is continuous. Remark 5.4.9 Weak continuity of a linear map does not always imply continuity. In fact: For a locally convex space F the following are equivalent:

5.4 Weakly bounded sets

227

(α) For each locally convex space E and each linear map T : E−→F such that T : (E, σ (E, E )) −→(F, σ (F, F )) is continuous it follows that T is continuous. (β) F is of finite type. Proof. Thanks to 5.3.4 the implication (β) =⇒ (α) is almost trivial. From that corollary we also have that if F is not of finite type then its initial and weak topologies are different, so (α) fails for E := (F, σ (F, F )) and T : E−→F the identity. Hence (α) =⇒ (β). For a metrizable space the property “every weakly bounded set is bounded” characterizes polarity, as we will show in 5.4.11. First a preliminary lemma. Lemma 5.4.10 Let E be a vector space, let τ1 , τ2 be locally convex topologies on E both induced by countably many seminorms. Suppose τ1 -bounded = τ2 -bounded for subsets of E. Then τ1 = τ2 . Proof. It suffices to show that τ1 ≤ τ2 . Let p1 , p2 , . . . be a sequence of seminorms defining τ1 and let q1 , q2 , . . . be a sequence of seminorms defining τ2 . First, we prove that p1 ≤ C qk for some constant C > 0 and some k ∈ N. Suppose not. Then p1 ≤ C q1 is true for no C > 0, so there exists a sequence x11 , x12 , . . . in E with q1 (x1n ) ≤ 1 while p1 (x1n ) ≥ n for each n. Also, p1 ≤ C q2 is true for no C > 0, so there exists a sequence x21 , x22 , . . . in E with q2 (x2n ) ≤ 1 and p1 (x2n ) ≥ n for each n, etc. Inductively we find a double sequence x11 , x21 , .. .

x12 , x22 , .. .

x13 , x23 , .. .

... ... .. .

in E such that qk (xkn ) ≤ 1 while p1 (xkn ) ≥ n for each k, n. We see that the diagonal sequence x11 , x22 , . . . is qk -bounded for each k, so it is τ2 -bounded. But p1 (xnn ) ≥ n for each n, which implies that the sequence x11 , x22 , . . . is not τ1 -bounded, a contradiction. In a similar way as above we can prove that p2 , p3 , . . . are dominated by positive multiples of some qk . It follows that τ1 ≤ τ2 . Theorem 5.4.11 Let E be a metrizable space such that every weakly bounded set is bounded. Then E is polar. Proof. By 3.5.2 there is a sequence p1 , p2 , . . . of seminorms inducing the topology τ of E. For each n ∈ N, p˜ n := sup{|f | : f ∈ E ∗ , |f | ≤ pn } is a polar seminorm on E. Let τpol be the topology on E generated by the sequence

228

The weak topology

of seminorms p˜ 1 , p˜ 2 , . . . . Then (E, τpol ) is a polar metrizable space such that σ (E, E ) ≤ τpol ≤ τ . (Notice that τpol is the polar topology associated to τ , see 4.5.13.) Together with the assumption this yields that τ -bounded = τpol bounded for subsets of E. Hence τ = τpol by 5.4.10, so E is polar. Remarks 5.4.12 (a) It follows from 5.4.11 that any non-polar metrizable space has weakly bounded sets that are not bounded. In fact, in [193], 4.N an example is constructed which is a Banach space, and even dual-separating. (b) Let K be not spherically complete. Then 5.4.11 is false if metrizability is dropped, i.e., there are non-polar spaces for which every weakly bounded set is bounded. In fact, let E := ∞ /c0 equipped with the locally convex topology τ generated by the canonical quotient norm on E and the seminorms |f |, with f running through E ∗ . Clearly τ ≤ υ, where υ is the strongest locally convex topology on E (see 3.4.22). Further, (E, τ ) = E ∗ = (E, υ) , so (E, τ ) and (E, υ) have the same weak topologies. Now let X ⊂ E be a weakly bounded subset of E. By polarity of (E, υ) (4.4.8(b)) and 5.4.5 X is υ-bounded, hence τ -bounded. Therefore, every weakly bounded subset of E is bounded. However, we have seen in 4.5.6 that E is not polar (but of course E is dual-separating).

5.5 Weakly convergent sequences It is well known that every weakly convergent sequence in the classical space 1 is convergent ([144], 22.4.(2)). In the non-Archimedean case the role of 1 is taken over by c0 . Theorem 5.5.1 In c0 every weakly convergent sequence is convergent. Proof. Let x1 , x2 , . . . be a sequence in c0 converging weakly to 0. Suppose this sequence does not converge to 0 in the norm topology; we derive a contradiction. We may assume that there is an ε > 0 such that xn ∞ ≥ ε for all n. For each n, the set J (n) := {i ∈ N : |λni | ≥ ε} is non-empty and finite, where xn = (λni )i∈N . On the other hand, as x1 , x2 , . . . tends to 0 coordinatewise, for each i ∈ N the set {n ∈ N : i ∈ J (n)} is also finite. Therefore, there is a strictly increasing sequence m1 , m2 , . . . in N such that the sets J (m1 ), J (m2 ), . . . are pairwise disjoint. Take a set J ⊂ N that has exactly one point in common with each J (mk ). Then f : c0 → K, (µ1 , µ2 , . . .) → i∈J µi is an element in c0 such that f (xmk ) ≥ ε for all k ∈ N, a contradiction.

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229

Theorem 5.5.1, which is of interest in its own right, is the stepping stone for the following general result that deviates much from the classical case. Theorem 5.5.2 Let E be a locally convex space. Suppose either: (i) K is spherically complete, or (ii) E is of countable type. Then every weakly convergent sequence in E is convergent. Proof. First, we prove (i) and (ii) when E is a normed space. Let x1 , x2 , . . . be a sequence in E converging weakly to 0, let D := [x1 , x2 , . . .]. By 4.1.2 (in case (i)) or 4.2.5 (in case (ii)) the weak topology on D is the restriction to D of the weak topology on E. Hence x1 , x2 , . . . converges to 0 in the weak topology of D. Since D is of countable type it is isomorphic to a subspace of c0 (2.3.9). It follows from 5.5.1 that x1 , x2 , . . . is norm convergent to 0. Now let E be a locally convex space as in (i) or (ii). It is easily seen that if x1 , x2 , . . . is a sequence in E converging weakly to 0 then, for each continuous seminorm p on E, the sequence πp (x1 ), πp (x2 ), . . . converges weakly to 0 in the normed space Ep (which is of countable type in case (ii)). Applying the first part of the proof we obtain that limn p(xn )) = 0, i.e., limn p(xn ) = 0. So x1 , x2 , . . . tends to 0 in E. Remark 5.5.3 For extensions see 5.6.1, 5.8.3 and 8.4.7. Theorem 5.5.4 Let E be a Hausdorff locally convex space. (i) Suppose E is polar. If E is weakly sequentially complete then E is sequentially complete. (ii) Suppose every weakly convergent sequence in E is convergent. Then E is weakly sequentially complete if and only if E is sequentially complete. Proof. The proof of (ii) is straightforward. To prove (i), let x1 , x2 , . . . be a Cauchy sequence in E. Then it is also weakly Cauchy. Let x be the weak limit of x1 , x2 , . . .. (Note that by 5.1.3 and 5.1.6 σ (E, E ) is Hausdorff.) By polarity and 4.4.14 the weak topology is closely related to the initial one on E. If follows from 3.4.17 that x1 , x2 , . . . converges to x in E. In 5.5.7 we will show that when K is not spherically complete there are polar spaces for which the conclusions of 5.5.2 and 5.5.4(ii) are false. To this end, it is useful to have a description of the dual of ∞ . We have seen in 2.5.11 that c0 ∞ . Surprisingly we are able to show that ( ∞ ) c0 but only if K is not spherically complete! (for K spherically complete we do not have a satisfactory description of ( ∞ ) ).

230

The weak topology

As in 2.5.11, B : c0 × ∞ −→K is the bilinear form defined by B(x, y) :=

∞

λn µn

(x = (λ1 , λ2 , . . .) ∈ c0 , y = (µ1 , µ2 , . . .) ∈ ∞ ).

n=1

Theorem 5.5.5 Let K be not spherically complete. Then the dual of ∞ is isometrically isomorphic to c0 . In fact, the map x → B(x, .) is an isometrical isomorphism c0 −→( ∞ ) . Proof. Like in 2.5.11 we conclude that the above map is a well-defined linear isometry. To see surjectivity, take f ∈ ( ∞ ) . By restricting f to c0 and applying 2.5.11 we obtain an x = (λ1 , λ2 , . . .) ∈ ∞ such that f (y) = ∞ n=1 λn µn for all y = (µ1 , µ2 , . . .) ∈ c0 . We first prove that x ∈ c0 . Take ε > 0 and let J := {n ∈ N : |λn | ≥ ε}. For z = (ν1 , ν2 , . . .) ∈ ∞ (J ), define z = (ν 1 , ν 2 , . . .) ∈ ∞ by −1 λn νn if n ∈ J ν n := 0 otherwise. The linear map z → f (z) is in ( ∞ (J )) . If z ∈ c0 (J ) then z ∈ c0 and f (z) = n∈J νn . By (β) =⇒ (α) of 4.1.13 it follows that J is finite. Hence x ∈ c0 . Now the linear map ∞ −→K, y → f (y) − B(x, y) is an element of ( ∞ ) that vanishes on c0 . By 4.1.12 it must vanish everywhere. Thus, f (y) = B(x, y) for all y ∈ ∞ , and surjectivity follows. Remark 5.5.6 If K is spherically complete the above proof does not work and also the conclusion of 5.5.5 is false. In fact, it follows from 4.1.4 that ( ∞ ) is not of countable type. In particular, this space cannot be isomorphic to c0 . Examples 5.5.7 Let K be not spherically complete. Then ∞ is a polar Banach space that is weakly sequentially complete. However, we have the following: (i) There exists a weakly convergent sequence in ∞ that is not convergent. (ii) There exists a complete subspace E of ∞ that is not weakly sequentially complete. Proof. Firstly note that since ∞ is a dual space (2.5.11) we have by the Banach–Steinhaus Theorem 2.1.21 that ∞ is weakly sequentially complete. (i). It follows from 5.5.5 that the unit vectors converge weakly to 0. But clearly this sequence is not norm convergent. (ii). The construction of the subspace E satisfying (ii) is more laborious. Let N be the collection of all sets J ⊂ N for which lim n

(J ∩ [1, n]) =0 n

5.5 Weakly convergent sequences

231

and let E be the set of all y = (µ1 , µ2 , . . .) ∈ ∞ such that for every ε > 0, {n ∈ N : |µn | ≥ ε} ∈ N . Clearly E ⊃ c0 . Also, E is a closed subspace of ∞ , so E is a Banach space with the induced norm, and E is polar (4.4.16(i)). We are going to see that E is not weakly sequentially complete. For that we first describe E . Take f ∈ E . By restricting f to c0 and applying 2.5.11 we obtain an x = (λ1 , λ2 , . . .) ∈ ∞ such that f (y) = ∞ n=1 λn µn for all y = (µ1 , µ2 , . . .) ∈ c0 . We prove that x ∈ c0 . If ε > 0 and Jε := {n ∈ N : |λn | ≥ ε} is an infinite set then Jε contains an infinite set J , which is an element of N . For z ∈ ∞ (J ), we define z ∈ E as in the proof of 5.5.5. As in that theorem, the linear map z → f (z) is an element of ( ∞ (J )) whose value at z = (ν1 , ν2 , . . .) ∈ c0 (J ) is n∈J νn . But this contradicts the implication (β) =⇒ (α) of 4.1.13. Hence for every ε > 0 the set Jε is finite, that is, x ∈ c0 . It follows easily that c0 E through the same isometry as the one given in 5.5.5 for ( ∞ ) . Therefore, the sequence y1 , y2 , . . . defined by yn := e1 + · · · + en is weakly Cauchy in E. If y = (µ1 , µ2 , . . .) ∈ E would be the limit of this sequence in the weak topology σ (E, E ) then y1 , y2 , . . . would be cordinatewise convergent to y. So µm = 1 for all m, conflicting y ∈ E. Theorem 5.5.5 can be also applied to give the examples, already announced in 3.8.30(a) and 3.8.40, showing the relevance of the metrizability condition in (i) of 3.8.29 and in 3.8.35, 3.8.38. Example 5.5.8 There exists a complete, absolutely convex, edged and compactoid set A and a continuous linear injection T such that: (i) (T (A))e is not closed, (ii) T |A is not a homeomorphism. Proof. Let K be not spherically complete. Let E := (c0 , σ (c0 , ∞ )), A := Bc0 . By 5.4.2 A is a compactoid in E, and clearly A is edged. Let us prove that A is complete. If (xi )i is a Cauchy net in A then (B(xi , .))i (B as in 5.5.5) is a Cauchy net in ( ∞ ) with respect to the topology of pointwise convergence. By completeness of K, for each y ∈ ∞ , f (y) := limi B(xi , y) exists. Since B(xi , .) = xi ∞ ≤ 1 for all i we have f ∈ ( ∞ ) and f ≤ 1. By 5.5.5 there is an x ∈ A such that B(x, .) = f . It follows that xi → x in E. Thus, A is complete. Now let F := K N (equipped with the product topology). The inclusion map T : E−→F is in L(E, F ), yet (T (A))e is not closed. (The sequence of partial sums of the unit vectors is in T (A) and converges in F to (1, 1, 1, . . .), which is not in c0 , hence not in (T (A))e .) So (i) holds.

232

The weak topology

Notice also that the sequence of unit vectors e1 , e2 , . . . is in T (A), and one verifies T (en ) = en → 0 in F , but en 0 in E. Thus, T |A is not a homeomorphism and we have (ii).

5.6 Weakly (pre)compact sets and “orthogonality” In this section we apply compactoidity theory to obtain several new results related to the weak topology. More applications of that theory will be presented along the book. First we prove that the conclusion of 5.5.2 can be strengthened as follows. Theorem 5.6.1 Let E be a Hausdorff locally convex space. Suppose either K is spherically complete or E is of countable type. Then each weakly compact subset of E is compact. If, in addition, K is not locally compact, each weakly precompact subset of E is precompact. Proof. Let X ⊂ E be weakly precompact. By 4.4.3 E is polar, so by 5.2.12 it suffices to show that X is a compactoid if either X is weakly compact or K is not locally compact. To this end, first notice that X is weakly bounded, hence bounded (5.4.5). Suppose X is not a compactoid; we will derive a contradiction. By 3.9.16 there is an “orthogonal” sequence x1 , x2 , . . . in X that does not converge to 0. So there is a continuous seminorm p such that x1 , x2 , . . . is orthogonal with respect to p and p(xn ) 0. We can select a subsequence e1 , e2 , . . . of x1 , x2 , . . . with inf n∈N p(en ) > 0. Applying 4.1.3 or 4.2.6 we find f1 , f2 , . . . ∈ E such that fm (en ) = δmn (m, n ∈ N). From these corollaries we also deduce the existence of a g ∈ E with g(en ) = 1 for all n. Set Y := {e1 , e2 , . . .}. Then by 5.2.4 D := [Y ] is weakly closed. Now we prove 1, 2, 3 below: σ

1. Y is weakly closed. Proof. Suppose we have x ∈ Y , x ∈ Y . Then x ∈ D, let x= ∞ n=1 λn en (λn ∈ K) be its expansion according to 3.9.13. If (eni )i is a net in Y converging weakly to x then (eni )i cannot be eventually constant, so limi fm (eni ) = 0 = λm for all m. It follows that x = 0. But then 1 = limi g(eni ) = g(x) = 0, impossible. 2. Y is not weakly compact. Proof. Direct verification yields that the sets 1 (n ∈ N) Un := x ∈ E : |fn (x) − fn (en )| < 2

5.6 Weakly (pre)compact sets and “orthogonality”

233

form a clopen covering of Y . But, as em ∈ Un if m = n, it has no finite subcovering. 3. If K is not locally compact then Y is not weakly precompact. Proof. There exist µ1 , µ2 , . . . ∈ K and δ > 0 such that δ ≤ µi − µj ≤ 1 (i = j ). Again by 4.1.3 and 4.2.6 there is an h ∈ E with h(en ) = µn for all n. Setting U := {x ∈ E : |h(x)| < δ}, then for m = n we have |h(em − en )| = |µm − µn | ≥ δ, so em − en ∈ U . It follows that Y cannot be covered by finitely many translates of the zero neighbourhood U . Finally, the desired contradiction follows from 1 and 2 if X is weakly compact and from 3 if K is not locally compact. Remarks 5.6.2 (a) For an extension see 5.8.5. (b) If K is locally compact the conclusion of the second part of the theorem is no longer true. In fact, in this case “weakly precompact” is identical to “weakly compactoid” (3.8.3) and, by 5.4.2 and 5.4.5, it is identical to “bounded”. (Observe that if K is locally compact then it is spherically complete.) Now we are in a position to give a negative answer to the question put by Gruson and van der Put in [100]. Let X be a compactoid in a locally convex space E. Does there exist a precompact Y ⊂ E such that X ⊂ aco Y ? Obviously, by taking Y = X, if K is locally compact, the answer is yes. Example 5.6.3 Let K be not locally compact, let X be the closed unit ball of c0 . Then, for the weak topology, X is a compactoid but is not contained in the closed absolutely convex hull of a precompact set. Proof. Suppose there is a weakly precompact set Y in c0 such that X is contained in the weak closed absolutely convex hull of Y . By 5.6.1 Y is precompact, hence compactoid, in c0 equipped with its canonical norm topology. If follows from 5.2.13 that X is a norm-compactoid in c0 , a contradiction with 3.8.6. The next example shows that when K is not spherically complete there are polar spaces for which 5.6.1 fails. Example 5.6.4 Let K be not spherically complete. Then there exists a weakly compact subset of the polar normed space ∞ that is not precompact. Proof. If e1 , e2 , . . . are the unit vectors of ∞ , it follows from 5.5.5 that this sequence converges weakly to 0, hence {0, e1 , e2 , . . .} is weakly compact. But

234

The weak topology

clearly this sequence has no convergent subsequences, so {0, e1 , e2 , . . .} is not precompact. Next we apply compactoidity theory to derive some non-Archimedean translations of the Bessaga–Pelczynski Selection Principle (see [46], p. 42). Theorem 5.6.5 Let E be a polar space. Let x1 , x2 , . . . be a sequence in E such that xn → 0 weakly but xn 0 in E. Then x1 , x2 , . . . contains an “orthogonal” subsequence. Proof. By weak convergence the set {x1 , x2 , . . .} is weakly bounded, so bounded by 5.4.5. If x1 , x2 , . . . had no “orthogonal” subsequence then {x1 , x2 , . . .} would be a compactoid by 3.9.16, and so would be aco {x1 , x2 , . . .}. Hence the weak topology and the initial one coincide on aco {x1 , x2 , . . .} (5.2.12), which contradicts the assumptions on the sequence x1 , x2 , . . . . Theorem 5.6.6 Let E be a metrizable space. Then the following are equivalent: (α) E ∧ is dual-separating. (β) Let x1 , x2 , . . . be a bounded sequence such that xn → 0 weakly but xn 0 in E. Then x1 , x2 , . . . contains an “orthogonal” subsequence. Proof. To prove (α) =⇒ (β) we may assume that E is complete. Suppose x1 , x2 , . . . has no “orthogonal” subsequence; we derive a contradiction. By boundedness and 3.9.16 the set {x1 , x2 , . . .} is a compactoid, hence so is A := aco {x1 , x2 , . . .} (3.8.4(ii)). A is a complete absolutely convex metrizable compactoid in E. By (α) and 5.1.6 σ (E, E ) is Hausdorff, so by 3.8.38 the weak topology and the initial one coincide on A, which contradicts the assumptions on the sequence x1 , x2 , . . . . To prove (β) =⇒ (α), we will arrive at a contradiction by assuming that there exists a z ∈ E ∧ , z = 0 such that f (z) = 0 for all f ∈ (E ∧ ) . By metrizability there exist x1 , x2 , . . . ∈ E converging to z. Then x1 , x2 , . . . is Cauchy, hence {x1 , x2 , . . .} is a compactoid (3.8.4(iii)). As xn → 0 weakly and xn 0 in E we deduce from (β) that x1 , x2 , . . . contains an “orthogonal” subsequence y1 , y2 , . . . . By 3.9.16 yn must tend to 0. But also yn tends to z, so z = 0, a contradiction. Remarks 5.6.7 (a) Obviously 5.6.5 is of interest only for polar spaces having weakly convergent sequences that are not convergent (such as ∞ over a non-spherically complete K, see 5.5.7(i)). (b) If E is a normed space one may drop boundedness of x1 , x2 , . . . in (β) of 5.6.6. In fact, if x1 , x2 , . . . is unbounded one can select λ1 , λ2 , . . . ∈ K,

5.7 Admissible topologies and the Mackey topology

235

|λn | ≤ 1 for all n, such that λ1 x1 , λ2 x2 , . . . is bounded and inf n∈N λn xn > 0. When K is not spherically complete there exist dual-separating normed spaces whose completion is not dual-separating, see the chapter notes. (c) If E is polar and Hausdorff then its completion is also polar (4.4.16(ii)) and so E satisfies (α) of 5.6.6. In 4.5.6 we gave an example of a (nonmetrizable) dual-separating space that is not polar. There exist also nonpolar dual-separating Banach spaces, for an example see [193], 4.N.

5.7 Admissible topologies and the Mackey topology Definition 5.7.1 Let E be a locally convex space. A locally convex topology τ on E is called admissible if (E, τ ) = E . Clearly the initial and the weak topologies on E are admissible and so are all topologies that are in between. Also, it is straightforward to check that: Theorem 5.7.2 The weak topology is the smallest admissible topology on a locally convex space. Remark 5.7.3 From 5.3.4 we obtain that the weak topology of a locally convex space E is the only admissible topology of finite type on E. As a direct consequence of 5.2.1, 5.2.2, 5.4.5, 5.5.2 and 5.6.1 we obtain: Corollary 5.7.4 Let E be a locally convex space. Then we have the following: (i) All the polar admissible topologies on E have the same bounded sets. (ii) If K is spherically complete then all the admissible topologies on E have the same closed convex sets, the same convergent sequences and the same compact sets; and also the same precompact sets if, in addition, K is not locally compact. (iii) For any scalar field K the conclusion of (ii) holds when we replace “admissible topologies” by “admissible topologies of countable type” and “convex” by “convex edged”. Next we discuss the existence of the finest admissible topology. Definition 5.7.5 If there exists a strongest polar admissible topology on a locally convex space E it is called the Mackey topology. If, in addition, this topology coincides with the original one on E then E is called a Mackey space.

236

The weak topology

We will prove that for spherically complete fields K the Mackey topology exists. (Notice that for such a K all the locally convex topologies are polar, see 4.4.3(i).) The following lemma will be useful. Lemma 5.7.6 (Decomposition Lemma 1) Let p1 , . . . , pn be seminorms on a vector space E. Let f ∈ E ∗ be such that |f | ≤ max(p1 , . . . , pn ). (i) If K is spherically complete then there exist f1 , . . . , fn ∈ E ∗ such that f = f1 + · · · + fn and |fi | ≤ pi for all i ∈ {1, . . . , n}. (ii) If p1 , . . . , pn are of countable type then, for each ε > 0 there exist f1 , . . . , fn ∈ E ∗ such that f = f1 + · · · + fn and |fi | ≤ (1 + ε) pi for all i ∈ {1, . . . , n}. Proof. (i). Let F := E n , let D be the diagonal of F , i.e., the subspace of F defined by D := {(x1 , . . . , xn ) ∈ F : x1 = x2 = · · · = xn }. Then p : F −→R, (x1 , . . . , xn ) → max1≤i≤n pi (xi ) is a seminorm on F , g : D−→K, (x, x, . . . , x) → f (x) (x ∈ E) is a linear functional on D and, by assumption, |g| ≤ p on D. By the analytic form of the Hahn–Banach Theorem 4.1.1 there is an extension g ∈ F ∗ of g such that |g| ≤ p on F . Now the formulas f1 (x) := g(x, 0, . . . , 0), f2 (x) = g(0, x, 0, . . . , 0), . . . , fn (x) = g(0, . . . , 0, x) (x ∈ E) define linear functionals on E satisfying the requirements of (i). To prove (ii) we follow the same reasoning as in (i), applying in this case the analytic form of the Hahn–Banach Theorem 4.2.4. (Observe that if p1 , . . . , pn are of countable type the seminorm p on F defined in the proof of (i) is also of countable type.) Remark 5.7.7 In the chapter notes we will give a slight generalization called Decomposition Lemma 2 (see 5.8.7). Theorem 5.7.8 Let K be spherically complete. Then for each locally convex space E the Mackey topology exists. It is induced by the collection P of all seminorms p on E having the following property: f ∈ E ∗ , |f | ≤ p =⇒ f ∈ E .

(5.2)

Proof. Let τP be the locally convex topology on E induced by P. If τ is an admissible topology on E then clearly any τ -continuous seminorm p on E satisfies (5.2). Hence τ ≤ τP , and E ⊂ (E, τP ) . So only the inclusion (E, τP ) ⊂ E remains to be proved. For that, let f ∈ (E, τP ) . Then there exist λ ∈ K × and p1 , . . . , pn ∈ P with |f | ≤ |λ| max(p1 , . . . , pn ). By 5.7.6(i) we find f1 , . . . , fn ∈ E ∗ such that f = f1 + · · · + fn and |fi | ≤ |λ| pi for all i ∈ {1, . . . , n}. Since pi ∈ P it follows from (5.2) that fi ∈ E . Thus, f ∈ E.

5.7 Admissible topologies and the Mackey topology

237

With an easy adaptation of the proof given in the above theorem, this time by using 5.7.6(ii), we obtain: Theorem 5.7.9 For each locally convex space E the finest admissible topology of countable type on E exists. It is induced by the collection Q of all seminorms of countable type p on E satisfying (5.2). Remark 5.7.10 Let K be not spherically complete. The problem of existence of the Mackey topology in a polar space is unsolved, see the chapter notes. However, we are going to see in 5.7.13 that when K is not spherically complete there exist polar spaces for which the finest admissible topology (polar or not) does not exist (compare with 5.7.8). To this end, we need the following lemmas. Lemma 5.7.11 Let E be a vector space, let D be a finite-codimensional subspace of E, let p be a seminorm on E, and let f ∈ D ∗ be such that |f | ≤ p ∗ onD. 1Then for each t ∈ (0, 1) there is an extension f ∈ E of f such that f ≤ p on E. t Proof. By proceeding recurrently it suffices to prove the result when D is one-codimensional. So let E = D ⊕alg Ky for some y ∈ E, y ∈ D. Let π : E−→E/D be the canonical map, let pπ be the quotient seminorm of p on E/D (see 3.4.4). If pπ (π (y)) = 0 then y is in the p-closure of D, so D is p-dense in E. In this case a standard argument leads to an extension f ∈ E ∗ of f satisfying the required condition for each t ∈ (0, 1). Now suppose pπ (π(y)) = 0. There is a z ∈ E with π(z) = π (y) and t p(z) ≤ pπ (π (y)). Then E = D ⊕alg Kz and f : E−→K, x = d + λ z → f (d) (d ∈ D, λ ∈ K) is in E ∗ and extends f . To prove that f ≤ 1t p on E, observe that for d ∈ D, λ ∈ K p(d + λ z) ≥ pπ (π(λ z)) ≥ t p(λ z). So by the Principle of van Rooij (2.2.1, which also works when . is replaced by a seminorm) we obtain p(d + λ z) ≥ t max(p(d), p(λ z)). Hence, for x = d + λ z ∈ E, f (x) = |f (d)| ≤ max(p(d), p(λ z)) ≤ 1 p(d + λ z) = 1 p(x). t t 1 Therefore, f (x) ≤ t p(x).

238

The weak topology

Lemma 5.7.12 Let E be a Hausdorff locally convex space, let D be a finitecodimensional closed subspace of E, and let f ∈ E ∗ be such that f = 0 on D. Then f is continuous. Proof. Let π : E−→E/D be the canonical map. Since D ⊂ Ker f there is a unique linear functional f1 : E/D−→K such that f1 ◦ π = f . Now E/D is a finite-dimensional space, which is Hausdorff (3.4.8), hence normable (3.4.24). By 2.1.14(ii) f1 is continuous. Hence f is also continuous. Example 5.7.13 Let K be not spherically complete. Then the finest admissible topology on ∞ does not exist. Proof. It is enough to prove that there are two locally convex topologies τ1 , τ2 on ∞ such that τ1 and τ2 are admissible but sup(τ1 , τ2 ) is not. To this end, take h ∈ ( ∞ /c0 )∗ , h = 0 and x0 ∈ ∞ /c0 , x0 = 0 with h(x0 ) = 2. Then the linear map T : ∞ /c0 −→ ∞ /c0 , y → y − h(y) x0 satisfies T 2 = I , so T is bijective. Let π : ∞ −→ ∞ /c0 be the canonical map and let . be the quotient norm on ∞ /c0 . Let τ1 (resp. τ2 ) be the locally convex topology on ∞ generated by all weakly continuous seminorms on ∞ and the seminorm p (resp. q) defined, for every x ∈ ∞ , by p(x) := π (x) (resp. q(x) := T (π (x))). Since τ1 is between the weak and the norm topology on ∞ , it is admissible. Also, as τ2 is bigger than the weak topology the inclusion ( ∞ ) ⊂ ( ∞ , τ2 ) holds. Now we prove the opposite inclusion (then τ2 is also admissible). For that, consider a τ2 -continuous linear functional f on ∞ , let us prove its norm continuity. There exist c > 0 and f1 , . . . , fn in ( ∞ ) such that, for all x ∈ ∞ , |f (x)| ≤ c max(|f1 (x)| , . . . , |fn (x)| , T (π(x))). Then |f (d)| ≤ c T (π (d)) for all d in the finite-codimensional (norm closed) subspace D := 1≤i≤n Ker fi . By 5.7.11 there is a C > 0 and an extension f ∈ ( ∞ )∗ of f |D such that f ≤ C |T (π (x)) for all x ∈ ∞ . We arrive at the desired norm continuity of f as soon as we see that f = 0. (In fact, from f = 0 we obtain f = 0 on D and, by 5.7.12, f ∈ ( ∞ ) .) In order to prove that f = 0, observe that f = 0 on c0 , so there is a unique f 1 ∈ ( ∞ /c0 )∗ such that f 1 ◦ π = f . Also, f 1 (y) ≤ C T (y)|| for all y ∈ ∞ /c0 and, since T 2 = I , we derive f 1 (T (y)) ≤ C y|| for all y ∈ ∞ /c0 . Hence f 1 ◦ T ∈ ( ∞ /c0 ) . By 4.1.12 we have f 1 ◦ T = 0 and, by bijectivity of T , f 1 = 0, i.e., f = 0. We have already constructed two admissible topologies, τ1 and τ2 , on ∞ . Next we prove that sup(τ1 , τ2 ) is not admissible. For this, let f := h ◦ π. Then f is not in ( ∞ ) . Indeed, suppose yes. Then h ∈ ( ∞ /c0 ) , so h = 0 (4.1.12), a contradiction. However, f is sup(τ1 , τ2 )-continuous. In fact, by the definitions

5.8 Notes

239

of T and f , x0 |f (x)| = π(x) − T (π (x)) ≤ max(π(x), T (π (x))), for all x ∈ ∞ . But x → π (x) = p(x) and x → T (π (x)) = q(x) are sup(τ1 , τ2 )-continuous seminorms on ∞ . Thus, f is continuous for this supremum topology. Although the proof of the example is finished here we feel that, after 5.7.10, we should show that ( ∞ , τ1 ) and ( ∞ , τ2 ) are not polar. We prove non-polarity of τ1 . Similar arguments work for τ2 . Suppose τ1 is polar; we derive a contradiction. By 4.4.14 there is a τ1 continuous polar seminorm r on E such that p ≤ r. As σ ( ∞ , ( ∞ ) ) and p generate the topology τ1 there exist c > 0 and f1 , . . . , fn ∈ ( ∞ ) with r ≤ c max(|f1 | , . . . , |fn | , p). Set D := 1≤i≤n Ker fi . Then on D we have p ≤ r ≤ c p. By polarity of r and 4.4.5 we can find a g ∈ ( ∞ , τ1 ) = ( ∞ ) with |g| ≤ 1c r and g|D = 0. Then |g| ≤ p on D. Applying 5.7.11 there is a C > 0 and an extension g ∈ ( ∞ )∗ of g|D such that |g(x)| ≤ C p(x) = C π (x) for all x ∈ ∞ . Then π (x) → g(x) defines a non-trivial continuous linear functional ∞ /c0 −→K, which contradicts 4.1.12.

5.8 Notes Spaces of finite type were introduced by Schikhof in [204], where several descriptions of such spaces can be found, see also [174] and, for spherically complete K, see [62]. In c0 closed convex edged sets and closed compactoid sets are weakly closed (5.2.2, 5.2.14(i)). If K is spherically complete then even each closed convex subset of c0 is weakly closed (5.2.1). On the other hand, if K is not spherically complete we constructed in 5.2.3 a closed convex subset of c0 that is not weakly closed. But the answer to the following problem is unknown. Problem Characterize the weakly closed convex subsets of c0 when K is not spherically complete. From 5.2.4 we know that all closed subspaces of a locally convex space are weakly closed when either the scalar field is spherically complete or the space is of countable type. It was proved by De Grande-De Kimpe and Perez-Garcia in [89] that for a Hausdorff locally convex space E the following are equivalent: (α) Every closed subspace of E is weakly closed. (β) Every closed subspace of E has the weak Hahn–Banach property (see the notes to Chapter 4).

240

The weak topology

Observe that the space E := ∞ equipped with the strongest locally convex topology satisfies (α), (β) (since E = E ∗ , 3.4.22(ii)) but, by 2.5.15, E is not of countable type. Theorem 5.2.12 was applied in Section 4 of [205] to define barycenters of compact sets in a locally convex space. As in the notes to Chapter 4 (4.5.8), we have also the following result on separation of convex sets (see 4.5.7) in polar spaces, as a refinement of the geometric form of the Hahn–Banach Theorem given in 5.2.16. Corollary 5.8.1 (Separation of convex sets) Suppose the residue class field k of K has at least three elements. Let E be a polar space, let C1 , C2 be convex subsets of E such that C1 ∩ C2 = ∅ and C1 − C2 is closed, edged and compactoid. Then there is a closed hyperplane in E separating C1 and C2 . Remark 5.8.2 The example given in 4.5.9 shows that the assumption on k in 5.8.1 cannot be dropped. A locally convex space for which every weakly convergent sequence is convergent is called an Orlicz–Pettis space ((O.P.)-space). With this formulation 5.5.2, whose first part was proved, for Banach spaces, by Monna in Section V.2 of [160], reads as follows: spaces over spherically complete fields or spaces of countable type are (O.P.)-spaces. But there are other (O.P.)-spaces, as we will see below. We say that a locally convex space has property (*) if the weak Hahn– Banach property holds for all subspaces of countable type (see the notes to Chapter 4). Spaces over spherically complete K and spaces of countable type have property (*) (4.1.3, 4.2.6). Also, one verifies the following. Theorem 5.8.3 Every locally convex space with property (*) is an (O.P.)-space. Proof. Let E be a locally convex space with property (*). Let x1 , x2 , . . . be a sequence in E converging weakly to 0, let D := [x1 , x2 , . . .]. D is of countable type, so by property (*) σ (D, D ) = σ (E, E )|D. Hence x1 , x2 , . . . is weakly convergent to 0 in D. From 5.5.2(ii) we conclude that x1 , x2 , . . . converges to 0. Now, for the required example, observe that for an uncountable set I the Banach space c0 (I ) is not of countable type (2.5.8) but has property (*) (2.5.9(ii)), hence is an (O.P.)-space by 5.8.3. (O.P.)-spaces were studied by Perez-Garcia and Schikhof in [178] where, among other things, it was shown that every metrizable (O.P.)-space is polar. Combining this with 5.8.3 yields 5.8.4.

5.8 Notes

241

Corollary 5.8.4 Every metrizable space with property (*) is polar. Theorem 5.6.1 admits the following extension for spaces with property (*), which was proved in [80]. Its version for Banach spaces already appeared in [191] (the part for weakly compact sets) and in [200] (the part for weakly precompact sets). Theorem 5.8.5 ([80], 6.9, 6.11) Let E be a Hausdorff polar space with property (*). Then each weakly compact subset of E is compact. If, in addition, K is not locally compact, each weakly precompact subset of E is precompact. Continuing in the above spirit one may ask about a non-Archimedean counter˘ part of the classical Eberlein–Smulian Theorem. For a subset X of a Banach space, consider the following statements: (α) X is weakly compact. (β) X is weakly sequentially compact. (γ ) X is weakly countably compact. ˘ The classical Eberlein–Smulian Theorem ([144], 24.3.(9)) states that (α), (β) and (γ ) are equivalent, the interesting implications being (γ ) =⇒ (α) and (γ ) =⇒ (β) (since (α) =⇒ (γ ) and (β) =⇒ (γ ) are true in general). In the non-Archimedean case, the following was proved by Kiyosawa and Schikhof in [142]. Theorem 5.8.6 ([142], 2.2, 2.3) Let X be a subset of a polar Banach space E. If (i) K is spherically complete or (ii) E has property (*) or (iii) E is of countable type or (iv) [X] is of countable type, then (α), (β), (γ ) are equivalent. But the story does not stop here; there are strong improvements, see [142]: if, in addition, K is not locally compact then X is weakly compact if and only if f (X) is compact for every f ∈ E . That this last peculiar result does not hold for locally compact K (neither in the Archimedean case) is easily seen by taking E := K 2 , X := {x ∈ E : 0 < x ≤ 1}. If (i) or (ii), then X is weakly compact if and only if X is norm compact (5.8.5). If neither (i), (ii), (iii) nor (iv) are satisfied, then the conclusion of 5.8.6 fails, see [142] for counterexamples. A first proof of the existence of the Mackey topology (5.7.8) was given by van Tiel in 1965, [227]. The non-existence of the finest admissible topology on ∞ for non-spherically complete K (5.7.13) was proved by Ka¸kol in [105]. However, the following problem remains. Problem For a polar Hausdorff space over a non-spherically complete field, does there exist a Mackey topology?

242

The weak topology

Curiously one may describe the only possible candidate for this Mackey topology (see 5.8.8). For that we need a new Decomposition Lemma (compare with 5.7.6). Its proof involves the topology on the algebraic dual E ∗ of a vector space E induced by the collection of seminorms f → |f (x)| (x ∈ E, f ∈ E ∗ ). This topology is clearly Hausdorff and of finite type, and is usually denoted by σ (E ∗ , E). It is a particular case of the so-called weak∗ topologies that will be extensively studied later in Chapter 7, where among other things we will see (7.3.12) that (E ∗ , σ (E ∗ , E)) = E in the sense that ϕ ∈ (E ∗ , σ (E ∗ , E)) if and only if there is an x ∈ E such that ϕ(f ) = f (x) for all f ∈ E ∗ (observe that such an x is unique). It is customary to identify (E ∗ , σ (E ∗ , E)) and E. Lemma 5.8.7 (Decomposition Lemma 2) Let p1 , . . . , pn , q be seminorms on a vector space E, p1 , . . . , pn of countable type, q polar, let f ∈ E ∗ be such that |f | ≤ max(p1 , . . . , pn , q). Then for each ε > 0 there exist f1 , . . . , fn , fn+1 ∈ E ∗ such that f = f1 + · · · + fn + fn+1 and |fi | ≤ (1 + ε) pi for all i ∈ {1, . . . , n}, |fn+1 | ≤ (1 + ε) q. Proof. By 5.7.6 we may assume that K is not spherically complete (hence densely valued) and that n = 1, call p1 := p. Let U := {x ∈ E : p(x) ≤ 1}, V := {x ∈ E : q(x) ≤ 1}. Then U ◦ and V ◦ are absolutely convex, edged, complete and compactoids in (E ∗ , σ (E ∗ , E)) (the polars are taken in E ∗ ). By assumption we have U ◦◦ = U , V ◦◦ = V , so f ∈ (U ∩ V )◦ = (U ◦◦ ∩ V ◦◦ )◦ = (U ◦ + V ◦ )◦◦ . As p is of countable type we have that U ◦ is σ (E ∗ , E)-metrizable. (If x1 , x2 , . . . ∈ E are such that its linear hull is p-dense in E and xn ∈ U for all n, then the formula d(g, h) := maxn∈N |g(xn ) − h(xn )| 2−n (g, h ∈ U ◦ ), defines an ultrametric on U ◦ and a standard procedure shows that the dtopology on U ◦ equals σ (E ∗ , E)|U ◦ , see also the proof of 7.6.10.) Since V ◦ is σ (E ∗ , E)-complete we conclude from 3.8.29(ii) that (U ◦ + V ◦ )e is σ (E ∗ , E)-complete so, by 5.2.7, (U ◦ + V ◦ )e = (U ◦ + V ◦ )◦◦ . It follows that f ∈ (U ◦ + V ◦ )e . By density of the valuation we can choose a λ ∈ K, 1 < |λ| ≤ 1 + ε. Then f ∈ λ (U ◦ + V ◦ ), hence f = f1 + f2 , f1 ∈ λ U ◦ , f2 ∈ λ V ◦ , i.e., |f1 | ≤ |λ| p ≤ (1 + ε) p and |f2 | ≤ |λ| q ≤ (1 + ε) q. Theorem 5.8.8 Let E be a locally convex space. Suppose the Mackey topology on E exists. Then it is induced by the collection R of all polar seminorms p on E satisfying f ∈ E ∗ , |f | ≤ p =⇒ f ∈ E (Compare with 5.7.8.)

(5.3)

5.8 Notes

243

Proof. Let us call the Mackey topology on E. Let τR be the locally convex topology on E generated by R. Since is admissible any -continuous polar seminorm p on E satisfies (5.3). Hence ≤ τR . Now we prove τR ≤ , i.e., every r ∈ R is -continuous. This will be done as soon as we have admissibility of the topology τQr induced by r and the family Q of all seminorms of countable type on E satisfying (5.3). In order to see this admissibility observe that, by 5.7.9, E ⊂ (E, τQr ) . For the opposite inclusion, let f ∈ (E, τQr ) . There exist λ ∈ K, |λ| > 1 and q1 , . . . , qn ∈ Q such that |f | ≤ |λ| max(q1 , . . . , qn , r). By 5.8.7 there are f1 , . . . , fn , fn+1 ∈ E ∗ such that f = f1 + · · · + fn + fn+1 , |fi | ≤ |λ|2 qi for all i ∈ {1, . . . , n}, |fn+1 | ≤ |λ|2 r. As r and each qi satisfy (5.3) it follows that f1 , . . . , fn , fn+1 ∈ E . Thus, f = f1 + · · · + fn + fn+1 ∈ E . The Decomposition Lemmas 5.7.6 and 5.8.7 were applied by Perez-Garcia and Schikhof in [183] to obtain the following Hahn–Banach-type Theorem. Theorem 5.8.9 (Approximation Theorem for linear forms, [183], Section 2) Let E be a locally convex space. Suppose either K is spherically complete or E is of countable type. Let A ⊂ E be absolutely convex, let f ∈ E ∗ be such that f |A is continuous. Then, for each ε > 0 there is a g ∈ E such that |g − f | ≤ ε on A. Several characterizations of spherical completeness of K were given in 4.1.13. Below we present a few new ones of a different kind. Corollary 5.8.10 The following properties on K are equivalent: (α) K is spherically complete. (β) Every closed subspace of a polar normed space is weakly closed. (γ ) For every polar normed space the finest admissible topology exists. Proof. (α) ⇐⇒ (β) follows from 5.2.4 and 5.2.6; (α) ⇐⇒ (γ ) follows from 5.7.8 and 5.7.13. It was proved by Ka¸kol in [105] that the above properties (α)–(γ ) are equivalent to: (δ) The completion of a dual-separating normed space is dual-separating. Hence, completions of dual-separating spaces may not be dual-separating when K is not spherically complete (see also 5.1.8).

6 C-compactness

In this chapter we describe an alternative approach to compactoidity theory. It will lead to results that are more powerful than the ones of Section 3.8 but, on the other hand, it works for spherically complete scalar fields only. In Section 3.8 we took the “convexification” of precompactness as a starting point and viewed convex complete compactoids as the counterparts of the classical convex compact sets. In this chapter we will “convexify” the well-known intersection property of closed subsets of a compact set. It brings about the notion of c-compactness. The connection with compactoidty will be explained in 6.1.13; see also 6.3.2.

6.1 Basics Definition 6.1.1 Let C be a convex subset of a Hausdorff locally convex space E. C is called c-compact (the prefix “c” standing for “convex”) if the following holds. Let C be a collection of non-empty relatively closed convex subsets of C with the binary intersection property, i.e., such that C1 , C2 ∈ C implies C1 ∩ C2 ∈ C. Then C = ∅. From the definition it follows that c-compactness does not depend on the embedding space. We first prove some immediate consequences. From now on in this chapter E, F are Hausdorff locally convex spaces. Clearly the empty set and singleton sets are c-compact. Translates of c-compact sets are c-compact. We also have the following.

244

6.1 Basics

245

Theorem 6.1.2 (i) A c-compact set is complete. (ii) Closed convex subsets of c-compact sets are c-compact. (iii) Let T ∈ L(E, F ). If C ⊂ E is c-compact then so is T (C). Proof. (i). Let C ⊂ E be c-compact. Since C is also a c-compact subset of the completion of E it suffices to show that C is closed. Let x ∈ C and consider C := {U ∩ C : U is a convex neighbourhood of x}. Then C has the properties stated in 6.1.1, so there is a point in its intersection, which can only be x, and we have x ∈ C. (ii). Let C be a collection of non-empty (relatively) closed convex subsets of C1 with the binary intersection property, where C1 is a closed convex subset of a c-compact set C. All members of C are closed in C, so by c-compactness of C we have C = ∅. (iii). Let C be a collection of non-empty relatively closed subsets of T (C) with the binary intersection property. Then T −1 (C) := {T −1 (V ) ∩ C : V ∈ C} is a collection of non-empty closed convex subsets of C with the binary inter section property, so there is an x ∈ T −1 (C), i.e., x ∈ C and T (x) ∈ V for all V ∈ C, hence T (x) ∈ C proving that T (C) is c-compact. The next theorem and its corollary reveal the restrictive character of c-compactness announced in the introduction. Theorem 6.1.3 The following statements on K are equivalent: (α) (β) (γ ) (δ)

K is spherically complete. K is c-compact. BK is c-compact. BK is spherically complete.

Proof. Each (closed) non-empty convex subset of K is either a ball, a singleton, or K itself. This proves (α) ⇔ (β). (β) =⇒ (γ ) follows from 6.1.2(ii) and (γ ) =⇒ (δ) is obvious. Now we prove (δ) =⇒ (α). Let B(µ1 , |λ1 |) ⊃ B(µ2 , |λ2 |) ⊃ · · · be a nest of “closed” balls in K, where λn ∈ K for each n ∈ N, |λ1 | > |λ2 | > · · · . By boundedness there is a λ ∈ K such that |µn | ≤ |λ|, |λn | ≤ |λ| for each n. Then B(λ−1 µ1 , λ−1 λ1 ) ⊃ B(λ−1 µ2 , λ−1 λ2 ) ⊃ · · · is a nest of balls in BK , so by (δ) there is an x in the intersection. But then λ x ∈ B(µn , |λn |) for each n. Corollary 6.1.4 Suppose K is not spherically complete. Then the only nonempty c-compact sets in E are the singletons.

246

C-compactness

Proof. It suffices to show that each c-compact absolutely convex set A equals {0}. If not, we had an x ∈ A, x = 0. Then BK x is closed in A, hence c-compact (6.1.2(ii)). By looking at the map BK −→BK x given by λ → λ x we obtain easily that BK is c-compact, hence K is spherically complete by the previous theorem, a contradiction. The next step is to prove that the product of c-compact sets is again c-compact (a convexification of Tychonov’s Theorem). For this it is most convenient to use the notion of convex filters, which we will introduce below. For a basic introduction to filters, see Appendix A. Definition 6.1.5 Let C be a convex set in E. A convex filter (on C) is a filter on C that has a base of convex sets. In other words, a convex filter on C is a non-empty collection F of subsets of C such that: (i) (ii) (iii) (iv)

∅ ∈ F, V , W ∈ F =⇒ V ∩ W ∈ F, V ⊂ W ⊂ C, V ∈ F =⇒ W ∈ F, For each V ∈ F there is a convex set W ∈ F for which V ⊃ W .

By Zorn’s Lemma each convex filter on C can be extended to a maximal one (i.e., a convex filter F on C such that for all convex filters F on C with F ⊃ F we have F = F). Theorem 6.1.6 Let F be a convex filter on a convex set C. Then the following are equivalent: (α) F is maximal. (β) For each convex C ⊂ C we have C ∈ F ⇐⇒ C ∩ V = ∅ for all V ∈ F. Proof. (α) =⇒ (β). Let C ∈ F. Then clearly C ∩ V = ∅ for all V ∈ F (this holds for each filter F on C). Conversely, let C ⊂ C be convex such that C ∩ V = ∅ for all V ∈ F. Let F := {W ⊂ C : W ⊃ C ∩ V for some V ∈ F}. Then F is a convex filter on C and F ⊃ F. By maximality we have F = F, so C ∈ F = F. (β) =⇒ (α). Let F be a convex filter on C, F ⊃ F. Let C ∈ F . There is a convex set C ∈ F such that C ⊃ C . For all V ∈ F one verifies C ∩ V = ∅. Hence by (β) we have C ∈ F, so also C ∈ F. Thus F = F and F is maximal. Theorem 6.1.7 Let x be a cluster point of a maximal convex filter F on a convex set C. Then F → x.

6.1 Basics

247

Proof. Let U be a convex neighbourhood of x in E; we have to prove that U ∩ C ∈ F. In fact, from the definition of cluster point we obtain (U ∩ C) ∩ V = ∅ for each V ∈ F. Then apply 6.1.6. Now we can formulate c-compactness in terms of convex filters. Theorem 6.1.8 Let C be a convex subset of E. Then the following are equivalent: (α) C is c-compact. (β) Each convex filter on C has a cluster point. (γ ) Each maximal convex filter on C converges. Proof. (α) =⇒ (β). Let F be a convex filter on C. Let C be the collection of all members of F that are closed and convex (note that by 6.1.2(i) we have C ∈ C). Then by (α) and 6.1.1 there exists an x ∈ C. Now let V ∈ F. Then there is a convex set C ∈ F with V ⊃ C . Hence C ∈ C, so that x ∈ C ⊂ V . We see that x is a cluster point of F. (β) =⇒ (γ ). This is 6.1.7. (γ ) =⇒ (α). Let C be a collection of non-empty relatively closed convex subsets of C with the binary intersection property. Then {W ⊂ C : W contains a member of C} is a convex filter on C that can be extended to a maximal one which by (γ ) converges to some point x ∈ C. Then for each V ∈ C we have x ∈ V ∩ C = V , so that x ∈ C. As an application of the previous theory we derive some more properties of c-compactness, among which we have a new characterization of spherical completeness of K (6.1.10). Theorem 6.1.9 Let C1 ⊂ E, C2 ⊂ F be convex sets, let T : E−→F be a linear map for which T (C1 ) = C2 . Let F be a maximal convex filter on C1 . Then T (F ) is a maximal convex filter on C2 . Proof. That T (F) is a convex filter on C2 is obvious; we prove maximality. Let C ⊂ C2 be convex such that C ∩ T (V ) = ∅ for all V ∈ F. By 6.1.6 it suffices to show that C ∈ T (F). Now from C ∩ T (V ) = ∅ it follows that (T −1 (C ) ∩ C1 ) ∩ V = ∅. By maximality of F and 6.1.6 we obtain T −1 (C ) ∩ C1 ∈ F. Hence C = T (T −1 (C ) ∩ C1 ) ∈ T (F). Theorem 6.1.10 Properties (α) − (δ) of 6.1.3 are equivalent to: (ε) Each maximal convex filter on K is an ultrafilter. Proof. (β) =⇒ (ε). Let F be a maximal convex filter on K. By assumption and 6.1.8 it converges to some x ∈ K. We first claim that x belongs to every

248

C-compactness

V ∈ F. In fact, V contains a convex set W ∈ F. If x ∈ W then, since W is closed, there is a ball B about x for which B ∩ W = ∅. But B ∈ F (since F → x), a contradiction. Now F = {V ⊂ K : x ∈ V } is an ultrafilter, is convex (as {x} ∈ F ), and contains F. By maximality, F = F . So, F is an ultrafilter and we get (ε). (ε) =⇒ (α). Suppose K is not spherically complete; let us prove that then (ε) does not hold. Let B1 ⊃ B2 ⊃ · · · be a decreasing sequence of balls in K with n∈N Bn = ∅. Let F := {V ⊂ K : V ⊃ Bn for some n}. It is easily seen that F is a convex filter on K. We now show that F is a maximal convex filter on K. By 6.1.6 it suffices to check that C ⊂ K, C convex, C ∩ V = ∅ for all V ∈ F =⇒ C ∈ F. So, let C satisfy the left-hand side. Then C ∩ Bn = ∅ for all n. But C ⊂ Bn for all n is impossible, because n∈N Bn = ∅. Thus, there is an m ∈ N for which C ⊂ Bm . Therefore, since C is a ball, applying the mercury drop behaviour (see Chapter 1) we obtain that C ⊃ Bm , implying C ∈ F. Finally, assume that F is an ultrafilter; we derive a contradiction. Choose x1 ∈ B1 , x2 ∈ B2 , . . . and put W := {x1 , x2 , . . .}. Clearly we have W ∩ Bn = ∅ for all n, so W ∩ V = ∅ for all V ∈ F. By assumption that F is an ultrafilter we must have W ∈ F. Then W ⊃ Bn for some n. But W is countable and Bn is not, a contradiction. Theorem 6.1.11 Every weakly c-compact set in E is c-compact. All the admissible topologies on E have the same c-compact sets. Proof. It suffices to show the first assertion, and we may assume (6.1.4) that K is spherically complete. Let C ⊂ E be a weakly c-compact set and let F be a convex filter on C with a base B of convex sets. By 6.1.8 there is an x ∈ E which is in the weak closure of each member V of B. Since by 5.7.4(ii) the closure of V coincides with its weak closure, x is a cluster point of F. So again by 6.1.8 it follows that C is c-compact. Theorem 6.1.12 Let {Ei : i ∈ I } be a collection of Hausdorff locally convex spaces. For each i ∈ I , let Ci ⊂ Ei be c-compact. Then i∈I Ci ⊂ i∈I Ei is c-compact. Proof. Let F be a maximal convex filter on C := i∈I Ci ; we prove (6.1.8) that F converges. For each i ∈ I , let πi : i∈I Ei −→Ei be the coordinate map. By 6.1.9 the convex filter πi (F) on Ci is maximal, so it converges to

6.1 Basics

249

some ci ∈ Ci (6.1.8). Now let c := (ci )i∈I ∈ C; we prove that F → c. Let U be a neighbourhood of c in C. To show that U ∈ F we may suppose that U = i∈I Ui , where Ui = Ci for all i outside a finite set J ⊂ I and where Ui is a convex neighbourhood of ci in Ci for i ∈ J . Now let i ∈ J . Since πi (F ) → ci we have Ui ∈ πi (F), so that Ui ∩ πi (V ) = ∅ for all V ∈ F, i.e., (C ∩ πi−1 (Ui )) ∩ V = ∅ for all V ∈ F. From maximality of F and 6.1.6 we obtain C ∩ πi−1 (Ui ) ∈ F. Hence also Z := C i∈J πi−1 (Ui ) ∈ F. We claim that Z ⊂ U (then we will be done). In fact, let x = (xi )i∈I ∈ Z. Then x ∈ C and for each i ∈ J , xi = πi (x) ∈ Ui . As Ui = Ci for i ∈ I \ J it follows that x ∈ i∈I Ui = U . After these preparations we are able to prove the announced connection between c-compactness and compactoidity. Theorem 6.1.13 Let K be spherically complete. For a convex set C in E the following are equivalent: (α) C is c-compact and bounded. (β) C is a complete compactoid. Proof. (α) =⇒ (β). 6.1.2(i) takes care of completeness. To prove compactoidity we may assume that C is absolutely convex. By boundedness and 3.9.16 it suffices to show that each “orthogonal” sequence e1 , e2 , . . . in C tends to 0. Consider the sets Cn := e1 + · · · + en + aco {en+1 , en+2 , . . .}. We have C ⊃ C1 ⊃ C2 ⊃ · · · and each Cn is convex and closed. By c compactness there is an x ∈ n∈N Cn . Now from 3.9.13 it follows that x ∞ has a unique expansion i=1 λi ei , where λi ∈ K, λi ei → 0. But, for each n, x ∈ Cn so x has an expansion e1 + · · · + en + i>n µi ei (µi ∈ K, µi ei → 0). ∞ It follows that λi = 1 for all i, so that x = i=1 ei and we deduce that ei → 0. (β) =⇒ (α). Define a map T : E−→K E by T (x) := (f (x))f ∈E

(x ∈ E).

Then T is linear, continuous and injective as E is dual-separating (5.1.4). To show that T maps C homeomorphically onto T (C), let (xi )i be a net in C such that T (xi ) → T (x) for some x ∈ C. This means that f (xi ) → f (x) for all f ∈ E , i.e., that xi → x weakly. But the weak and the initial topologies coincide on C (5.2.12), so that xi → x. Now T (C) is the linear homeomorphic image of the complete set C, so T (C) itself is complete, hence closed in K E . Now 6.1.3 and 6.1.12 tell us that K E is c-compact. Hence so are T (C) (6.1.2(ii)) and C.

250

C-compactness

Corollary 6.1.14 Let K be spherically complete. Then we have the following: (i) Any finite-dimensional convex subset of E is c-compact. (ii) E is finite-dimensional if and only if it has a bounded c-compact zero neighbourhood (and then E is c-compact). In particular, a normed space E is finite-dimensional if and only if E has a c-compact zero neighbourhood. (iii) If C ⊂ E is c-compact and bounded then [C] is of countable type. (iv) If K is locally compact then a convex set C ⊂ E is c-compact and bounded if and only if C is compact. Proof. (i). A finite-dimensional subspace D of E is isomorphic to some K n , n ∈ N (3.4.24), so D is c-compact by 6.1.3 and 6.1.12. Also, the convex subsets of D are closed (3.4.22(i)), hence c-compact by 6.1.2(ii). (ii). The “only if” of the first part follows from (i). Also, to get the “if” apply 3.8.5 and 6.1.13. The rest of (ii) is clear. Finally, (iii) (resp. (iv)) follows from 4.2.2 (resp. 3.8.3) and 6.1.13.

6.2 Permanence properties Theorem 6.1.13 enables us to combine c-compactness and compactoidity leading to the following permanence properties, which are more general than the ones given in Section 3.8. Theorem 6.2.1 (Compare with 3.8.38 and 5.5.8(ii)) Let K be spherically complete, let C be a complete convex compactoid in a Hausdorff locally convex space (E, τ ). If τ1 is a weaker Hausdorff locally convex topology on E then τ1 = τ on C. Proof. We may assume that C is absolutely convex. Let (xi )i be a net in C τ1 τ for which xi −→ 0. To prove xi −→ 0, let U be an absolutely convex τ -zero neighbourhood in E. Let ρ ∈ K, 0 < |ρ| < 1. By compactoidity and 3.8.9 there are x1 , . . . , xn ∈ C such that ρ C ⊂ U + aco {x1 , . . . , xn }. Then also ρ C ⊂ U ∩ C + aco {x1 , . . . , xn }. Now U ∩ C is τ -closed in C and further, by 6.1.13, C is c-compact with respect to τ . Hence so is U ∩ C. Now applying 6.1.2 to the identity map (E, τ )−→(E, τ1 ), U ∩ C is also c-compact with respect to τ1 , hence τ1 -closed. τ1 By 3.8.22 we have, since ρ xi ∈ ρ C and ρ xi −→ 0, that ρ xi ∈ ρ −1 (U ∩ C)

6.2 Permanence properties

251

τ

for large i, that is, ρ 2 xi ∈ U ∩ C ⊂ U . It follows that ρ 2 xi −→ 0, hence τ xi −→ 0. Remark 6.2.2 At this point it is a natural question to ask whether a continuous linear map T (not necessarily injective) restricted to a c-compact set C is an open mapping C−→T (C). But we have seen in 3.8.36 that this is not true in general. Theorem 6.2.3 (Compare with 3.8.29(i), 3.8.31(i) and 5.5.8(i)) Let T ∈ L(E, F ), let C ⊂ E be c-compact. Then we have the following: (i) T (C) is complete. (ii) C e is c-compact and T (C e ) = (T (C))e . In particular, if C is edged then so is T (C). Proof. To get (i) apply 6.1.2. (ii). We may assume that the valuation of K is dense and that C is absolutely convex. For c-compactness of C e use again 6.1.2. The inclusion T (C e ) ⊂ (T (C))e is clear. To prove the opposite inclusion let y ∈ (T (C))e . Let λ ∈ K, |λ| > 1. Then y ∈ λ T (C) = T (λ C), so T −1 ({y}) ∩ λ C is non-empty and closed in λ C. By c-compactness of λ C there is an x ∈ 1<|µ|≤|λ| T −1 ({y}) ∩ µ C. Hence T (x) = y and x ∈ {µ C : |µ| > 1} = C e . The rest of (ii) is obvious. From 6.1.2(iii) and 6.1.12 it follows that, if C and D are c-compact in E then C + D is also c-compact (C + D is the image of the c-compact set C × D ⊂ E × E under addition). But we have more. Theorem 6.2.4 (Compare with 3.8.29(ii), 3.8.31(ii) and 4.1.9) Let C, D ⊂ E, where C is c-compact and where D is convex and closed. Then C + D is closed. Proof. Let U be the collection of all absolutely convex zero neighbourhoods in E, let x ∈ C + D. For each U ∈ U, (x + U ) ∩ (C + D) = ∅, so that CU := {c ∈ C : x − c ∈ D + U } (= {c ∈ C : (x + U ) ∩ (c + D) = ∅}) is non-empty, convex and closed. Since CU ∩ CV ⊃ CU ∩V for U, V ∈ U we have by 6.1.8 that {CU : U ∈ U} = ∅, let z be in the intersection. Then z ∈ C and x − z ∈ D + U for all U ∈ U. Hence x − z ∈ D = D and x ∈ z + D ⊂ C + D. Remarks 6.2.5 (a) If in 6.2.4 the set D is, in addition, complete then C + D is complete. (b) Let K be spherically complete, not locally compact. Then 6.2.4 fails when we drop the convexity condition of D. In fact, let λ1 , λ2 , . . . , µ1 , µ2 , . . .

252

C-compactness and X be as in 3.8.30(b). Take C := {(λ, 0) : λ ∈ BK }, D := X . Then C is absolutely convex and c-compact in K 2 and X is a complete compactoid in K 2 . We claim that C + D is not closed. In fact, we have C ∩ D = ∅, so 0 ∈ D − C = C + D. But, for each n ∈ N, (0, µn ) = (λn , µn ) − (λn , 0) ∈ C + D, so that 0 ∈ C + D.

Theorem 6.2.6 (Compare with 3.1.6(iv) and 3.8.32) Let C, D ⊂ E, where C is c-compact and where D is convex and closed. Then (C + D)e = C e + D e . Proof. We may assume that the valuation of K is dense and that C, D are absolutely convex edged sets; we prove (C + D)e ⊂ C + D. Then the conclusion follows from 3.1.6 and 6.2.3(ii). Let y ∈ (C + D)e . For each λ ∈ K, |λ| > 1 we have y ∈ λ (C + D), i.e., Cλ := (y − λ D) ∩ λ C is non-empty, convex and closed in λ C. By c-compactness there is an x ∈ 1<|µ|≤|λ| Cµ . Then x ∈ {µ C : |µ| > 1} = C e = C and x ∈ y − µ D for all µ ∈ K with |µ| > 1, i.e., y − x ∈ D e = D and y ∈ C + D.

6.3 Notes C-compactness was introduced by Springer in 1965, [226]. For a survey on references (e.g. [192] for normed spaces and [62], [99], [227] for general locally convex spaces) and on results about c-compactness until 1986, see [198]. In [37] and [38] some applications of c-compactness to differential operators were obtained. C-compact sets need not be bounded. In fact, if K is spherically complete, K is a c-compact space over K; more generally, any product K I × BKJ is a non-bounded c-compact subset of K I × K J . Throughout 6.3.1−6.3.3 K is spherically complete and E := (E, τ ) is a Hausdorff locally convex space (over K ). In [62] De Grande-De Kimpe gave the following characterizations of c-compact spaces (compare (γ ) with 6.1.14(ii)). Theorem 6.3.1 ([62], Propositions 10, 11) The following are equivalent: (α) E is c-compact. (β) E is isomorphic to a closed subspace of some power of K. (γ ) E has a c-compact zero neighbourhood.

6.3 Notes

253

Observe that (β) above equals 5.3.11(β), from which we obtain that c-compact spaces are exactly those that are complete and of finite type. Also, the following unbounded versions of 6.1.13 and 6.2.1 respectively were given by Schikhof in [197]. Theorem 6.3.2 ([197], 2.2) For a convex set C in E the following are equivalent: (α) C is c-compact. (β) C is complete and for every zero neighbourhood U in E there exists a finite-dimensional subspace D of E such that C ⊂ U + D. Theorem 6.3.3 ([197], 3.2) Let C be a c-compact set in E. If τ1 is a Hausdorff locally convex topology on E weaker than τ , then τ1 = τ on C. One may wonder what happens if we approach compactoidity by means of “convexifying” the definition of compactness by open coverings. For an instructive example take BK− , where K is densely valued. Then {ρ BK : |ρ| < 1} is an open absolutely convex covering of BK− but there is no finite subcovering. So we could think of introducing (c∗ ): Let E be a locally convex space, let X ⊂ E. We say that X has (c*) if it satisfies the following property. For each λ ∈ K, λ := 1 if the valuation of K is discrete, |λ| > 1 otherwise and each open covering {Ui : i ∈ I } of X, there is a finite set J ⊂ I such that λ−1 X ⊂ co ( Ui : i ∈ J ). But this does not lead to a new concept. Theorem 6.3.4 X is a compactoid if and only if X has (c*). Proof. Let X be a compactoid. To prove (c*) we may assume that 0 ∈ X and that all Ui are convex. We have 0 ∈ Ui0 for some i0 ∈ I . By compactoidity and 3.8.9 there are x1 , . . . , xn ∈ X such that λ−1 X ⊂ Ui0 + aco {x1 , . . . , xn } = co ({x1 , . . . , xn } ∪ Ui0 ). Now there are i1 , . . . , in ∈ I such that xm ∈ Uim for m ∈ {1, . . . , n}. Then co ({x1 , . . . , xn } ∪ Ui0 ) ⊂ co ( Uij : j ∈ {0, 1, . . . , n}) and (c*) is proved. Conversely, let X have (c*). Let λ be as above. Let U be an absolutely convex zero neighbourhood. Then so is λ−1 U . All additive cosets of λ−1 U cover X. So by (c*) there are y1 , . . . , yn ∈ E such that λ−1 X ⊂ co ( yi + λ−1 U : i ∈ {1, . . . , n}) ⊂ λ−1 U + aco {y1 , . . . , yn }. Hence, X ⊂ U + aco {λ y1 , . . . , λ yn }, so X is a compactoid.

7 Barrelledness and reflexivity

We first follow the “classical” path by developing the notion of barrelledness, a key tool for the theory of reflexivity to be treated in Section 7.4. Indeed, we will prove (7.1.3) that Fr´echet spaces are barrelled, by using the Baire Category Theorem. Nevertheless, for a proper characterization of reflexivity (see 7.4.13) we need a modification of the notion of barrelledness by introducing the wider concept of “polar barrelledness” in 7.1.6. The fact that it is not identical to “ordinary” barrelledness (7.1.10) is a typical non-Archimedean feature. Despite of this, the proofs of the hereditary properties of (polar) barrelledness for quotients (7.1.12(i)), for locally convex direct sums and inductive limits (7.1.13), for products (7.1.15) and completions (7.1.17) are basically “classical”. In the same spirit we introduce in Section 7.3 the weak star and strong topologies on the dual E , define reflexivity in 7.4.1 and prove the characterization 7.4.13 of reflexivity. Products and locally convex direct sums of reflexive spaces are reflexive (7.4.23). However, after this point our results are going to diverge from the classical ones; therefore, we mention a few facts. For spherically complete K we show in 7.4.19 that every reflexive space is semi-Montel (i.e., each bounded set is a compactoid, see 8.4.1(i) and 8.4.5(δ)), so that reflexive Banach spaces are finite-dimensional. This is in contrast to 7.4.30 showing that every Fr´echet space of countable type over a nonspherically complete K is reflexive. If K is spherically complete, closed subspaces of reflexive Fr´echet spaces are reflexive (7.4.27), but if K is not, the reflexive Banach space ∞ contains a closed non-reflexive subspace (7.4.18(i)). In Sections 7.2 and 7.5 we test our examples on (polar) barrelledness and reflexivity, respectively. Section 7.6 treats various metrizability properties of E and E and is a stepping stone for Chapter 8. 254

7.1 Polar barrelledness, hereditary properties

255

7.1 Polar barrelledness, hereditary properties We start by taking over the classical notions. Definition 7.1.1 A barrel in a locally convex space is an absolutely convex, closed, absorbing set. Any absolutely convex zero neighbourhood is a barrel. The example {(λ1 , λ2 , . . .) ∈ c00 : |λn | ≤ n1 for each n} shows that the converse is not true, which leads to the following. Definition 7.1.2 A locally convex space is barrelled if each barrel is a zero neighbourhood. Thus, the normed space c00 is not barrelled. As in the classical case we have: Theorem 7.1.3 Fr´echet spaces, in particular Banach spaces, are barrelled. Proof. Let B be a barrel in a Fr´echet space E, let λ1 , λ2 , . . . ∈ K, |λn | > n for each n. Then E = n∈N λn B and by the Baire Category Theorem we have that λn B has a non-empty interior for some n. By convexity λn B is open (3.3.11(ii)), hence so is B = λ−1 n (λn B). Remark 7.1.4 We will see in 7.1.19 that, despite of the above result, completeness and barrelledness in general are not related. Barrelledness can be expressed in terms of seminorms. Theorem 7.1.5 A locally convex space is barrelled if and only if the supremum of each pointwise bounded non-empty collection of continuous seminorms is a continuous seminorm. Proof. To prove the “if”, let B be a barrel in a locally convex space E. By 4.1.6 for each x ∈ E \ B there is a continuous seminorm px for which px (x) = 1 but px (B) < 1. Since B is absorbing the collection P := {px : x ∈ E \ B} is pointwise bounded, so by assumption p := sup P is continuous and from {y ∈ E : p(y) < 1} ⊂ B we infer that B is a zero neighbourhood. To prove the “only if”, let P be a pointwise bounded non-empty collection of continuous seminorms, let p := sup P. Then {x ∈ E : p(x) ≤ 1} = {x ∈ E : q(x) ≤ 1} q∈P

is a barrel, hence a zero neighbourhood by assumption. It follows that p is continuous. We now introduce a weaker form of barrelledness that in non-Archimedean duality theory is more useful than barrelledness. (See 7.4.13.)

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Definition 7.1.6 A locally convex space is polarly barrelled if each polar barrel is a zero neighbourhood. We leave the proof of the following characterizations (that are in the spirit of 7.1.5) to the reader. As usual, if E, F are locally convex spaces, a set F ⊂ L(E, F ) is called pointwise bounded if for each x ∈ E the set {f (x) : f ∈ F} is bounded in F . Theorem 7.1.7 For a locally convex space E the following are equivalent: (α) E is polarly barrelled. (β) The supremum of each pointwise bounded non-empty collection of polar continuous seminorms is a (polar) continuous seminorm. (γ ) Each pointwise bounded non-empty set F ⊂ E is equicontinuous, i.e., the seminorm sup{|f | : f ∈ F} is continuous. (δ) For each polar locally convex space F , each pointwise bounded non-empty set in L(E, F ) is equicontinuous. Remarks 7.1.8 (a) (α) =⇒ (δ) is an extension of the Uniform Boundedness Principle 2.1.20 to polarly barrelled spaces, promised in 3.5.11. (b) It is obvious from (α)=⇒ (δ) that the Banach–Steinhaus Theorem 2.1.21 (let T1 , T2 , . . . ∈ L(E, F ) be such that T (x) := limn Tn (x) exists for every x ∈ E. Then T ∈ L(E, F )) holds for E polarly barrelled, locally convex, F polar, locally convex. This was also announced in 3.5.11. We now compare the notions “barrelled” and “polarly barrelled” (it is not hard to see that their Archimedean counterparts coincide). Theorem 7.1.9 (i) Barrelledness implies polar barrelledness. (ii) If K is spherically complete or the space is of countable type then “barrelledness” and “polar barrelledness” coincide. Proof. We only have to establish (ii). Let B a barrel in a polarly barrelled space E. By 5.2.9 the set B e is polar, hence a zero neighbourhood. Let ρ ∈ K, 0 < |ρ| < 1. Then ρ B e ⊂ B, so B is a zero neighbourhood. It takes some effort to find an example of a polarly barrelled space that is not barrelled. The following construction produces such a space which is, in addition, normed and polar. Example 7.1.10 Let K be not spherically complete. Then there is a subspace E of ∞ that is polarly barrelled but not barrelled.

7.1 Polar barrelledness, hereditary properties

257

Proof. For each n ∈ N let Sn := {X ⊂ N : k −→

(X ∩ {1, . . . , k}) √ n k is bounded}. k

Straightforward verification shows that S1 is the collection of all finite subsets of N and that S1 ⊂ S2 ⊂ · · · is strictly increasing. For n ∈ {0, 1, . . .} let En be the Banach space of all (λ1 , λ2 , . . .) ∈ ∞ such that for each ε > 0 the set {m ∈ N : |λm | ≥ ε} belongs to Sn+1 . Then the sequence c0 = E0 ⊂ E1 ⊂ · · · is strictly increasing. Now put E :=

En .

n∈N

To prove that E -with the (norm) topology inherited from ∞ -meets the requirements we use the following six steps: 1. If n ∈ N, f ∈ En , f = 0 on c0 then f = 0. Proof. Fix x := (λ1 , λ2 , . . .) ∈ En and ε > 0. It is enough to prove that |f (x)| ≤ ε f . To this end set X := {m ∈ N : |λm | ≥ ε}. Then X ∈ Sn+1 , so for the closed subspace D of En defined by D := {(η1 , η2 , . . .) ∈ ∞ : ηm = 0 if m ∈ N \ X} we have dist (x, D) ≤ ε and it therefore suffices to show that f = 0 on D. This is clear if X is finite (then D ⊂ c0 ), so let X = {s1 , s2 , . . .} with s1 < s2 < · · · . The formula φ((µ1 , µ2 , . . .)) := (η1 , η2 , . . .) where ηm := 0 if m ∈ X, ηm := µj if m = sj for some j , defines a linear surjective isometry φ : ∞ −→D for which φ(c0 ) = D ∩ c0 . It follows that ∞ /c0 is isometrically isomorphic to φ( ∞ )/φ(c0 ) = D/D ∩ c0 . Now 4.1.12 tells that (D/D ∩ c0 ) = {0}, so we must have f = 0 on D. 2. Let n ∈ N, f ∈ En∗ . Then f ∈ En if and only if there exists a (unique) y := (µ1 , µ2 , . . .) ∈ c0 such that f ((λ1 , λ2 , . . .)) =

∞

λm µ m

m=1

for all (λ1 , λ2 , . . .) ∈ En , and we have f = y∞ .

(7.1)

258

3.

4.

5.

6.

Barrelledness and reflexivity

Proof It suffices to show the “only if”. Since c0 ⊂ En , there exists by 2.5.11 a y = (µ1 , µ2 , . . .) ∈ ∞ such that (7.1) holds for all (λ1 , λ2 , . . .) ∈ c0 . We now prove that actually y ∈ c0 . (Once we have this we will be done since then (7.1) holds for all (λ1 , λ2 , . . .) ∈ En as c0 is weakly dense in En (by step 1 and 5.2.5) and both sides of (7.1) define elements of En .) Suppose y ∈ c0 ; we derive a contradiction. There exists an ε > 0 such that Y := {m ∈ N : |µm | ≥ ε} is infinite and therefore it must contain a set X := {s1 , s2 , . . .} ∈ Sn+1 where s1 < s2 < · · · . Then φ : ∞ −→En , defined in a similar way as in step 1, is a linear isometry and f ◦ φ ∈ ( ∞ ) . By 5.5.5 there exists an ∞ (f ◦φ)((λ x = (α1 , α2 , . . .) ∈ c0 such that 1 , λ2 , . . .)) = m=1 λm αm for all ∞ (λ1 , λ2 , . . .) ∈ . But then αj = µsj ≥ ε for all j ∈ N, a contradiction. The equality f = y∞ is straightforward. Let n ∈ N, f ∈ En . Then f extends uniquely to an element g ∈ En+1 , and g = f . Proof. The statement follows directly from steps 1 and 2. Let p be a polar seminorm on E such that p|En is continuous for each n ∈ N. Then p is continuous. Proof. Let F := {f ∈ E ∗ : |f | ≤ p}. Since p|E1 is continuous there is a C > 0 such that p(x) ≤ C x∞ for all x ∈ E1 . Then |f (x)| ≤ C x∞ for all x ∈ E1 , f ∈ F. By step 3 we arrive inductively at |f (x)| ≤ C x∞ for all x ∈ n∈N En = E, f ∈ F. Then for all x ∈ E we have p(x) = sup{|f (x)| : f ∈ F} ≤ C x∞ , proving continuity of p. E is polarly barrelled. Proof. Let P be a pointwise bounded non-empty family of polar continuous seminorms on E. Then p := sup P is a polar seminorm on E. Since each En is a Banach space, hence barrelled by 7.1.3, p|En is continuous for each n and by step 4, p is continuous. E is not barrelled. Proof. By the Baire Category Theorem E = n∈N En is not complete, hence not closed in ∞ ; let y ∈ E \ E, let λ1 , λ2 , . . . ∈ K, 0 < |λ1 | < |λ2 | < · · · , limn |λn | = ∞. For each x ∈ E we have dist (x, En ) = 0 eventually, so the formula

p(x) := max n∈N

|λn | dist (x, En ) dist (y, En )

defines a seminorm p on E. If E were barrelled, p would be continuous and could be extended to a continuous seminorm p on E. By choosing a

7.1 Polar barrelledness, hereditary properties

259

sequence y1 , y2 , . . . in E tending to y we get for each n, p(y) = lim p(ym ) ≥ lim m

m

|λn | dist (ym , En ) = |λn | , dist (y, En )

an impossibility as limn |λn | = ∞.

Remark 7.1.11 The construction above tells us that the topology on E is the strongest polar locally convex topology such that all injections En −→E are continuous. It is quite remarkable that this “polar inductive limit” is a normable and non-complete space. In Chapter 11 we will deal with these features more thoroughly. We continue with some hereditary properties. Theorem 7.1.12 (i) Quotients of (polarly) barrelled spaces are (polarly) barrelled. (ii) Let E be a (polarly) barrelled space and let P ∈ L(E) be a continuous projection. Then P (E) is (polarly) barrelled. Proof. The proof of (i) is straightforward. To see that (ii) is a special case of (i), let D := Ker P and let π : E−→E/D be the quotient map. We have the natural factorization / P (E) EC CC w; P1 ww CC w C ww π CC ! ww E/D P

and the fact that π|P (E) is the inverse of P1 , which shows that P1 is a homeomorphism, implying that P (E) ∼ E/D. Theorem 7.1.13 (i) The locally convex direct sum of a collection {Ei : i ∈ I } of locally convex spaces is (polarly) barrelled if and only if each Ei is (polarly) barrelled. (ii) Inductive limits of inductive systems of (polarly) barrelled spaces are (polarly) barrelled. Proof. Immediate from the definitions and 7.1.12.

To prove that (polar) barrelledness is stable for products we need a classical lemma. Lemma 7.1.14 Let B be a barrel in a locally convex space E, let A ⊂ E be absolutely convex and bounded. Suppose that [A] is Hausdorff and that A is complete. Then A ⊂ λ B for some λ ∈ K.

260

Barrelledness and reflexivity

Proof. let τ1 be the restriction topology on [A], let τ2 be the norm topology on [A] induced by the Minkowski function pA . By τ1 -boundedness of A we have τ1 ≤ τ2 . The scalar multiples of A form a τ2 -neighbourhood base of zero, consisting of τ1 -closed sets, so τ1 is closely related to τ2 in the sense of 3.4.16. By 3.4.17, ([A], τ2 ) is a Banach space containing the barrel [A] ∩ B, which by 7.1.3 is a τ2 -neighbourhood of zero. So, for some λ ∈ K, A ⊂ λ ([A] ∩ B) ⊂ λ B. Theorem 7.1.15 The product of a collection {Ei : i ∈ I } of locally convex spaces is (polarly) barrelled if and only if each Ei is (polarly) barrelled. Proof. If the product is (polarly) barrelled we can use the canonical projections πi : E−→Ei and 7.1.12(ii) to conclude that each Ei is (polarly) barrelled. Now suppose that each Ei is (polarly) barrelled. Let P be a pointwise bounded non-empty family of (polar) continuous seminorms on E := i∈I Ei . By 7.1.5 and 7.1.7 it suffices to show that p := sup P is continuous. To this end we first prove that there is a finite set J ⊂ I such that p = 0 on i∈J Ei (considered as a subspace of E in the natural way). For that, since Ker p is closed, it is enough to show that p = 0 on Ei (again considered as a subspace of E) for almost all i ∈ I . Suppose the contrary. Then there exist i1 , i2 , . . . ∈ I , x1 ∈ Ei1 , x2 ∈ Ei2 , . . . such that p(xn ) ≥ n for all n. Let A be the set of all (yi )i∈I ∈ E such that yi = 0 if i ∈ I \ {i1 , i2 , . . .} and yi ∈ BK xn if n ∈ N, i = in . Then [A] is Hausdorff, A is complete, bounded and absolutely convex. By 7.1.14 there is a λ ∈ K such that A ⊂ λ {x ∈ E : p(x) ≤ 1} conflicting the fact that p is unbounded on A. Thus, we have proved that there is a finite set {j1 , . . . , jm } ⊂ I such that p = 0 on i∈{j1 ,...,jm } Ei . For x := (xi )i∈I ∈ E we then have p(x) = p(xj1 + · · · + xjm ) ≤ max1≤k≤m p(xjk ) = (max1≤k≤m p ◦ πjk )(x). Now, for each k ∈ {1, . . . , m}, p ◦ πjk = (p|Ejk ) ◦ πjk . By (polar) barrelledness of Ejk the seminorm p|Ejk , hence also p ◦ πjk is continuous. Then p, being dominated by a maximum of finitely many continuous seminorms, is again continuous. Corollary 7.1.16 (i) The spaces K (I ) and K I are barrelled for each index set I . (ii) Each complete space of finite type is barrelled. (Compare with 7.2.13.) Proof. The proof of (i) follows directly from 7.1.13(i) and 7.1.15. For (ii), use 5.3.11,(α) =⇒ (γ ) and apply (i). The following fact is easy to prove.

7.1 Polar barrelledness, hereditary properties

261

Theorem 7.1.17 A locally convex space having a dense (polarly) barrelled subspace is itself (polarly) barrelled. In particular, the completion of a Hausdorff (polarly) barrelled space is (polarly) barrelled. Proof. Let D be a dense (polarly) barrelled subspace of a locally convex space E. Let P be a pointwise bounded non-empty collection of continuous (polar) seminorms on E; we prove that p := sup P is continuous. By assumption p|D is continuous, so it extends uniquely to a continuous seminorm q on E. For each r ∈ P we have r ≤ q on D, hence on E and it follows that p ≤ q (in fact we have p = q) and continuity of p follows. As for the stability of subspaces, we have, other than 7.1.12(ii), the following (see 7.1.20 for further discussion). Theorem 7.1.18 Let D be a subspace of finite codimension of a locally convex space E. Then E is (polarly) barrelled if and only if D is (polarly) barrelled. Proof. If suffices to consider the case where D has codimension 1. First, suppose that D is (polarly) barrelled. If D is dense then E is (polarly) barrelled by 7.1.17, so we may assume that D = E implying that D = D, i.e., that D is closed. By 3.4.27 and 3.4.28 we have that E is isomorphic to Kx ⊕ D, where x ∈ E \ D. Then (polar) barrelledness follows from 7.1.13(i). Conversely, suppose E is (polarly) barrelled. Whether D is closed or dense, each f ∈ D can be extended to an element of E (apply 3.4.27 in the first case). This implies that, if B is a polar set in D (4.4.10) then B = B ◦◦ ∩ D, where B ◦◦ is the bipolar of B with respect to E. Now let B be a polar barrel (resp. barrel) in D. We consider two cases: 1. B ◦◦ = B (resp. B is closed in E). Then choose x ∈ E \ D and form B1 := B + aco {x}. We have that B1 is absorbing. We claim that B1 is polar (resp. B1 is closed). In fact, by polarity of B there is certainly an f ∈ E with f (x) = 0, so that aco {x} is weakly closed and edged. The latter also holds for B. Then, since B ∩ aco {x} = {0}, we have that B1 is edged. By 3.8.20, B1 is weakly closed. So by 5.2.8, B1 is polar (resp. from 3.8.19(ii), B1 is closed). By assumption B1 is a zero neighbourhood in E, so that B = B1 ∩ D is a zero neighbourhood in D. 2. B ◦◦ = B (resp. B is not closed in E). Then take an x ∈ B ◦◦ \ B (resp. x ∈ B \ B). Since B ◦◦ ∩ D = B (resp. B ∩ D = B) % we & have x ∈ D, so E = Kx + D = [aco {x} + B] ⊂ [B ◦◦ ] (resp. E ⊂ B ). Thus, B ◦◦ (resp. B) is absorbing and therefore, by assumption, a zero neighbourhood in E. Then B = B ◦◦ ∩ D (resp. B = B ∩ D) is a zero neighbourhood in D.

262

Barrelledness and reflexivity

We now are in a position to show that “barrelledness” and “completeness” are, in general, independent. (Compare with 7.1.3 and 7.1.16(ii).) Examples 7.1.19 (i) There exist complete spaces, of countable type, that are not polarly barrelled. (ii) There are non-complete normed spaces, of countable type, that are barrelled. Proof. (i). Let E := ∞ with the topology induced by the seminorms (λ1 , λ2 , . . .) → maxn∈N |λn µn |, where (µ1 , µ2 , . . .) runs through c0 . A standard reasoning shows that E is complete and the linear span of the unit vectors e1 , e2 , . . . is dense in E. For each (λ1 , λ2 , . . .) ∈ E we have that (λ1 , λ2 , . . .)∞ = supn∈N (λ1 , . . . , λn , 0, 0, . . .)∞ , so that . ∞ is the supremum of a collection of polar continuous seminorms. But . ∞ is not continuous as limn en = 0 and en ∞ = 1 for all n. It follows that E is not polarly barrelled. (ii). Let f ∈ c0∗ \ c0 . Then Ker f is not complete, but barrelled by 7.1.18. Remark 7.1.20 From 3.4.10 it follows that every complete locally convex space is a closed subspace of a product of Banach spaces which is barrelled by 7.1.3 and 7.1.15. Thus, the above example (i) shows that subspaces – even closed ones – of (polarly) barrelled spaces need not be (polarly) barrelled. For polarly barrelled spaces the Mackey topology (see 5.7.5) exists. In fact, we even have: Theorem 7.1.21 Any polar, polarly barrelled space is Mackey. Proof. Let (E, τ ) be polar, polarly barrelled, let τ1 be a polar locally convex topology on E for which (E, τ1 ) = (E, τ ) . Let p be a polar τ1 -continuous seminorm. Then p = sup{|f | : f ∈ E , |f | ≤ p}, so that by 7.1.7(α) =⇒ (γ ), p is τ -continuous and we have τ1 ≤ τ .

7.2 Examples of (polarly) barrelled spaces 7.2.1 Immediate examples All the examples of Banach spaces that we met in Subsections 2.5.1–2.5.7 are barrelled by 7.1.3. The Fr´echet spaces A(r) (3.7.26), O(A) if A has a fundamental sequence n ∞ (X), BCW (X) of complete bounded sets (3.7.31), BC ∞ (X) (3.7.47), BCW

7.2 Examples of (polarly) barrelled spaces

263

where W is the collection of all polynomial functions X−→K (3.7.55), C0∞ (X) (3.7.57), S(X) (3.7.64) are barrelled by 7.1.3. The space A† (r) (3.7.27) is an inductive limit of Banach spaces and therefore barrelled by 7.1.3 and 7.1.13(ii).

7.2.2 Barrelledness of spaces of continuous functions We now discuss the polar barrelledness of the spaces (C(X), τs ) and (C(X), τc ) (3.7.1 and 3.7.3). It is convenient to start with τc . Throughout Subsection 7.2.2 X is a non-empty zero-dimensional Hausdorff topological space. Recall (Subsection 3.7.1) the notation pY (f ) := sup{|f (x)| : x ∈ Y } for ∅ = Y ⊂ X, f : X−→K, p∅ (f ) := 0. We start by studying the elements of the dual space (C(X), τc ) , sometimes called measures. First we define the support of such a measure in a natural way. Recall (2.5.20) that (X) is the ring of all clopen subsets of X. Definition 7.2.1 Let ϕ ∈ (C(X), τc ) . A U ∈ (X) is called ϕ-null set if ϕ(f ) = 0 for all f ∈ C(X) having their support in U . Let Nϕ be the union of all ϕ-null sets. The support of ϕ, supp ϕ, is defined as X \ Nϕ . Clearly, if U, V ∈ (X), U ⊂ V and V is a ϕ-null set then so is U . If U, V are ϕ-null sets and supp f ⊂ U ∪ V for some f ∈ C(X) then writing f = f ξU + (f − f ξU ) and observing that supp f ξU ⊂ U , supp (f − f ξU ) ⊂ V we conclude that ϕ(f ) = 0, so that U ∪ V is ϕ-null. It follows that finite unions of ϕ-null sets are ϕ-null. The following lemma discusses infinite unions. Lemma 7.2.2 Let ϕ ∈ (C(X), τc ) , let U ∈ (X) be a union of ϕ-null sets. Then U is ϕ-null. Proof. Let f ∈ C(X) with supp f ⊂ U and let U = i∈I Ui where each Ui is ϕ-null. For each finite subset J of I we set UJ := i∈J Ui . Then UJ is ϕ-null, so that ϕ(f ξUJ ) = 0. Now the f ξUJ , where J runs through the collection of finite subsets of I , form a net in the usual way. We will arrive at the desired conclusion ϕ(f ) = 0 once we can prove that f ξUJ → f with respect to τc . To that end, let Z ⊂ X be compact. Since Z ∩ U is compact there is a finite set J ⊂ I such that Z ∩ U ⊂ UJ . Hence f ξUJ = f on Z ∩ U . If x ∈ Z \ Z ∩ U then both f ξUJ (x) and f (x) are 0. Thus, we find f ξUJ = f

264

Barrelledness and reflexivity

on Z, and obviously f ξUJ = f on Z for all finite sets J with J ⊂ J ⊂ I . It follows that pZ (f − f ξUJ ) → 0. As this holds for each compact Z ⊂ X we τc have f ξUJ −→ f . Corollary 7.2.3 Each clopen subset of Nϕ is a ϕ-null set. Proof. Let U ⊂ Nϕ be clopen, let Nϕ = i∈I Ui , where each Ui is ϕ-null. Then U = i∈I (Ui ∩ U ) and each Ui ∩ U is ϕ-null. Now apply 7.2.2. Theorem 7.2.4 Let ϕ ∈ (C(X), τc ) . Then supp ϕ is compact and there is a C > 0 such that |ϕ(f )| ≤ C psupp ϕ (f ) for all f ∈ C(X). Proof. Nϕ is open, so supp ϕ is closed. There are a compact set Z ⊂ X and a C > 0 such that |ϕ(f )| ≤ C pZ (f ) for all f ∈ C(X). We shall prove: (i) supp ϕ ⊂ Z (implying compactness of supp ϕ), (ii) |ϕ(f )| ≤ C psupp ϕ (f ) for all f ∈ C(X). To prove (i), let U ∈ (X), U ∩ Z = ∅. For f ∈ C(X) with support in U we have pZ (f ) = 0, so that ϕ(f ) = 0. We see that U is ϕ-null. Hence X \ Z is a union of ϕ-null sets and therefore contained in Nϕ . This means that supp ϕ ⊂ Z. To establish (ii), let f ∈ C(X), ε > 0; we prove that |ϕ(f )| ≤ C (psupp ϕ (f ) + ε). For this, consider U := {x ∈ X : |f (x)| > psupp ϕ (f ) + ε}. Then U ∈ (X), U ∩ supp ϕ = ∅, so U ⊂ Nϕ and U is ϕ-null by 7.2.3. Then |ϕ(f )| = ϕ(f ξX\U ) ≤ C pZ (f ξX\U ) ≤ C pX (f ξX\U ). But, for every x ∈ X \ U we have |f (x)| ≤ psupp ϕ (f ) + ε. We see that pX (f ξX\U ) ≤ psupp ϕ (f ) + ε and it follows that |ϕ(f )| ≤ C (psupp ϕ (f ) + ε). Corollary 7.2.5 Let ϕ ∈ (C(X), τc ) . If f ∈ C(X) vanishes on supp ϕ then ϕ(f ) = 0. Now we prove the key lemma. Lemma 7.2.6 Suppose that for each closed but non-compact Y ⊂ X there is a clopen partition U1 , U2 , . . . of X such that Un ∩ Y = ∅ for each n. Then, for any pointwise bounded non-empty family F ⊂ (C(X), τc ) , the set Z := {supp ϕ : ϕ ∈ F} is compact.

7.2 Examples of (polarly) barrelled spaces

265

Proof. Suppose Z is not compact; we derive a contradiction. By assumption there is a clopen partition U1 , U2 , . . . of X such that Un ∩ Z = ∅ for each n. Then also, for each n, there is a ϕn ∈ F such that Un ∩ (supp ϕn ) = ∅. Hence Un is not ϕn -null, so there is an fn ∈ C(X) with support in Un and ϕn (fn ) = 0. Now set xn := (ϕn (f1 ), ϕn (f2 ), . . .). Since supp ϕn is compact and supp fm ⊂ Um , we can apply 7.2.5 to obtain that, for each n, ϕn (fm ) = 0 for large m and we see that xn ∈ K (N) . We equip K (N) with the strongest locally convex topology, so that we may identify (K (N) ) and K N in the usual way (see the comments before 5.3.11). Take (µ1 , µ2 , . . .) ∈ K N . Then g := ∞ m=1 µm fm ∈ C(X) and it is easily seen that the sum converges . Thus, by assumption, M := supn∈N |ϕn (g)| < ∞. Therefore, for each n, in τ c ∞ |ϕ µ ϕ (f ) = (g)| ≤ M and it follows that {x1 , x2 , . . .} is weakly n m=1 m n m bounded in K (N) , hence bounded by 5.4.5. Applying 3.6.6(a) we conclude that [x1 , x2 , . . .] is finite-dimensional, an impossibility as ϕn (fn ) = 0 for each n. Now we are ready for the main result on (C(X), τc ) (compare with [103], Section 11.7). Notice that (γ ) below does not depend on the choice of K. Theorem 7.2.7 The following are equivalent: (α) (C(X), τc ) is polarly barrelled. (β) If Y ⊂ X is closed but not compact then there exists an f ∈ C(X) that is unbounded on Y . (γ ) If Y ⊂ X is closed but not compact then there exists a clopen partition U1 , U2 , . . . of X such that Un ∩ Y = ∅ for each n. Proof. (α) =⇒ (β). Let Y ⊂ X be closed and suppose that each f ∈ C(X) is bounded on Y ; we prove that Y is compact. We have pY (f ) < ∞ for all f ∈ C(X), so pY is a seminorm on C(X) which is the supremum of the polar continuous seminorms f → |f (y)|, where y ∈ Y . So by (α) pY is continuous and there is a compact Z ⊂ X and a C > 0 such that pY ≤ C pZ . If y ∈ Y \ Z then there is a clopen neighbourhood U of y that does not meet Z and we would have 1 = pY (ξU ) ≤ C pZ (ξU ) = 0, a contradiction. So Y ⊂ Z and Y is compact. (β) =⇒ (γ ). Let Y ⊂ X be closed, not compact. By (β) there exists an f ∈ C(X) and points y1 , y2 , . . . ∈ Y such that 0 < |f (y1 )| < |f (y2 )| < · · · , |f (yn )| → ∞. Putting U1 := {x ∈ X : |f (x)| ≤ |f (y1 )|} and, for n ≥ 2, Un := {x ∈ X : |f (yn−1 )| < |f (x)| ≤ |f (yn )|} we find a clopen partition U1 , U2 , . . . of X and, for each n, yn ∈ Un ∩ Y .

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Barrelledness and reflexivity

(γ ) =⇒ (α). Let F be a pointwise bounded non-empty collection in (C(X), τc ) . By 7.1.7 it suffices to show that the seminorm p := sup{|ϕ| : ϕ ∈ F} is continuous. In 7.2.6 we have seen that {supp ϕ : ϕ ∈ F} Z := is compact. The restriction map R : (C(X), τc )−→(C(Z), . ∞ ) is obviously continuous, and surjective by 2.5.23. If f ∈ Ker R then f = 0 on Z and therefore, by 7.2.5, ϕ(f ) = 0 for all ϕ ∈ F. Thus, for each ϕ ∈ F we have the factorization C(X)D DD DD ϕ DDD "

R

K

/ C(Z) zz zz z z ϕ˜ |z z

To show continuity of each ϕ˜ first observe that by 7.2.4 there is a C > 0 such that |ϕ(f )| ≤ C psupp ϕ (f ) ≤ C pZ (f ) for all f ∈ C(X). Then |ϕ(R(f ˜ ))| = |ϕ(f )| ≤ C pZ (f ) = C R(f )∞ and continuity of ϕ˜ follows. Clearly {ϕ˜ : ϕ ∈ F} is pointwise bounded. Now C(Z) is a Banach space, hence barrelled, so there is an M > 0 such that |ϕ(g)| ˜ ≤ M g∞ for all g ∈ C(Z) and all ϕ˜ with ϕ ∈ F. Then, for all f ∈ C(X), ϕ ∈ F we have |ϕ(f )| = |ϕ(R(f ˜ ))| ≤ M R(f )∞ = MpZ (f ), showing that p ≤ M pZ , so that p is continuous. The above theorem raises the question about which spaces X satisfy (γ ). We will not enter a full discussion but only present some special cases and (counter)examples. Theorem 7.2.8 Let X be ultrametrizable. Then (C(X), τc ) is (polarly) barrelled. Proof. According to 4.3.2 (C(X), τc ) is of countable type, so (7.1.9) it suffices to prove polar barrelledness. Let d be an ultrametric inducing the topology on X. Let Y ⊂ X be closed, not compact; we prove (γ ) of 7.2.7. We distinguish two cases: (i) (Y, d) is complete. Then Y is not precompact. Hence there is an ε > 0 and y1 , y2 , . . . ∈ Y such that the balls B(y1 , ε), B(y2 , ε), . . . are pair wise disjoint. Thus, the sets U1 := B(y1 , ε) (X \ n∈N B(yn , ε)), U2 := B(y2 , ε), U3 := B(y3 , ε), . . . satisfy (γ ) of 7.2.7. (ii) (Y, d) is not complete. Denote the canonical ultrametric on the completion X∧ of X again by d. Then Y is not closed in X∧ , so there is a z ∈ X∧ \ X and

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267

a sequence y1 , y2 , . . . in Y such that d(z, y1 ) > d(z, y2 ) > · · · , d(z, yn ) → 0. Put U1 := {x ∈ X : d(x, z) ≥ d(x, y1 )} and, for n ≥ 2, Un := {x ∈ X : d(x, yn ) ≤ d(x, z) < d(x, yn−1 )}. Then, since z ∈ X, we have that U1 , U2 , . . . is a clopen partition of X and, for each n, yn ∈ Un ∩ Y = ∅. Remarks 7.2.9 (a) There are non-metrizable X for which (C(X), τc ) is polarly barrelled. In fact, it is not hard to see that for every quasicomplete locally convex space E, (C(E), τc ) is polarly barrelled, by using the fact that if Y ⊂ E is closed but not compact then Y is not precompact and by modifying part (i) of the proof of 7.2.8. (b) Let X be pseudocompact (i.e., each clopen partition of X into non-empty sets is finite), but not compact. (An example of such a space can be found in [52], 3.10.29.) Then X does not satisfy (γ ) of 7.2.7 and (C(X), τc ) is not polarly barrelled. (c) We do not know a characterization of barrelledness of (C(X), τc ) in the spirit of 7.2.7. Now the characterization of (polar) barrelledness of (C(X), τs ) is easy. Theorem 7.2.10 The following are equivalent: (α) (C(X), τs ) is (polarly) barrelled. (β) If Y ⊂ X is infinite then there is an f ∈ C(X) that is unbounded on Y . (γ ) If Y ⊂ X is infinite then there is a clopen partition U1 , U2 , . . . of X such that Un ∩ Y = ∅ for each n. Proof. For (α) =⇒ (β) follow the proof of (α) =⇒ (β) of 7.2.7 replacing “compact” by “finite” and “closed” by “infinite”. Similarly, for (β) =⇒ (γ ) use the corresponding proof of 7.2.7 starting with an infinite set Y ⊂ X. Finally, to prove (γ ) =⇒ (α) observe that (γ ) implies that compact subsets of X are finite, so that τs = τc . Then (γ ) of 7.2.7 holds, hence (C(X), τs ) = (C(X), τc ) is polarly barrelled and by 7.1.9 we have barrelledness as τs is of countable type (text preceding 4.3.2). Of course, discrete spaces X satisfy (α)–(γ ) above. If X is ultrametrizable the converse is true. Theorem 7.2.11 Let X be ultrametrizable. Then (C(X), τs ) is (polarly) barrelled if and only if X is discrete.

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Proof. Let X be not discrete; we prove that (C(X), τs ) is not polarly barrelled. There is a sequence x1 , x2 , . . . ∈ X and an x ∈ X such that d(x1 , x) > d(x2 , x) > · · · , d(xn , x) → 0, where d is an ultrametric inducing the topology. Then Y := {x, x1 , x2 , . . .} is an infinite compact subset of X, so (γ ) of 7.2.10 does not hold and (C(X), τs ) is not polarly barrelled. On the other hand, we have the following. Example 7.2.12 There exist non-discrete spaces X for which (C(X), τs ) is barrelled. Proof. Let X be the space of 2.5.33 (X := R, U ⊂ X is by definition open if 0 ∈ U or X \ U is countable). Let Y ⊂ X be infinite. Then there are pairwise distinct y1 , y2 , . . . in Y with yn = 0 for all n. Set U1 := {y1 } ∪ (X \ {y1 , y2 , . . .}) and, for n ≥ 2, Un := {yn }. Then U1 , U2 , . . . is a clopen partition of X and Un ∩ Y = ∅ for each n. Thus, (C(X), τs ) is barrelled by 7.2.10. Remark 7.2.13 The above example furnishes another space that is barrelled but not complete (3.7.2(iii)), compare with 7.1.19(ii). It is also interesting to observe that (C(X), τs ) is of finite type, showing that the natural converse of 7.1.16(ii) is false. Next, we consider the strict topology β0 (see 3.7.14). Theorem 7.2.14 (BC(X), β0 ) is (polarly) barrelled if and only if X is compact. Proof. If X is compact then (BC(X), β0 ) equals C(X) with the uniform topology, so it is a Banach space, hence barrelled by 7.1.3. Conversely, let (BC(X), β0 ) be polarly barrelled. Then B := {f ∈ BC(X) : f ∞ ≤ 1}, being a polar barrel in (BC(X), β0 ), is a β0 -neighbourhood of 0. Thus, the uniform topology on BC(X) is weaker than β0 , but it is also trivially stronger than β0 , hence they are equal. Therefore, (BC(X), β0 ) is normable so, by 3.7.17, X is compact. Now we treat the space (Cc (X), τi ), see 3.7.21. Theorem 7.2.15 (Cc (X), τi ) is barrelled. Proof. We first prove that for each compact Y ⊂ X the space (GY , . ∞ ) is Banach where, as in 3.7.21, GY := {f ∈ C(X) : supp f ⊂ Y }. Let f1 , f2 , . . . be a Cauchy sequence in GY . Then it is also Cauchy in (BC(X), . ∞ ), so fn → f ∈ BC(X) uniformly. Since fn = 0 on X \ Y for all n, we have also that f = 0 on X \ Y , so f ∈ GY .

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Now the proof of barrelledness is standard. Let P be a non-empty pointwise bounded collection of τi -continuous seminorms on Cc (X), let p := sup P. Let Y ⊂ X be compact. Then {q | GY : q ∈ P} is a pointwise bounded collection of . ∞ -continuous seminorms on GY . Now (GY , . ∞ ) is a Banach space, hence barrelled, so p | GY is continuous. Thus, by the definition of τi , p is τi -continuous. We do not have characterizations of (polar) barrelledness of (BCWc (X), τWc ), where Wc (3.7.19) is the set of all functions X−→K that are bounded on compact sets. But we do have the following special case. Corollary 7.2.16 Let X be locally compact and metrizable. Then (BCWc (X), τWc ) is barrelled. Proof. By 3.7.20 and 3.7.22 we have Cc (X) = BCWc (X) and τi = τWc . Now apply 7.2.15.

7.2.3 Barrelledness of spaces of differentiable functions Throughout Subsection 7.2.3 X is a non-empty subset of K without isolated points. Recall from Subsection 3.7.3 that for a non-empty subset Y of X, n ∈ N ∪ {0}, f ∈ C n (X) we set, admitting the value ∞, pYn (f ) := max i f Y , 0≤i≤n

where

i f Y := sup{i f (t) : t ∈ Y i+1 }.

First, we consider the topology of pointwise convergence, see 3.7.42. Theorem 7.2.17 The spaces (C n (X), τsn ) (n ∈ {0, 1, . . .}) and (C ∞ (X), τs∞ ) are not polarly barrelled. Proof. Let x ∈ X, n ∈ {0, 1, . . .}. Since x is not isolated there are x1 , x2 , . . . ∈ X such that |x1 − x| > |x2 − x| > · · · , limi xi = x. Then Y := {x, x1 , x2 , . . .} is compact, so the seminorm pY := pY0 is well-defined on C n (X) (resp. C ∞ (X)) and it is a supremum of polar continuous seminorms pZ , where Z ⊂ Y , Z finite. So if (C n (X), τsn ) (resp. (C ∞ (X), τs∞ )) were polarly barrelled then pY would be τsn (resp. τs∞ )-continuous implying in both cases the existence of a constant C > 0, an m ∈ {0, 1, . . .}, and a finite setZ ⊂ X such that pY ≤ C pZm . But Y

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is infinite, so there is a point y ∈ Y \ Z. Let U be a clopen neighbourhood of y in X that does not meet Z. Then 1 = pY (ξU ) ≤ C pZm (ξU ) = 0, a contradiction. The case of uniform convergence on compact sets (3.7.36) is more difficult. For a restricted class of domains we have a simple answer. Theorem 7.2.18 Let X be locally compact and separable. Then (C n (X), τcn ) (n ∈ {0, 1, . . .}) and (C ∞ (X), τc∞ ) are barrelled. Proof. From 3.7.37 and 3.7.41 we obtain that the spaces under consideration are Fr´echet, hence barrelled by 7.1.3. As for general X not much is known. By 7.2.8 (C(X), τc ) is barrelled. It is shown in [206] that (C 1 (X), τc1 ) is barrelled as well. Problem Let X be a subset of K without isolated points. Are (C n (X), τcn ) (n ∈ {2, 3, . . .}) and (C ∞ (X), τc∞ ) (polarly) barrelled? Finally, we consider the only remaining case of Subsection 3.7.3, viz. the n ∞ spaces BCW (X) and BCW (X), where X is locally compact and W := n ∞ C (X). In 3.7.53 we have seen that BCW (X) = Ccn (X) := C n (X) ∩ Cc (X) ∞ (X) = Cc∞ (X) := C ∞ (X) ∩ Cc (X). and BCW Theorem 7.2.19 Let X be locally compact. Let W := C ∞ (X). Then (Ccn (X), τWn ) (n ∈ {0, 1, . . .}) is barrelled. Proof. In 3.7.53 it is proved that, for n ∈ {0, 1, . . .}, τWn is the strongest locally convex topology on Ccn (X) for which the obvious embeddings (C n (U ), . n )−→Ccn (X) (U ⊂ X, U open compact) are continuous. Now the (C n (U ), . n ) are Banach spaces and with the usual procedure (e.g. the second half of the proof of 7.2.15) we arrive at barrelledness of (Ccn (X), τWn ). The situation for (Cc∞ (X), τW∞ ) is different. To prove 7.2.21 we need the following lemma. Lemma 7.2.20 Let n ∈ N, ε > 0. Then there is an f ∈ C ∞ (X) such that n−1 (f ) ≤ ε. Dn f = 1 and pX Proof. We may suppose ε < 1. The equivalence relation x ∼ y if |x − y| ≤ ε yields a partition (Bi )i∈I of X into balls of radius ε. Choose ai ∈ Bi (i ∈ I ). The assignment f (x) := (x − ai )n

(x ∈ Bi )

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defines an f ∈ C ∞ (X). We have Dn f = 1. We now prove inductively that, for j ∈ {0, 1, . . . , n − 1}, j f ∞ ≤ ε n−j (which leads to the desired result n−1 0 (f ) ≤ max(ε n , εn−1 , . . . , ε) = ε). We have pX (f ) = f ∞ ≤ εn . For the pX j +1 step j − 1 → j let (x1 , . . . , xj +1 ) ∈ X . If all x1 , . . . , xj +1 are in the same Bi then, by a translated version of rule X of 2.5.78, we have j f ∞ ≤ εn−j . If not, we may assume by symmetry that x1 , x2 are in different Bi . Then |x1 − x2 | > ε and by using the induction hypothesis we get j f (x1 , . . . , xj +1 ) = (x1 − x2 )−1 (j −1 f (x1 , x3 , . . . , xj +1 ) − j −1 f (x2 , x3 , . . . , xj +1 )) ≤ ε−1 j −1 f ∞ ≤ ε −1 εn−j +1 = ε n−j . Theorem 7.2.21 Let X be locally compact. Let W := C ∞ (X). Then (Cc∞ (X), τW∞ ) is (polarly) barrelled if and only if X is compact. Proof. If X is compact then (Cc∞ (X), τW∞ ) = BC ∞ (X) is Fr´echet by 3.7.47, hence barrelled by 7.1.3. Now let X be not compact and suppose (Cc∞ (X), τW∞ ) is polarly barrelled; we derive a contradiction. By 1.1.3 X admits an infinite partition into compact balls Bi (i ∈ I ). Without loss, assume N ⊂ I . Choose an ∈ Bn for each n ∈ N. The seminorms p1 , p2 , . . . defined by the formula pn (f ) := |Dn f (an )| (f ∈ Cc∞ (X)) are clearly τW∞ -continuous and polar. For each f ∈ Cc∞ (X), supp f meets only finitely many Bn , so that {p1 , p2 , . . .} is pointwise bounded and by assumption there exist m ∈ N and φ ∈ C ∞ (X) such that for all n ∈ N, all f ∈ Cc∞ (X), m |Dn f (an )| ≤ pX (f φ),

in particular m |Dm+1 f (am+1 )| ≤ pX (f φ).

Now consider an f ∈ Cc∞ (X) with support in Bm+1 . By 3.7.34 and the product m rule for pBmm+1 we have pX (f φ) ≤ d m pBmm+1 (f φ) ≤ d m pBmm+1 (φ) pBmm+1 (f ) for some constant d. Putting M := d m pBmm+1 (φ) + 1 we find |Dm+1 g(am+1 )| ≤ M pBmm+1 (g) for all g ∈ C ∞ (Bm+1 ). But by 7.2.20, applied to Bm+1 , there is a g ∈ C ∞ (Bm+1 ) with Dm+1 g = 1 and pBmm+1 (g) < M1 , a contradiction.

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7.3 The weak star and the strong topology on the dual Throughout Sections 7.3 and 7.4 E is a locally convex space over K . We introduce two “classical” topologies on E . Definition 7.3.1 The weak star topology or topology of pointwise convergence on E is the weakest topology on E for which the evaluation f → f (x) (f ∈ E ) is continuous for each x ∈ E. This topology is denoted by σ (E , E). We sometimes write Eσ instead of (E , σ (E , E)). The strong topology or topology of uniform convergence on bounded sets on E is the locally convex topology induced by the seminorms f → sup{|f (x)| : x ∈ B}, where B runs through the collection of all bounded subsets of E. This topology is denoted by b(E , E). We sometimes write Eb instead of (E , b(E , E)). Clearly σ (E , E) is locally convex and Hausdorff (whether E is Hausdorff or not) and is induced by the seminorms f → sup{|f (x)| : x ∈ B}, where B runs through the collection of all finite subsets of E. It follows that Eσ is of finite type and that σ (E , E) is weaker than b(E , E). This last topology is Hausdorff and polar but in general not of countable type. Also notice that σ (E , E) is weaker than the weak topology of Eb ; we will describe in 7.4.9 the situations where they are equal. If E is normed the topology b(E , E) is induced by the single norm f → sup{|f (x)| : x ≤ 1}, which is easily seen to be equivalent to the Lipschitz norm introduced in the text following 2.1.4. Now that we have topologies on the dual of a locally convex space we can introduce the adjoint of a continuous linear map and consider continuity properties of this adjoint. Definition 7.3.2 Let F be a locally convex space, let T ∈ L(E, F ). The adjoint of T is the linear map T : F −→E , f → f ◦ T . Clearly T is well-defined. Also, with respect to continuity we have the following result. Its proof is straightforward and classical and is left to the reader. Theorem 7.3.3 Let F be a locally convex space, let T ∈ L(E, F ). Then T : Fσ −→Eσ (resp. T : Fb −→Eb ) is continuous. Next, we compare bounded sets in Eσ and Eb .

7.3 The weak star and the strong topology on the dual

273

Theorem 7.3.4 Each equicontinuous set in E is strongly bounded. Each strongly bounded set in E is weak star bounded. Proof. Let H ⊂ E be equicontinuous, let B ⊂ E be bounded. Then U := {x ∈ E : |f (x)| ≤ 1 for all f ∈ H } is a zero neighbourhood, so there is a λ ∈ K such that B ⊂ λ U . We see that the seminorm f → sup{|f (x)| : x ∈ B} is bounded on H . The second statement is obvious. Corollary 7.3.5 E is polarly barrelled if and only if, for subsets of E , the properties “equicontinuity”, “strong boundedness” and “weak star boundedness” coincide. Proof. Apply 7.1.7(α) ⇔ (γ ) and 7.3.4.

Here are non-Archimedean versions of the well-known Alaoglu–Bourbaki Theorem. Theorem 7.3.6 (i) In Eσ each bounded set is a compactoid. (ii) In Eσ each closed equicontinuous set is complete. (iii) The closure in Eσ of an equicontinuous set is equicontinuous. (iv) If K is spherically complete then in Eσ each closed convex equicontinuous set is c-compact. Proof. (i). Eσ is of finite type, use 5.4.1. (ii). Let H ⊂ E be closed and equicontinuous, let (fi )i be a weak star Cauchy net in H . Then, for each x ∈ E, (fi (x))i is a Cauchy net in K, so f (x) := limi fi (x) defines an f ∈ E ∗ . By equicontinuity there is a zero neighbourhood U in E for which |fi | ≤ 1 on U for all i. Thus, |f | ≤ 1 on U implying f ∈ E . Then fi → f in Eσ and, since H is closed, we have f ∈ H . The proof of (iii) is straightforward, and (iv) follows from (i), (ii) and 6.1.13. Corollary 7.3.7 If E is polarly barrelled then Eσ is quasicomplete. Proof. Combine 7.3.5 and 7.3.6(ii).

Remark 7.3.8 The converse is not true. In fact, let E := (c0 , σ (c0 , ∞ )). Then E = c0 but since σ (c0 , ∞ ) is strictly weaker than the norm topology, E is not Mackey, so E is not polarly barrelled by 7.1.21. On the other hand, Eσ = (c0 )σ is quasicomplete as c0 is polarly barrelled.

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Corollary 7.3.9 The unit ball of the dual of a Banach space is a weak star complete compactoid (c-compact if K is spherically complete). We can also prove the following (compare with 7.3.7). Theorem 7.3.10 If E is polarly barrelled then Eb is quasicomplete. Proof. Let (fi )i be a bounded Cauchy net in Eb . Then it is also a bounded Cauchy net in Eσ , so by 7.3.7 there is an f ∈ E with fi → f with respect to σ (E , E). Now σ (E , E) is closely related to b(E , E) and by 3.4.17 we have fi → f with respect to b(E , E). We continue by studying the biduals (Eσ ) and (Eb ) . Clearly we have (Eσ ) ⊂ (Eb ) . Definition 7.3.11 We define the natural map jE : E−→(Eb ) by the formula (x ∈ E, f ∈ E ).

jE (x)(f ) := f (x)

Theorem 7.3.12 jE maps E linearly onto (Eσ ) . Proof. Linearity and the fact that jE (E) ⊂ (Eσ ) are obvious. To complete the proof, let ∈ (Eσ ) ; we prove that has the form f → f (x) for some x ∈ E. By the very definition of the weak star topology there are a constant C > 0 and finitely many points x1 , . . . , xn ∈ E such that, for each f ∈ E , |(f )| ≤ C max(|f (x1 )| , . . . , |f (xn )|). Let the linear map ϕ : E −→K n be given by the formula ϕ(f ) := (f (x1 ), . . . , f (xn )). Then Ker ϕ ⊂ Ker , so we have a factorization E A AA AA AA

ϕ

K

/ Kn | | || || 1 | ~|

where 1 is linear. Now, letting e1 , . . . , en be the canonical base of K n , we have for each f ∈ E , n n (f ) = 1 (ϕ(f )) = 1 ( f (xi ) ei ) = f ( 1 (ei ) xi ),

and we may take x :=

n i=1

i=1

1 (ei ) xi .

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7.4 Reflexivity Definition 7.4.1 E is called reflexive if jE is a surjective homeomorphism E−→(Eb )b , and semireflexive if jE is surjective. For simplicity we will often write E instead of (Eb )b . Remark 7.4.2 It is customary to call a normed space (E, . ) reflexive if jE is an isometrical isomorphism E−→E (where E carries the canonical norm). This “geometric” reflexivity does not make sense in the locally convex setting. It is, however, not hard to see that if . is a polar norm then (E, . ) is “geometrically” reflexive if and only if E is reflexive in the sense of 7.4.1. Clearly finite-dimensional Hausdorff spaces are reflexive. In 2.5.11 and 5.5.5 we have seen that, if K is not spherically complete, c0 and ∞ are reflexive. We quote an extension. For the proof, see [193], 4.21, 4.M. For the notion of a small set see the glossary of terms in Appendix A.1. Theorem 7.4.3 Let K be not spherically complete, let I be an index set. If I is small then c0 (I ) and ∞ (I ) are reflexive. If I is not small then neither c0 (I ) nor ∞ (I ) are reflexive. For more examples of reflexive spaces over any K, see Section 7.5. The main goal of 7.4 will be the characterization of reflexivity, 7.4.13. As a stepping stone we first prove a density result (7.4.7), valid without any conditions on E. (One should keep in mind that in general jE need not be injective, surjective, continuous or open.) To this end we introduce a notation. Recall (4.4.10) that for X ⊂ E its polar X ◦ is defined as {f ∈ E : |f (X)| ≤ 1}. Then X◦ , considered as a subset of Eb , has a polar in E . To avoid confusion with X ◦◦ (which is a subset of E, see 4.4.10) we follow the notation of [103]. Definition 7.4.4 For X ⊂ E we set X◦• := { ∈ E : |(X ◦ )| ≤ 1}. Lemma 7.4.5 E =

◦• {B : B is a bounded subset of E}.

Proof. Let, ∈ E . By continuity of : Eb −→K there is a bounded set B ⊂ E such that |(B ◦ )| ≤ 1, i.e., ∈ B ◦• . Theorem 7.4.6 Let A ⊂ E be absolutely convex. Then σ (E ,E ) e jE (A) = A◦• .

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Proof. By the Bipolar Theorem 5.2.7, applied to G := (E , σ (E , E )) we σ (G,G ) e have jE (A) = jE (A)◦◦ (where the last bipolar is taken in G). By 7.3.12, applied to Eb , each g ∈ G is of the form jE (f ) for some f ∈ E , so that σ (G, G ) = σ (E , E ), and also

jE (A)◦ = {jE (f ) : f ∈ E , |jE (f )(jE (A))| ≤ 1} = {jE (f ) : f ∈ E , |f (A)| ≤ 1} = jE (A◦ ). Thus, jE (A)◦◦ = {h ∈ G : |jE (A◦ )(h)| ≤ 1} = {h ∈ G : |h(A◦ )| ≤ 1} = A◦• . Corollary 7.4.7 jE (E) is σ (E , E )-dense in E . Even more, for each ∈ E there exists a bounded net (xi )i in E such that (f ) = limi f (xi ) for all f ∈ E. Proof. It suffices to prove the second assertion. Let ∈ E . By 7.4.5 there is a bounded set B ⊂ E with ∈ B ◦• , we may assume B absolutely convex. Let σ (E ,E ) , so there is λ ∈ K, |λ| > 1. Then by 7.4.6 we have ∈ B ◦• ⊂ jE (λ B) a net (xi )i in the bounded set λ B such that jE (xi ) → in σ (E , E ), and we are done. For polar normed spaces we can even be more specific. Corollary 7.4.8 (Goldstine Theorem) Let (E, . ) be a normed space, where . is a polar norm. Then jE maps the open unit ball of E onto a σ (E , E )dense subset of the open unit ball of E . σ (E ,E )

Proof. Let A := BE− and C := jE (A) . Since, by polarity, jE is an isometry, jE (A) ⊂ BE− and since BE− is σ (E , E )-closed, C ⊂ BE− . So it suffices to show that C ⊃ BE− . If the valuation of K is discrete, by polarity we obtain A◦• = BE− , hence by 7.4.6, C = C e = BE− . Now let the valuation of K be dense and let ∈ BE− . Then there is a λ ∈ K, |λ| > 1 such that λ ∈ BE− ⊂ A◦• so, by 7.4.6, λ ∈ C e , hence ∈ C. Theorem 7.4.9 E is semireflexive if and only if σ (E , E) = σ (E , E ). Proof. Suppose σ (E , E) = σ (E , E ) and let ∈ E . Then is σ (E , E )continuous, hence σ (E , E)-continuous, so ∈ (Eσ ) = jE (E) by 7.3.12. It follows that E is semireflexive. The other implication is trivial. We now add the conditions that E is polar and Hausdorff.

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Lemma 7.4.10 Let E a polar Hausdorff space. Then we have the following: (i) jE maps (E, σ (E, E )) homeomorphically into (E , σ (E , E )). (ii) jE : E−→E is open. Proof. (i). Injectivity of jE follows from the fact that E is dual-separating (5.1.3). Let (xi )i be a net in E. Then xi → 0 weakly ⇐⇒ f (xi ) → 0 for all f ∈ E ⇐⇒ jE (xi )(f ) → 0 for all f ∈ E ⇐⇒ jE (xi ) → 0 in σ (E , E ), completing the proof of (i). For (ii), let (xi )i be a net in E for which jE (xi ) → 0 in E . Let p be a polar continuous seminorm on E. Then B := {f ∈ E : |f | ≤ p} is equicontinuous, hence bounded in Eb by 7.3.4. Thus, jE (xi ) → 0 uniformly on B. But, since p = sup{|f | : f ∈ B} we have p(xi ) → 0. Theorem 7.4.11 Let E be polar and Hausdorff. Then we have the following: (i) If E is semireflexive then Eb is polarly barrelled. (ii) E is weakly quasicomplete if and only if E is semireflexive. (iii) If E is weakly quasicomplete then E is quasicomplete. Proof. (i). By semireflexivity and 7.1.7 it suffices to see that if {jE (x) : x ∈ X} (X ⊂ E) is pointwise bounded in E then it is equicontinuous. In fact, pointwise boundedness means that X is weakly bounded, hence bounded by 5.4.5. Now equicontinuity follows easily. (ii). Let E be weakly quasicomplete. Let ∈ E . By 7.4.7 there is a bounded net (xi )i in E such that jE (xi ) → in σ (E , E ). Then (jE (xi ))i is Cauchy in σ (E , E ) and according to 7.4.10(i) the net (xi )i is weakly Cauchy. By assumption there is an x ∈ E such that xi → x weakly. Then, again by using 7.4.10(i), we obtain jE (x) = limi jE (xi ) = and surjectivity of jE is proved. Conversely, assume E is semireflexive. By (i) Eb is polarly barrelled. Then applying 7.3.7 to Eb we obtain that (E , σ (E , E )) is quasicomplete. From 7.4.10(i) and surjectivity of jE we conclude that E is weakly quasicomplete. (iii). Let (xi )i be a bounded Cauchy net in E. Then (xi )i is also bounded and Cauchy in (E, σ (E, E )), hence xi → x weakly for some x ∈ E. Now apply 3.4.17 to conclude that xi → x in the initial topology of E. We need one more simple observation. Theorem 7.4.12 Let E be reflexive. Then Eb is reflexive. Proof. let (jE ) : E −→E be the adjoint of jE (see 7.3.2). Since jE is a homeomorphism then by 7.3.3 also (jE ) is one (where both E and E carry the strong topology). Now (jE ) ◦ jE : Eb −→Eb is the identity and it follows that jE is a surjective homeomorphism Eb −→(E )b . Now we are ready for the main result of this section.

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Theorem 7.4.13 For a locally convex space E the following are equivalent: (α) E is reflexive. (β) E is Hausdorff, polar, polarly barrelled and weakly quasicomplete. Proof. (α) =⇒ (β). E is isomorphic to a strong dual space, hence E is Hausdorff and polar. By 7.4.12 Eb is reflexive, so E is polarly barrelled by 7.4.11(i), hence so is E. Weak quasicompleteness follows from 7.4.11(ii). (β) =⇒ (α). From 7.4.11(ii) it follows that jE is surjective and from 7.4.10 that jE is injective and open. It remains to be shown that jE is continuous. For this, let (xi )i be a net in E converging to 0. Let B be a bounded set in Eb . Then using polar barrelledness of E and 7.3.5 B is equicontinuous, so p := sup{|f | : f ∈ B} is continuous. Thus, p(xi ) → 0, which means that jE (xi ) → 0 uniformly on B, i.e., jE (xi ) → 0 in E . Corollary 7.4.14 Reflexive spaces are quasicomplete. Metrizable reflexive spaces are Fr´echet. Proof. Apply 7.4.11(iii) and 7.4.13.

Corollary 7.4.15 Let E be Hausdorff and polar. Then E is reflexive if and only if E is polarly barrelled and semireflexive. Proof. Apply 7.4.11(ii) and 7.4.13.

Corollary 7.4.16 Let (E, . ) be a Banach space whose norm . is polar. Then E is reflexive if and only if the closed unit ball of E is weakly complete. Proof. Let E be reflexive. Since . is polar the closed unit ball is polar (4.4.11) hence weakly closed, hence weakly complete by 7.4.13. Conversely, to prove reflexivity, by 7.1.3 and 7.4.13 we only have to establish weak quasicompleteness. So, let X ⊂ E be (weakly) bounded and weakly closed. Then there is a λ ∈ K, λ = 0 such that λ X is contained in the weakly complete unit ball. Therefore, λ X is weakly complete, and so is X. Corollary 7.4.17 Let E be reflexive and let P ∈ L(E) be a continuous projection. Then P (E) is reflexive. Proof. Clearly P (E) is Hausdorff, polar and, by 7.1.12(ii), polarly barrelled, so according to 7.4.13 we only have to show weak quasicompleteness. Let (xi )i be a (weakly) bounded net, weakly Cauchy in P (E). Then (xi )i is bounded and weakly Cauchy in E, so there is an x ∈ E for which xi → x with respect to σ (E, E ). Hence xi = P (xi ) → P (x) in σ (E, E ), so that x = P (x) ∈ P (E). To show that also xi → x in σ (P (E), P (E) ), let f ∈ P (E) . Then f ◦ P ∈ E , so f (xi ) = (f ◦ P )(xi ) → (f ◦ P )(x) = f (x), proving the assertion.

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279

In general, (closed) subspaces and (polar) quotients of reflexive spaces need not be reflexive, even when the spaces involved are Banach (contrary to the “classical” situation). In fact, we have the following. Examples 7.4.18 Let K be not spherically complete. (i) There exist non-reflexive closed subspaces of ∞ . (ii) Assuming that K is small, there is a reflexive Banach space having a quotient that is a polar Banach space, but not reflexive. Proof. (i). Let E1 ⊂ E2 ⊂ . . . be the sequence of Banach subspaces of ∞ constructed in 7.1.10; we prove that none of them are reflexive. Since En c0 for each n, the bounded sequence (1, 0, 0, . . .), (1, 1, 0, 0, . . .), (1, 1, 1, 0, 0, . . .), . . . is in En and converges weakly to (1, 1, 1, . . .) ∈ ∞ \ En . We see that En is not weakly quasicomplete. Now apply 7.4.13 to conclude that En is not reflexive. (ii). Let E1 be as in (i). Then BE1 ≤ ∞ ≤ K N and we see that BE1 is small. Now 2.5.6 and its proof show that there is a set I with I = BE1 such that E1 is a quotient of c0 (I ). Finally, c0 (I ) is reflexive (7.4.3) but E1 is not by (i) above. For spherically complete scalar fields we can characterize reflexivity in terms of the initial topology only. Theorem 7.4.19 Consider the following statements: (α) E is Hausdorff, polar, polarly barrelled. Each closed bounded subset of E is a complete compactoid. (β) E is reflexive. Then (α) =⇒ (β). If, in addition, K is spherically complete we also have (β) =⇒ (α). Proof. (α) =⇒ (β). By 7.4.13 it suffices to check weak quasicompleteness, so let X ⊂ E be bounded, weakly closed. Then X is also closed, so by (α) it is a complete compactoid. But the weak and initial topologies coincide on compactoids (5.2.12), so that X is weakly complete. (β) =⇒ (α). By 7.4.13 it suffices to show that if X ⊂ E is a closed bounded set then X is a complete compactoid. Now A := aco X is weakly compactoid, absolutely convex and weakly closed (5.2.1), so weakly complete by 7.4.13. Then applying 6.1.13 we conclude that A is weakly c-compact, hence (6.1.11) c-compact. So, again by 6.1.13, A is a complete compactoid, and therefore so is X. This leads to a surprising corollary.

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Corollary 7.4.20 A reflexive Banach space over a spherically complete field K is finite-dimensional. Proof. The closed unit ball is a compactoid by the above theorem. Then finitedimensionality follows from 3.8.6. Remark 7.4.21 If K is not spherically complete (β) =⇒ (α) of 7.4.19 is not true even when E is of countable type. In fact, just take E := c0 . We proceed by showing that products and locally convex direct sums of reflexive spaces are reflexive. Theorem 7.4.22 Let {Ei : i ∈ I } be a collection of Hausdorff locally con vex spaces. Then there are natural isomorphisms ( i∈I Ei ) ∼ i∈I Ei and ( i∈I Ei ) ∼ i∈I Ei (where the occurring dual spaces are assumed to carry the strong topology). Proof. We will treat both cases more or less simultaneously. Denote the canon ical embeddings Ei −→ i∈I Ei , Ei −→ i∈I Ei by φi , and the canonical projections i∈I Ei −→Ei , i∈I Ei −→Ei by πi . We will show that in both cases the formula f → (f ◦ φi )i yields the required linear homeomorphism. If f ∈ ( i∈I Ei ) we have f ◦ φi = 0 for all but finitely many i ∈ I , hence (f ◦ φi )i ∈ i∈I Ei and we have the formula f = f ◦ φi ◦ πi (7.2)

i∈I

as a finite sum. If f ∈ ( i∈I Ei ) then clearly (f ◦ φi )i ∈ i∈I Ei and we again have (7.2) pointwise (this time, for each x ∈ i∈I Ei , the set {i ∈ I : (f ◦ φi ◦ πi )(x) = 0} is finite). It is straightforward to verify now that f → (f ◦ φi )i is a linear bijection in both cases. Also, one checks easily, by using the description of bounded sets in i∈I Ei and i∈I Ei (see 3.6.4), and formula (7.2) that the strong topologies on both ( i∈I Ei ) and ( i∈I Ei ) are induced by the seminorms

f −→ sup{|(f ◦ φi )(Bi )| : i ∈ I , Bi bounded in Ei for each i ∈ I }, whereas the seminorms (gi )i −→ sup{|gi (Bi )| : i ∈ I , Bi bounded in Ei for each i ∈ I } induce the topologies on i∈I Ei and i∈I Ei . We conclude that f → (f ◦ φi )i maps a base of continuous seminorms onto a base of continuous seminorms and therefore it is a homeomorphism.

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281

Corollary 7.4.23 Products and locally convex direct sums of reflexive spaces are reflexive. Proof. Let Ei be a reflexive space for each i ∈ I . Then E := i∈I Ei is Hausdorff, polar (4.4.16(iv)) and polarly barrelled (7.1.15), so to prove reflexivity it suffices to settle weak quasicompleteness (7.4.13). Let j → x j be a bounded j σ (E, E )-Cauchy net in E, where x j = (yi )i for each j . Then, for each i ∈ I , j j → yi is a bounded and σ (Ei , Ei )-Cauchy net in Ei , hence it converges weakly to some yi ∈ Ei . Then x = (yi )i ∈ E and for each f ∈ E we have by j formula (7.2) of the proof of 7.4.22, limj f (x j ) = i∈I limj (f ◦ φi )(yi ) = j i∈I (f ◦ φi )(yi ) = i∈I (f ◦ φi ◦ πi )(x) = f (x). Thus, (x )j converges in σ (E, E ) and reflexivity of i∈I Ei is proved. For the direct sum, observe that i∈I Ei ∼ i∈I Ei ∼ ( i∈I Ei ) (7.4.22). Now, the latter space is reflexive by 7.4.12 and the first part of this proof. Remark 7.4.24 By 7.4.17 also the converse holds. If i∈I Ei or i∈I Ei is reflexive then so are the Ei . Corollary 7.4.25 Let I be an index set. Then K I and K (I ) are reflexive. If K is not spherically complete then c0I , ( ∞ )I , c0(I ) , ( ∞ )(I ) are reflexive. The next result tells us about the reflexivity of subspaces of reflexive spaces, compare with 7.4.18. Clearly, every reflexive (sub)space is polarly barrelled and quasicomplete (7.4.13, 7.4.14). Under certain circumstances the converse holds. Theorem 7.4.26 Let E be reflexive and assume that K is spherically complete or E is of countable type. Then every polarly barrelled quasicomplete subspace of E is reflexive. Proof. Let D be a quasicomplete polarly barrelled subspace of E; we prove that D is weakly quasicomplete (and by 7.4.13 we are done). Let (xi )i be a bounded σ (D, D )-Cauchy net in D. Then it is also σ (E, E )-Cauchy, so by reflexivity of E and 7.4.13 we have xi → x with respect to σ (E, E ), for some x ∈ E. To show that x ∈ D consider A := (aco {xi : i ∈ I })e , where the closure is taken in the relative topology of D. By assumption A is complete, hence closed in E. Also, A is absolutely convex and edged, so by 5.2.9 A is a polar set in E, hence σ (E, E )-closed so that x = σ (E, E )-limi xi ∈ A ⊂ D. Finally, since every element of D can be extended to an element of E (4.1.3, 4.2.6) we have σ (E, E )|D = σ (D, D ) and therefore x = σ (D, D )-limi xi . Corollary 7.4.27 Let K be spherically complete. Then every closed subspace of a reflexive Fr´echet space is reflexive.

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Remark 7.4.28 However, in 9.8.2(ii) we will see that there exist reflexive Fr´echet spaces of countable type over a spherically complete K, having quotients that are not reflexive. For non-spherically complete K we have the following surprising consequence, again a characterization of reflexivity (for spaces of countable type) in terms of the initial topology only, compare with 7.4.19. Theorem 7.4.29 Let E be of countable type, let K be not spherically complete. Then E is reflexive if and only if E is polarly barrelled and quasicomplete. Proof. If E is reflexive then (7.4.13, 7.4.14) E is polarly barrelled and quasicomplete. To prove the converse, let I be a set such that E can be viewed as a subspace of c0I (4.2.14). Now c0I is reflexive (7.4.25) and after applying 7.4.26 we arrive at reflexivity of E. Corollary 7.4.30 Every Fr´echet space of countable type over a non-spherically complete K is reflexive. Remark 7.4.31 In Chapter 8 we will continue studying reflexivity.

7.5 Examples of reflexive spaces In Section 7.5 we will discuss the reflexivity of spaces we have met previously, in those cases where we can apply the theory introduced in this chapter. Reflexivity of spaces such as A(r) and A† (r) (3.7.25 and 3.7.27) will be considered later on (9.7.5) when we have developed more powerful tools. Also, new examples will appear in the following chapters. Throughout Section 7.5 X is a non-empty zero-dimensional Hausdorff topological space. (X) is the ring of all clopen subsets of X (2.5.20).

7.5.1 Reflexivity of Banach spaces By 7.4.20 this subject is interesting only when K is not spherically complete. Therefore, Throughout Subsection 7.5.1 K is not spherically complete. Recall that 7.4.3 states that c0 (I ) and ∞ (I ) are reflexive if and only if I is small.

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283

Theorem 7.5.1 (i) Let X be compact. Then C(X) is reflexive if and only if X is small. (ii) Let X be locally compact. Then C0 (X) is reflexive if and only if X is small. (iii) P C(X) is reflexive if and only if X is small. Proof. (i) is a special case of (ii). To prove (ii) first apply 2.5.34 to see that C0 (X) has an orthonormal base {ei : i ∈ I }, where the ei are characteristic functions of open compact sets. Thus, C0 (X) c0 (I ). Now suppose that C0 (X) is reflexive. Then by 7.4.3 the set I is small, hence so is {0, 1}I . The formula x → (ei (x))i defines an injection X−→{0, 1}I and it follows that X is small. Conversely, if X is small then so are (X) and I . Therefore, C0 (X) c0 (I ) and, by 7.4.3, C0 (X) is reflexive. To prove (iii), recall (2.5.37) that P C(X) C(Xζ ), so we will be done once we show that X is small implies X ζ is small. But this follows from X ζ ⊂ {0, 1}(X) (see the construction of Xζ , text preceding 2.5.37). We do not have a characterization of reflexivity of BC(X); we only quote without proof a result from [193], 7.27. For the concept of N-compact space, see the glossary of terms in Appendix A.5. Theorem 7.5.2 If X is a small N-compact k0 -space then BC(X) is reflexive. The following on valued field extensions is immediate from 2.5.42, 7.4.3 and 7.4.30. Theorem 7.5.3 Let K be algebraically closed with residue class field k. Then the valued field extension K(a) of 2.5.42 is reflexive except when dist (a, K) ∈ |K| is attained and k is not small. For Banach spaces of power series and analytic elements we can directly apply 7.4.3 together with 2.5.55, 2.5.59, 2.5.60, 2.5.68, 2.5.69 and 2.5.72 to obtain the next result. Theorem 7.5.4 Let K be algebraically closed with residue class field k. Then we have the following: (i) For each fenced disk B in K the space P S(B) is reflexive. (ii) For each disk B in K the space BP S(B) is reflexive. (iii) For each bounded disk A = B(a, r) in K the H (A) is reflexive. space − × (iv) For each bounded disk A = B(a, r ) (r ∈ K ) in K the space H (A) is reflexive if and only if k is small. (v) If K is separable then for each bounded closed set A ⊂ K the space H (A) is reflexive. We regret to admit that we do not have a characterization of reflexivity of BC n (X) (n ∈ N, X ⊂ K, X no isolated points), apart from the obvious remark

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that for compact X, BC n (X) = C n (X) c0 (2.5.84), hence it is reflexive. (For the (locally convex) space BC ∞ (X), see 7.5.22 and 7.5.23.) For the Banach space C0n (X) of 3.7.57, as a direct consequence of 4.3.21(ii) and 7.4.30 we obtain the following. Theorem 7.5.5 Let X ⊂ K, X without isolated points. Suppose balls in X are compact. Then, for each n ∈ {0, 1, . . .}, C0n (X) is reflexive.

7.5.2 Reflexivity of locally convex spaces of continuous functions We start with the topology of pointwise convergence on C(X), see 3.7.1. Theorem 7.5.6 (C(X), τs ) is reflexive if and only if X is discrete. Proof. Let (C(X), τs ) be reflexive. Then it is quasicomplete (7.4.14), hence X is discrete by 3.7.2(iii). If conversely, X is discrete then (C(X), τs ) is isomorphic to K X which is reflexive by 7.4.25. For the topology τc of compact convergence on C(X) (3.7.3) it matters whether K is spherically complete or not. In the next two theorems, for a compact Z ⊂ X, we fix a linear isometry T : (C(Z), . ∞ )−→ (P C(X), . ∞ ) such that R ◦ T is the identity on C(Z), where R is the restriction map C(X)−→C(Z) (2.5.23). Notice that T also maps (C(Z), . ∞ ) homeomorphically into (C(X), τc ). Theorem 7.5.7 Let K be spherically complete. Then (C(X), τc ) is reflexive if and only if X is discrete. Proof. If X is discrete then (C(X), τc ) = (C(X), τs ) is reflexive by 7.5.6. Conversely, let (C(X), τc ) be reflexive. Let Z ⊂ X be compact, let B be the unit ball of C(Z). Then T (B) is bounded in (C(X), τc ), so T (B) is a compactoid (7.4.19). Therefore, B = (RT )(B) is a compactoid, so that by 3.8.6 C(Z) is finite-dimensional implying finiteness of Z. Thus, each compact set in X is finite, i.e., τc = τs , so, by 7.5.6 X is discrete. For non-spherically complete K the question is more interesting. Theorem 7.5.8 Let K be not spherically complete. Then (C(X), τc ) is reflexive if and only if (i), (ii), (iii) below hold: (i) If Y ⊂ X is closed but not compact then there exists a clopen partition U1 , U2 , . . . of X such that Un ∩ Y = ∅ for every n. (ii) X is a k0 -space. (iii) Each compact subset of X is small.

7.5 Examples of reflexive spaces

285

Proof. Suppose (C(X), τc ) is reflexive. Then it is polarly barrelled (7.4.13) and we have (i) by 7.2.7. Further, (C(X), τc ) is quasicomplete (7.4.14) and we obtain (ii) from 3.7.9(iii). To prove (iii), let Z ⊂ X be compact; it suffices to check that the closed unit ball of (C(Z), . ∞ ) is weakly complete (7.4.16 and 7.5.1(i)). So, let (fi )i be a weak Cauchy net in the closed unit ball of (C(Z), . ∞ ). Then (T (fi ))i is a bounded weak Cauchy net in (C(X), τc ) and by reflexivity and 7.4.13 there is an f ∈ C(X) such that T (fi ) → f weakly in (C(X), τc ). Then fi = (RT )(fi ) → R(f ) weakly in (C(Z), . ∞ ). For each z ∈ Z we have (Rf )(z) = limi fi (z), so that R(f )∞ ≤ 1. Conversely, suppose that (i), (ii), (iii) hold. To prove reflexivity we check (β) of 7.4.13. Clearly (C(X), τc ) is Hausdorff and polar. Polar barrelledness follows from (i) and 7.2.7. To verify weak quasicompleteness let (fi )i be a (weakly) bounded weak Cauchy net in (C(X), τc ). Then, for each x ∈ X, the net (fi (x))i converges in K, so fi → f pointwise for some f : X−→K. We will show that f ∈ C(X) and that fi → f weakly. Let Z ⊂ X be compact. Then (R(fi ))i is bounded and weakly Cauchy in (C(Z), . ∞ ). Since this last space is reflexive by (iii) and 7.5.1(i) we have by 7.4.13 that R(fi ) → g weakly in (C(Z), . ∞ ) for some g ∈ C(Z), which must be R(f ). This implies continuity of f by (ii) and 3.7.7. Now let ϕ ∈ (C(X), τc ) , let Z := supp ϕ, which is compact by 7.2.4. If h ∈ C(X), R(h) = 0 then ϕ(h) = 0 (7.2.5), so we have the factorization R

(C(X), τIc ) II II I ϕ III I$

K

/ C(Z) z z zz z z ϕ˜ |zz

where ϕ˜ = ϕ ◦ T is continuous. Then ϕ(fi ) = ϕ(R(f ˜ ˜ )) = ϕ(f ), i )) → ϕ(R(f and we are done. We will next mention a special case. Theorem 7.5.9 Let K be not spherically complete, and let X be ultrametrizable. Then (C(X), τc ) is reflexive. Proof. (It is possible to verify (i), (ii) and (iii) of the above theorem, but for a change we follow an alternative route.) By 4.3.2 the space (C(X), τc ) is of countable type. Theorem 7.2.8 furnishes polar barrelledness. Quasicompleteness follows from 3.7.8 and 3.7.9(iii) and, finally, from 7.4.29 we obtain reflexivity. Next, we move to the strict topology (see 3.7.14).

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Theorem 7.5.10 (i) Let K be spherically complete. Then (BC(X), β0 ) is reflexive if and only if X is finite. (ii) Let K be not spherically complete. Then (BC(X), β0 ) is reflexive if and only if X is a small compact space. Proof. Let (BC(X), β0 ) be reflexive. Then it is polarly barrelled by 7.4.13, hence X is compact by 7.2.14, so (BC(X), β0 ) is isomorphic to the Banach space (C(X), . ∞ ). In case (i) we obtain finite-dimensionality from 7.4.20 implying finiteness of X. In case (ii) we can apply 7.5.1(i) to arrive at smallness of X. The converse in case (i) is trivial, whereas the converse in case (ii) is again an application of 7.5.1(i). For the next example (Cc (X), τi ) (3.7.21) we restrict ourselves to the case where X is locally compact and ultrametrizable. Then (3.7.20, 3.7.22) we have (Cc (X), τi ) = (BCWc (X), τWc ), where Wc is the set of all functions X−→K that are bounded on compact sets (3.7.19). It follows that (Cc (X), τi ) is polar. Theorem 7.5.11 Let X be locally compact and ultrametrizable. (i) If K is spherically complete then (Cc (X), τi ) is reflexive if and only if X is discrete. (ii) If K is not spherically complete then (Cc (X), τi ) is reflexive. Proof. By 1.1.3 X has a partition {Ui : i ∈ I } into open compact sets. A straightforward verification shows that the map f → S(f ) := (f |Ui )i∈I is a bijective linear homeomorphism S : (Cc (X), τi )−→ i∈I (C(Ui ), . ∞ ). Then (ii) follows from 7.5.9 (with Ui in place of X) and 7.4.23. To prove (i), suppose (Cc (X), τi ) is reflexive. Then by applying S and 7.4.24 we obtain reflexivity of each C(Ui ), which can happen only if Ui is finite (7.4.20) implying discreteness of X. Conversely, if X is discrete then by choosing the partition of X into singleton sets we obtain that (Cc (X), τi ) is isomorphic to K (X) , a reflexive space by 7.4.25. For the space O(A) of 3.7.29 we have the following. Theorem 7.5.12 Let K be algebraically closed and separable. If A ⊂ K has a fundamental sequence of complete bounded sets then O(A) is reflexive. In particular, O(Cp \ Qp ) is reflexive. Proof. The assumptions on A and K imply respectively that O(A) is Fr´echet (3.7.31) and of countable type (4.3.15). Since K is not spherically complete it follows from 7.4.30 that O(A) is reflexive.

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287

7.5.3 Reflexivity of locally convex spaces of differentiable functions Throughout Subsection 7.5.3 X is a non-empty subset of K without isolated points. For the topologies of pointwise convergence (3.7.42) we derive the following from previous results. Theorem 7.5.13 The spaces (C n (X), τsn ) (n ∈ {0, 1, . . .}) and (C ∞ (X), τs∞ ) are not reflexive.

Proof. Combine 7.2.17 and 7.4.13.

For the topologies τcn , τc∞ of compact convergence (3.7.36) we have only partial results on reflexivity. We first consider τcn . Theorem 7.5.14 Let K be spherically complete, let n ∈ {0, 1, . . .}. Then (C n (X), τcn ) is not reflexive. Proof. The conclusion follows from 7.4.19(β) =⇒ (α) and the following lemma. Lemma 7.5.15 Let n ∈ {0, 1, . . .}. Then (C n (X), τcn ) contains a bounded noncompactoid set. Proof. Let n = 0. Let Z ⊂ X be an infinite compact subset of X. Let T : (C(Z), . ∞ )−→ (C(X), τc ) be the homeomorphic embedding as in the preamble to 7.5.7. If B is the unit ball of C(Z) then T (B) is a bounded noncompactoid set in (C(X), τc ). Now let n ≥ 1. Let a ∈ X. There exist a1 , a2 , . . . ∈ X such that 1 > |a1 − a| > |a2 − a| > · · · and limm am = a. For m ∈ N, set Bm := B(a, |am − a|) and fm := (X − a)n ξBm

(7.3)

(recall that the symbol X stands for the function x → x). Then fm ∈ C ∞ (X). We first prove that {f1 , f2 , . . .} is bounded, even in the stronger topology of n the Banach space (BC n (X), pX ). To this end we shall prove inductively that for j ∈ {0, 1, . . . , n} j fm ∞ ≤ |a − am |n−j

(m ∈ N).

(7.4)

For the case j = 0 we have 0 fm ∞ ≤ sup{|x − a|n : x ∈ Bm } = |a − am |n , yielding (7.4). For the step j − 1 → j , let (x1 , x2 , . . . , xj +1 ) ∈ X j +1 . If x1 , . . . , xj +1 are all in X \ Bm then clearly j fm (x1 , . . . , xj +1 ) = 0.

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have by rule X of 2.5.78 that If x1 , . . . , xj +1 are all in Bm we j fm (x1 , . . . , xj +1 ) = j (X − a)n (x1 , . . . , xj +1 ) ≤ |a − am |n−j . In the remaining case we may suppose by symmetry that x1 ∈ Bm , x2 ∈ Bm . Then j fm (x1 , . . . , xj +1 ) f (x , x , . . . , x )− f (x , x , . . . , x ) j +1 j −1 m 2 3 j +1 j −1 m 1 3 = x1 − x2 ≤ |x1 − x2 |−1 j −1 fm ∞ ≤ |a − am |−1 |a − am |n−j +1 , where the last inequality holds because of the induction hypothesis. Hence we have (7.4). From (7.4) and the fact that |a − am | < |a − a1 | < 1 we get n (fm ) ≤ max{|a − a1 |n−j : j ∈ {0, 1, . . . , n}} ≤ 1 pX

for all m. To finish the proof, let us assume that {f1 , f2 , . . .} is a compactoid; we derive a contradiction. First let K be spherically complete. Then A := aco {f1 , f2 , . . .} is a closed absolutely convex compactoid and by completeness of (C n (X), τcn ) (3.7.37) and 6.2.1 we obtain that on A the topology τcn coincides with the τs topology τs of pointwise convergence. But we have fm −→ 0 (here we use τn

c n ≥ 1), whereas fm 0 since Dn fm (a) = 1 for all m, a contradiction. Now let K be not spherically complete. Let K ∨ be the spherical completion of K (see the comments after 1.2.16). Let C n∨ (X) be the K ∨ -vector space of all C n -functions X−→K ∨ equipped with the topology of compact convergence (3.7.36), which is generated by the family of (K ∨ -)seminorms {pYn : ∅ = Y ⊂ X, Y compact}. Then C n (X) is included, as a K-vector subspace, in C n∨ (X) (in fact, C n (X) = {f ∈ C n∨ (X) : f (X) ⊂ K}) and its topology, τcn , is defined by the family of K-seminorms {qYn : ∅ = Y ⊂ X, Y compact}, where each qYn is the restriction to C n (X) of pYn . By what we have just proved in the previous paragraph, it suffices to show that compactoidity of {f1 , f2 , . . .} in C n (X) implies compactoidity in C n∨ (X). In order to see this, suppose {f1 , f2 , . . .} is a compactoid in C n (X). Let ∅ = Y ⊂ X be compact and let ε > 0. Then there exists a finite set G in C n (X) such that

{f1 , f2 , . . .} ⊂ {f ∈ C n (X) : qYn (f ) < ε} + acoK G, where acoK G is the absolutely convex hull of G in the K-vector space C n (X). Then certainly {f1 , f2 , . . .} ⊂ {f ∈ C n∨ (X) : pYn (f ) < ε} + acoK ∨ G,

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where acoK ∨ G is the absolutely convex hull of G in the K ∨ -vector space C n∨ (X). Hence we obtain compactoidity of {f1 , f2 , . . .} in C n∨ (X). For non-spherically complete K the problem of reflexivity of (C n (X), τcn ) can be reduced to the open problem following 7.2.18. Theorem 7.5.16 Let K be not spherically complete, let n ∈ {0, 1, . . .}. Then (C n (X), τcn ) is reflexive if and only if (C n (X), τcn ) is polarly barrelled. Proof. The space (C n (X), τcn ) is complete (3.7.37), Hausdorff, and of countable type (4.3.16). Now apply 7.4.29. Corollary 7.5.17 Let K be not spherically complete. If X is locally compact and separable then (C n (X), τcn ) is reflexive for each n ∈ {0, 1, . . .}. Proof. Combine 7.2.18 and 7.5.16.

Next, we discuss the reflexivity of (C ∞ (X), τc∞ ). Lemma 7.5.18 Let ∅ = Y ⊂ X, Y compact, let n ∈ {0, 1, . . .}. Then {f ∈ C ∞ (X) : pYn+1 (f ) ≤ 1} is a compactoid in (C ∞ (X), pYn ). Proof. Let V := {f ∈ C ∞ (X) : pYn+1 (f ) ≤ 1}, H := {f ∈ C ∞ (X) : pYn (f ) = 0}. Let π : C ∞ (X)−→C ∞ (X)/H be the natural map. The formula f −→ (f |Y, 1 f |Y 2 , . . . , n f |Y n+1 ) defines a linear map : C ∞ (X)−→C(Y ) × C(Y 2 ) × · · · × C(Y n+1 ) that factorizes: / C(Y ) × C(Y 2 ) × · · · × C(Y n+1 ) C ∞ (X)L LLL jj4 LLL jjjj j j j jj 1 π LL L& jjjj ∞ C (X)/H

and it is almost obvious that 1 is a linear isometry (where C ∞ (X)/H carries the norm induced by pYn ). Now, V is a pYn -compactoid if and only if π (V ) is a compactoid in C ∞ (X)/H . It therefore suffices to show that (V ) is a i i+1 compactoid in n+1 :f ∈ i=1 C(Y ), i.e., that, for each i ∈ {0, . . . , n}, {i f |Y i+1 V } is a compactoid in (C(Y ). To prove this compactoidity, observe that by assumption, for all f ∈ V , i+1 f ≤ 1 on Y i+2 , so that i f (z) − i f (u) ≤ z − u∞ for all z, u ∈ Y i+1 . Therefore, {i f |Y i+1 : f ∈ V } is equicontinuous and, obviously, bounded, and is hence a compactoid by the Ascoli Theorem 3.8.2.

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Corollary 7.5.19 Each bounded set in (C ∞ (X), τc∞ ) is a compactoid. Corollary 7.5.20 (C ∞ (X), τc∞ ) is reflexive if and only if (C ∞ (X), τc∞ ) is polarly barrelled. Proof. The space (C ∞ (X), τc∞ ) is complete (3.7.37) and of countable type (4.3.16). For non-spherically complete K the result follows from 7.4.29. If K is spherically complete we can apply 7.5.19 together with 7.4.19. Corollary 7.5.21 If X is locally compact and separable then (C ∞ (X), τc∞ ) is reflexive. Proof. Combine 7.2.18 and 7.5.20.

Problem Characterize those X ⊂ K for which (C ∞ (X), τc∞ ) is reflexive. (See also the Problem following 7.2.18.) Next we discuss reflexivity of the Fr´echet space BC ∞ (X) of 3.7.46. Theorem 7.5.22 Let X be precompact. Then BC ∞ (X) is reflexive. Proof. According to 4.3.18 we may assume that X is compact. Then BC ∞ (X) = (C ∞ (X), τc∞ ) and the result follows from 7.5.21. For spherically complete K we have the converse. Theorem 7.5.23 Let K be spherically complete. Let X be not precompact. Then BC ∞ (X) is not reflexive. Proof. By assumption there is an r > 0 and a sequence B1 , B2 , . . . of mutually disjoint balls in X of radius r. Without loss, assume r < 1. Let V := {ξB1 , ξB2 , . . .}. We show that V is bounded but not a compactoid. (Then the result follows from 7.4.19.) A simple inductive procedure shows that n ξBi ∞ ≤ r −n for all i ∈ n {1, 2, . . .}, n ∈ {0, 1, . . .}. Then also pX (ξBi ) ≤ r −n for those i, n and boundedness of V follows. If V were a compactoid it would be also a compactoid for 0 = . ∞ . But ξB1 , ξB2 , . . . is easily seen to be . ∞ -orthonormal, hence pX ξBi ∞ → 0 by 3.9.6, a contradiction. Problem Let K be not spherically complete. If BC ∞ (X) is reflexive then does it follow that X is precompact? (Compare with 8.7.11.) Finally, we consider the W -topologies, see 3.7.48. We change the usual order of treatment of examples a bit and consider first the space C0∞ (X), see 3.7.57.

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Theorem 7.5.24 Suppose each ball in X is compact. Then we have the following: (i) If K is not spherically complete then C0∞ (X) is reflexive. (ii) Let K be spherically complete. Then C0∞ (X) is reflexive if and only if X is compact. Proof. (i). C0∞ (X) is a Fr´echet space, of countable type (4.3.21(ii)), hence reflexive by 7.4.30. (ii). If X is compact then C0∞ (X) = BC ∞ (X) and we can apply 7.5.22 to arrive at reflexivity. Now assume that X is not compact. From the assumption on X it follows easily that X is closed in K, hence X is not precompact. Let B1 , B2 , . . . be as in the proof of 7.5.23. Let i ∈ N. To show that ξBi ∈ C0∞ (X), let n ∈ {0, 1, . . .}. If B is a ball with radius R > 1, B ⊃ Bi then n ξB (z) ≤ R −n i for all z ∈ Xn+1 \ B n+1 , as a simple induction procedure shows. Hence, n ξBi ∈ C0 (X n+1 ) and therefore ξBi ∈ C0∞ (X), and we see that V := {ξB1 , ξB2 , . . .} lies in the subspace C0∞ (X) of BC ∞ (X). Further, it is shown in the proof of 7.5.23 that V is bounded but not compactoid in BC ∞ (X), therefore the same occurs in C0∞ (X). Now apply 7.4.19. Next we investigate the case W = C ∞ (X) for locally compact X. We have n (X) = Ccn (X) (n ∈ {0, 1, . . .}), where the topolseen in 3.7.53 that BCW ogy τWn is the strongest locally convex topology making the embeddings (C n (U ), . n )−→Ccn (X) (U ⊂ X, U open compact) continuous, and that these embeddings are homeomorphic. A straightforward device leads from this to the following. Theorem 7.5.25 Let X be locally compact and suppose {Ui : i ∈ I } is a partition of X into open compact sets. Then, for n ∈ {0, 1, . . .}, W := C ∞ (X), n (BCW (X), τWn ) ∼

(C n (Ui ), . n ). i∈I

As a corollary we obtain the following characterization of reflexivity. Theorem 7.5.26 Let X be locally compact, W := C ∞ (X), n ∈ {0, 1, . . .}. n (X), τWn ) is reflexive if and only if K is not spherically complete. Then (BCW Proof. By 1.1.3 there is a partition {Bi : i ∈ I } of X into compact balls. The Banach spaces (C n (Bi ), . n ) are of countable type (2.5.84). So, if K is not spherically complete they are reflexive (7.4.30), hence so is n n n i∈I (C (Bi ), . n ) by 7.4.23, and is (BCW (X), τW ) by 7.5.25. If K is

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n (X), τWn ) were reflexive then so would be also spherically complete and (BCW n the (C (Bi ), . n ) by 7.4.24, an impossibility by 7.4.20. ∞ (X), τW∞ ) the picture changes. For (BCW ∞ Theorem 7.5.27 Let X, W be as in 7.5.26. Then (BCW (X), τW∞ ) is reflexive if and only if X is compact. ∞ (X), τW∞ ) = BC ∞ (X) which is reflexive by Proof. If X is compact then (BCW ∞ 7.5.22. If X is not compact, (BCW (X), τW∞ ) is not polarly barrelled (7.2.21), so (7.4.13) certainly not reflexive.

Finally, we discuss the case where W is the collection of all polynomial funcn (X) tions X−→K, see 3.7.55. Let n ∈ {0, 1, . . .}. On the Fr´echet space BCW n the topology τW is induced by the norms . n m (m ∈ {0, 1, . . .}), where n f n m := max pX (f Xj ) = max{i (f Xj )∞ : 0 ≤ i ≤ n, 0 ≤ j ≤ m}. 0≤j ≤m

∞ (X). The norms . n m (n, m ∈ {0, 1, . . .}) induce the topology τW∞ on BCW ∞ ∞ Recall that (BCW (X), τW ) was called S(X) in 3.7.64 (when balls in X are compact).

Lemma 7.5.28 Let n, m ∈ {0, 1, . . .}, let B be a ball in X. Then there exists a n number C ≥ 1 such that, for all f ∈ BCW (X) with supp f ⊂ B, pBn (f |B) ≤ f n m ≤ C pBn (f |B). n Proof. Clearly we have for f ∈ BCW (X) that pBn (f |B) = pBn (f ) ≤ n max0≤j ≤m pBn (f Xj ) ≤ max0≤j ≤m pX (f Xj ) = f n m . Now let, in addition, supp f ⊂ B; we shall prove the second equality for C := d n max0≤j ≤m pBn (Xj ), where d := 1 if B = X, d := max(1, 1r ) if B = X, n (f Xj ) ≤ r= radius of B. From 3.7.34 we have, for j ∈ {0, 1, . . . , m}, pX d n pBn (f Xj ) ≤ d n pBn (f ) pBn (Xj ) ≤ C pBn (f ) = C pBn (f |B), and we are done. n (X), τWn ). Now we are ready for characterizations of reflexivity of (BCW

Theorem 7.5.29 Let balls in X be compact, let W be the collection of all polyn nomial functions X−→K, let n ∈ {0, 1, . . .}. Then (BCW (X), τWn ) is reflexive if and only if K is not spherically complete. Proof. Let B a compact ball in X. For f ∈ C n (B), the function f : X−→K given by f (x) = f (x) if x ∈ B, f (x) = 0 if x ∈ X \ B, is easily seen to be n (X) and from 7.5.28 above it follows directly that f → f is a linin BCW n (X), τWn ). Thus, the Banach space ear homeomorphism (C n (B), pBn )−→(BCW

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n (X), τWn ). (C n (B), pBn ) is isomorphic to a subspace of the Fr´echet space (BCW n n If K is spherically complete and (BCW (X), τW ) were reflexive then so would be (C n (B), pBn ) by 7.4.27, which is impossible by 7.4.20. n Now let K be not spherically complete. BCW (X) is a Fr´echet space of countable type (4.3.21(i)), hence reflexive by 7.4.30.

Next, we discuss reflexivity of S(X) = (BC ∞ (X), τW∞ ). If K is not spherically complete the same idea of the previous proof works also for S(X), yielding reflexivity. But for general K the proof is more involved. For a compact ball B in X the restriction map f → f |B sends S(X) into C ∞ (B) and the canonical extension map g → g (see the proof of 7.5.29) sends C ∞ (B) into the subspace HB := {f ∈ S(X) : supp f ⊂ B}. Clearly, f |B = f for every f ∈ HB . Lemma 7.5.30 Let B be a compact ball in X, let n, m ∈ {0, 1, . . .}. Then each . n+1, m -bounded set in HB is a . n m -compactoid. Proof. Let V ⊂ HB be . n+1, m -bounded. Let ε > 0; we shall prove that V ⊂ {f ∈ HB : f n m < ε} + aco G for some finite set G ⊂ HB . By 7.5.28 (with n replaced by n + 1) the set V |B is bounded in (C ∞ (B), pBn+1 ). By compactness of B and 7.5.18 it is a compactoid in (C ∞ (B), pBn ), so there exist g1 , . . . , gk ∈ C ∞ (B) such that V |B ⊂ {g ∈ C ∞ (B) : pBn (g) < C −1 ε} + aco {g1 , . . . , gk }, where C is the constant of 7.5.28. Applying the map g → g we obtain V = V |B ⊂ U + aco {g 1 , . . . , g k }, where U := {g : g ∈ C ∞ (B), pBn (g) < C −1 ε}. If g ∈ C ∞ (B), g ∈ U then g ∈ HB and pBn (g|B) < C −1 ε, so, by 7.5.28, gn m < C C −1 ε = ε. We see that U ⊂ {f ∈ HB : f n m < ε} and we find V ⊂ {f ∈ HB : f n m < ε} + aco G, where G := {g 1 , . . . , g k }. Now we arrive at the crucial next step. Theorem 7.5.31 Let balls in X be compact. Then, for all n, m ∈ {0, 1, . . .}, each . n+1, m+1 -bounded subset of S(X) is a . n m -compactoid. Proof. If suffices to prove that V := {f ∈ S(X) : f n+1, m+1 ≤ 1} is a . n m -compactoid. Let ε > 0; we prove that V ⊂ {f ∈ S(X) : f n m < ε} + aco G for some finite set G ⊂ S(X). Without harm, assume ε < 1 and B := {x ∈ X : |x| ≤ ε−1 } = ∅. Then B is a ball in X and compact by assumption.

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(i) As a first step we prove that f ξX\B n+1, m < ε for all f ∈ V . This is equivalent to i (f Xj )(z) < ε, for all i ∈ {0, 1, . . . , n + 1}, j ∈ {0, 1, . . . , m}, z ∈ Xi+1 \ B i+1 , f ∈ V . Let z = (x1 , . . . , xi+1 ). By symmetry of i we may suppose that xi+1 ∈ B, i.e., |xi+1 | > ε −1 . By rule V of 2.5.78 we have i (f Xj +1 )(z) = i (f Xj )(z) xi+1 + i−1 (f Xj )(x1 , . . . , xi ), where i−1 (f Xj ) := 0 if i = 0. Since f ∈ V the terms i (f Xj +1 )(z) and i−1 (f Xj )(x1 , . . . , xi ) are both ≤ 1 and we get i (f Xj )(z) | xi+1 | ≤ 1, implying i (f Xj )(z) < ε. (ii) Certainly V lies in the . n+1, m -unit ball of S(X), and so does V ξX\B by (i) and the fact that ε < 1. Thus, also V ξB lies in this unit ball. Further, since V ξB ⊂ HB we may apply 7.5.30 obtaining compactoidity of V ξB in (HB , . n m ). So there is a finite set G ⊂ HB such that V ξB ⊂ {f ∈ HB : f n m < ε} + aco G. (iii) Now we combine (i) and (ii). Let f ∈ V . Then by (i), f ξX\B n m ≤ f ξX\B n+1, m < ε whereas, by (ii), f ξB ∈ {h ∈ S(X) : hn m < ε} + aco G. We see that f = f ξB + f ξX\B ∈ {h ∈ S(X) : hn m < ε} + aco G. We obtain the following as a corollary. Theorem 7.5.32 Let balls in X be compact. Then S(X) is reflexive. Proof. S(X) is a Fr´echet space (3.7.55), hence barrelled (7.1.3). From the previous theorem it follows that each bounded set in S(X) is a compactoid. Now apply 7.4.19(α) =⇒ (β).

7.6 Metrizability considerations in duality theory This section may be viewed logically as an appendix to Chapter 7 but will be applied in the next chapter. We will not focus on reflexivity but derive some connections between E and E involving certain metrizability conditions on (subsets of) our spaces.

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Most results in this section do not have a typical non-Archimedean flavour. Throughout Section 7.6 E is a locally convex space. A fundamental sequence of equicontinuous sets in E is a sequence B1 , B2 , . . . in E of equicontinuous sets such that for each equicontinuous set B ⊂ E we have B ⊂ Bn for some n. Theorem 7.6.1 Let E be Hausdorff and polar. Then E is metrizable if and only if E has a fundamental sequence of equicontinuous sets. Proof. Suppose E is metrizable. By 3.5.2 there exist p1 ≤ p2 ≤ · · · polar seminorms defining the topology on E. Choose λ1 , λ2 , . . . ∈ K, |λ1 | < |λ2 | < · · · , limn |λn | = ∞. Let Bn := {f ∈ E : |f | ≤ |λn | pn }. Then each Bn is equicontinuous. Let B ⊂ E be equicontinuous. Then sup{|f | : f ∈ B} is a continuous seminorm, so there exist C > 0 and n ∈ N with |f | ≤ C pn for all f ∈ B. Thus, for m > n such that |λm | ≥ C we have |f | ≤ |λm | pm for all f ∈ B implying B ⊂ Bm , and we are done. Conversely, let B1 , B2 , . . . be a fundamental sequence of equicontinuous sets in E . Then the formula pn := sup{|f | : f ∈ Bn } defines a sequence p1 , p2 , . . . of continuous seminorms on E. Now let p be a polar continuous seminorm on E. We prove that p ≤ pn for some n (and by 3.5.2 we have metrizability). In fact, B := {f ∈ E : |f | ≤ p} is equicontinuous, hence B ⊂ Bn for some n, and we see that, by polarity, p = sup{|f | : f ∈ B} ≤ sup{|f | : f ∈ Bn } = pn . A fundamental sequence of bounded sets in E is a sequence B1 , B2 , . . . in E of bounded sets such that for each bounded set B ⊂ E we have B ⊂ Bn for some n. Then we have the following dual of 7.6.1. Theorem 7.6.2 Let E be a polar space. Then Eb is metrizable if and only if E has a fundamental sequence of bounded sets. Proof. The “if” is a direct consequence of 3.5.2. To prove the “only if”, again by 3.5.2, there exist B1 , B2 , . . . polar bounded sets in E such that the seminorms pn : f → sup |f (x)| : x ∈ Bn } generate the topology b(E , E). Then we

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may suppose that B1 ⊂ B2 ⊂ · · · (so p1 ≤ p2 ≤ · · · ). Choose λ1 , λ2 , . . . ∈ K such that |λ1 | < |λ2 | < · · · , limn |λn | = ∞. To show that {λn Bn : n ∈ N} is a fundamental sequence of bounded sets, let B ⊂ E be bounded, let p(f ) := sup{|f (x)| : x ∈ B}

(f ∈ E ).

Since p is b(E , E)-continuous, p ≤ C pn for some n ∈ N and C > 0. Hence, p ≤ |λm | pm for some m > n for which |λm | ≥ C. Then it follows easily that B ⊂ λm Bm◦◦ = λm Bm . Lemma 7.6.3 Let E be metrizable, let x1 , x2 , . . . be a sequence in E tending to 0. Then there are λ1 , λ2 , . . . ∈ K with limn |λn | = ∞ such that limn λn xn = 0. Proof. By 3.5.2 there are absolutely convex zero neighbourhoods U1 ⊃ U2 ⊃ · · · forming a neighbourhood base at 0. Choose λ ∈ K, |λ| > 1. There are m1 < m2 < · · · in N such that for each m, n ∈ N m > mn =⇒λn xm ∈ Un . Choose λ1 = λ2 = · · · = λm1 = 1, λm1 +1 = · · · = λm2 = λ, λm2 +1 = · · · = λm3 = λ2 , etc. One checks easily that limn λn xn = 0. Corollary 7.6.4 Let E be metrizable. Then bounded subsets of Eb are equicontinuous. Proof. Let B ⊂ E be bounded in Eb , and suppose B is not equicontinuous; we derive a contradiction. The set V := {x ∈ E : |f (x)| ≤ 1 for all f ∈ B} is not a zero neighbourhood in E, so there exists a sequence x1 , x2 , . . . in E tending to 0 for which xn ∈ V for all n. By 7.6.3 there are λ1 , λ2 , . . . ∈ K × such that limn |λn | = ∞ and limn λn xn = 0. Then {λ1 x1 , λ2 x2 , . . .} is bounded, so U := {f ∈ E : |f (λn xn )| ≤ 1 for all n} is a zero neighbourhood in Eb so that B ⊂ λ U for some λ ∈ K, which implies that |f (λn xn )| ≤ |λ| for all n, |λ| ≤ 1 for all f ∈ B implying all f ∈ B. Thus, for large n, |f (xn )| ≤ λ−1 n xn ∈ V for those n, a contradiction. Corollary 7.6.5 Let E be metrizable. Then jE : E−→E is continuous. Proof. Let (xi )i be a net in E converging to 0, let B be a bounded sets in Eb ; we have to prove that jE (xi ) → 0 uniformly on B, i.e., f (xi ) → 0 uniformly for f ∈ B. But 7.6.4 tells us that B is equicontinuous, and we are done. Corollary 7.6.4, combined with 7.6.1 and 7.6.2, leads to an interesting conclusion. Theorem 7.6.6 If E is polar and metrizable then so is E .

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Proof. Polarity of E is clear. Let us prove metrizability. 7.6.1 shows that E has a fundamental sequence of equicontinuous sets. From 7.3.4 and 7.6.4 we know that, for subsets of E , “equicontinuous” is the same as “bounded in Eb ”. Now by applying 7.6.2 to Eb in place of E we obtain metrizability of (Eb )b = E . The next two theorems state that the strong dual of a metrizable space is rarely metrizable, but always complete. Theorem 7.6.7 Let E be polar and suppose both E and Eb are metrizable. Then E is normable. Proof. Assume E is not normable; we derive a contradiction. By 7.6.2 E has a fundamental sequence B1 ⊂ B2 ⊂ · · · of bounded sets. By assumption and 3.6.2 no Bn is a zero neighbourhood. Thus, letting U1 ⊃ U2 ⊃ · · · be a neighbourhood base at 0 (3.5.2) we can find x1 , x2 , . . . ∈ E such that xn ∈ Un \ Bn for all n. Then xn → 0, so B := {x1 , x2 , . . .} is bounded, but B ⊂ Bn for no n, a contradiction. Remark 7.6.8 Clearly, if E is normable then so is Eb . But it may happen that Eb is normable, while E is not. In fact, consider E := (c0 , σ (c0 , c0 )). Then E is not normable but Eb = ∞ with the norm topology. Theorem 7.6.9 Let E be metrizable. Then Eb is complete. Proof. Let (fi )i be a Cauchy net in Eb . Then clearly fi → f uniformly on bounded sets, for some f ∈ E ∗ , and the restriction of f to bounded subsets of E is continuous. Thus, for a sequence x1 , x2 , . . . converging to 0 we have f (xn ) → 0. By metrizability of E, continuity of f follows. Next we move to spaces of countable type and treat them in a way similar to what we did for metrizability. Theorem 7.6.10 Let E be a polar space. Then E is of countable type if and only if each equicontinuous subset of E is σ (E , E)-metrizable. Proof. Suppose E is of countable type, let B ⊂ E be equicontinuous. To prove σ (E , E)-metrizability we may assume that B = {f ∈ E : |f | ≤ p} for some (polar) continuous seminorm p on E . By assumption there is a countable set {x1 , x2 , . . .} such that [x1 , x2 , . . .] is p-dense in E, without loss assume p(xn ) ≤ 1 for all n. For f, g ∈ B put d(f, g) := max |f (xn ) − g(xn )| 2−n . n∈N

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(Since |f (xn ) − g(xn )| 2−n ≤ p(xn ) 2−n ≤ 2−n → 0 this maximum exists.) It is not hard to see that d is an ultrametric on B. If (fi )i is a net in B converging to f ∈ B in the weak∗ -topology then a standard procedure shows that d(fi , f ) → 0. Conversely, suppose d(fi , f ) → 0 for some net (fi )i in B and f ∈ B. Let x ∈ E. To show fi (x) → f (x), let ε > 0 and let y ∈ [x1 , x2 , . . .] be such that p(x − y) < ε. We clearly have fi (xn ) → f (xn ) for each n, hence also fi (y) → f (y). Thus, for large i, |fi (x) − f (x)| ≤ max (|fi (x) − fi (y)| , |fi (y) − f (y)| , |f (y) − f (x)|) ≤ max (p(x − y), |fi (y) − f (y)|) < ε. To prove the converse, suppose equicontinuous sets in E are σ (E , E)metrizable, let p be a polar continuous seminorm on E. Then A := {f ∈ E : |f | ≤ p} is equicontinuous and p = sup{|f | : f ∈ A}. A is σ (E , E)-bounded, hence a σ (E , E)-compactoid, and by assumption σ (E , E)-metrizable. Then, by 3.8.25, A ⊂ acoσ (E ,E) {f1 , f2 , . . .}, where fn ∈ λ A for some |λ| > 1 and fn → 0 in σ (E , E). Hence {f1 , f2 , . . .} is equicontinuous and p(x) ≤ sup |fn (x)| ≤ |λ| p(x)

(x ∈ E).

n∈N

Therefore, T : Ep −→c0 , defined by T (πp (x)) = (f1 (x), f2 (x), . . .) ∈ c0

(x ∈ E)

is a linear homeomorphism of Ep into c0 . As c0 is of countable type then so is Ep . As an application of 7.6.10 we have: Theorem 7.6.11 For a polar space E the following are equivalent: (α) E is of countable type. Each equicontinuous sequence in E that converges in σ (E , E) converges in b(E , E). (β) Equicontinuous subsets of E are metrizable compactoids in Eb . Proof. (α) =⇒ (β). Let B ⊂ E be equicontinuous. By 7.6.10, B is σ (E , E)metrizable. B is also a σ (E , E)-compactoid. We show that σ (E , E) = b(E , E) on B. (Then by 3.8.13 we are done.) By metrizability it suffices to show that if f1 , f2 , . . . is a sequence in B converging pointwise to f ∈ B then fn → f in Eb . But this follows from (α). (β) =⇒ (α). Since b(E , E) is closely related to σ (E , E), from 3.8.23 and (β) we derive that σ (E , E) and b(E , E) coincide on equicontinuous subsets

7.6 Metrizability considerations in duality theory

299

of E . Then clearly the second condition of (α) holds, and the first one follows from 7.6.10. We also have dual forms of 7.6.10 and 7.6.11. Theorem 7.6.12 Let E be a Hausdorff polar space. Then Eb is of countable type if and only if each bounded subset of E is σ (E, E )-metrizable. Proof. Suppose Eb is of countable type. Let B be a bounded set in E. The seminorm p on E defined by the formula p(f ) = sup{|f (x)| : x ∈ B} is continuous for b(E , E). By assumption there are f1 , f2 , . . . in E such that [f1 , f2 , . . .] is p-dense in Eb . Without loss we may assume that p(fn ) ≤ 1 for all n. It takes a similar reasoning to the one done in 7.6.10 to show that d : B × B−→R, (x, y) → maxn∈N |fn (x) − fn (y)| 2−n is an ultrametric on B whose induced topology equals σ (E, E )|B. Conversely, suppose each bounded subset of E is σ (E, E )-metrizable. Let A ⊂ E be bounded, let us prove that the seminorm p on E defined by p(f ) = sup{|f (x)| : x ∈ A}, is of countable type. We may assume that A is absolutely convex. As A is σ (E, E )-compactoid, by the assumption is also σ (E, E )metrizable. Then, by 3.8.25, there is a sequence e1 , e2 , . . . in λ A for some |λ| > 1, converging weakly to 0, such that A ⊂ acoσ (E,E ) {e1 , e2 , . . .}. Then, with an analogous procedure as in the last part of the proof of 7.6.10 we arrive at a linear homeomorphism of (E )p into c0 , so (E )p is of countable type. Theorem 7.6.13 For a Hausdorff polar space E the following are equivalent: (α) Eb is of countable type. Each weakly convergent sequence in E is convergent. (β) Bounded subsets of E are metrizable and compactoid. Proof. (α) =⇒ (β). Let B ⊂ E be bounded. By 7.6.12 B is σ (E, E )metrizable. B is also σ (E, E )-bounded, hence a σ (E, E )-compactoid. By 3.8.25 there exists a sequence x1 , x2 , . . . in E tending weakly to 0 such that B ⊂ acoσ (E,E ) {x1 , x2 , . . .}. From (α) we obtain xn → 0, so that B1 := aco {x1 , x2 , . . .} is a metrizable compactoid in E (3.8.24). By 5.2.14(i) B1 is weakly closed, hence B ⊂ B1 and we have (β). (β) =⇒ (α). From (β) and 5.2.12 we have that the weak and the initial topologies coincide on bounded sets (= weakly bounded sets, by 5.4.5). From this we derive that weakly convergent sequences are convergent and that bounded sets in E are σ (E, E )-metrizable. Now apply 7.6.12 to conclude that Eb is of countable type. Remark 7.6.14 Observe that if K is spherically complete or E is of countable type, the second condition of (α) is automatically fulfilled (5.5.2).

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7.7 Notes For non-spherically complete K a first example of a non-reflexive closed subspace of ∞ was constructed by van Rooij in [193], 4.J; it differs from ours (7.4.18(i)). A more natural example was presented by Perez-Garcia and Schikhof in [180]. The fact that reflexive Banach spaces over spherically complete fields are finite-dimensional (7.4.20) was established by van Rooij in [193], 4.16 and by Schneider in [210], 11.1, but with different proofs. It has been a long-standing conjecture that, for non-spherically complete K, under certain cardinality assumptions, every dual Banach space is reflexive, but a counterexample has been given by Eda, Kiyosawa and Ohta in [50]. The main theory of barrelledness and reflexivity was developed by van Tiel in [227] for spherically complete K and by Schikhof in [196] for general K. For other weaker versions of barrelledness, see [181]. In [196] and [227] some attention was paid to semireflexivity, which also was studied by De Grande-De Kimpe in [62], for spherically complete K, and in [74] for general K. Open mapping and closed graph theorems with barrelled spaces involved were proved by Gilsdorf and Ka¸kol in [60]. Given a locally convex space (E, τ ) one may ask whether the barrelled topology associated to τ exists, analogous to the topology of countable type and the polar topology associated to τ (4.5.11, 4.5.13). The answer is yes, but we have to reverse the inclusions. In fact, the smallest barrelled (polarly barrelled) locally convex topology ≥ τ exists. The proof is straightforward. Theorem 7.2.7 follows from the more general result [124], 4.2, proved in a way different from ours.

8 Montel and nuclear spaces

In this chapter we study the class of the so-called semi-Montel spaces and two important subclasses consisting of the nuclear spaces and the Montel spaces, respectively. We will see that for members of these classes the duality and reflexivity theory becomes more powerful and varied than for arbitrary spaces. We first study compactoid operators (i.e., operators mapping some zero neighbourhood onto a compactoid set, 8.1.1) and compactifying operators (i.e., continuous operators that map bounded sets onto compactoids, 8.3.1). Compactoid operators are compactifying (8.3.2) but the converse does not hold (8.3.4). Basic properties of compactoid operators are listed in 8.1.3, the general form of a compactoid operator into c0 is given in 8.1.9(ii). As an application we derive in 8.2.1 and 8.2.2 that, if the valuation of K is dense, there is no continuous linear surjection ∞ −→c0 , and that ∞ does not have a base. The general form of a compactifying operator into c0 is given in 8.3.9. In Section 8.4 we treat semi-Montel spaces E, i.e., for each normed space F each T ∈ L(F, E) is compactoid, 8.4.1(i); equivalently, for each normed space F each T ∈ L(E, F ) is compactifying, 8.4.5(ε). It is also proved in 8.4.5(δ) that a space is semi-Montel if and only if each bounded set is a compactoid. For polar spaces (spaces of countable type) E we characterize semi-Montelness in terms of Eb in 8.4.8 (8.4.13). For this we provide a new characterization of compactoids in polar spaces (8.4.9). Hereditary properties of semi-Montel spaces and reflexive semi-Montel spaces, called Montel spaces (8.4.2), follow in 8.4.24, 8.4.25 and 8.4.26. By “reversing the arrows” in the definition of semi-Montelness we define a space E to be nuclear if for each normed space F each T ∈ L(E, F ) is compactoid (8.4.1(ii)). We show in 8.5.1(β) that nuclearity of E is equivalent to the “usual one”, i.e., for each continuous seminorm p there is a continuous seminorm q ≥ p such that the canonical map Eq −→Ep is compactoid. Nuclear spaces are semi-Montel (8.5.2), but the converse does not hold (8.5.4(a)). 301

302

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Nuclear spaces are of countable type (8.5.1(δ)), a normable nuclear space is finite-dimensional (8.5.6(i)). The class of nuclear spaces has rich permanence properties (8.5.7). In Section 8.6 we focus on metrizable semi-Montel spaces. We first prove that such a space is of countable type (8.6.1), that its strong dual is a nuclear Montel space (8.6.3, 8.6.4), and that its completion is Montel (8.6.4, 8.6.7). This leads to various characterizations of metrizable semi-Montel spaces (8.6.8). The classical counterpart of 8.6.1 states that every Fr´echet Montel space over R or C is separable (see e.g. [103], 11.6.2). Its proof does not simply carry over the non-Archimedean case because K may be not locally compact; so we have to deal with compactoids rather than compact sets. This modification is obstructing the classical proof, which is essentially based upon separability. It is here that the t-frames introduced in 3.10.4 come to the rescue, as we show in the proof of 8.6.1. Examples of semi-Montel and nuclear spaces are considered in Section 8.7.

8.1 Compactoid operators We will need the concept of a compactoid operator. In locally convex theory over R or C, a linear map between two locally convex spaces is called precompact if it maps some zero neighbourhood into a precompact set (see [103], Section 17.1). This leads to the following non-Archimedean counterpart. Definition 8.1.1 Let E, F be locally convex spaces (over K). A linear map T : E−→F is called compactoid if there is a zero neighbourhood U in E such that T (U ) is a compactoid in F . The collection of all compactoid maps E−→F is denoted C(E, F ). Remark 8.1.2 See the chapter notes for discussions on terminology and alternative concepts. Henceforth we shall use the expression “operator” to denote a linear map between K -vector spaces. A typical example of a compactoid operator is the “multiplication map” c0 −→c0 defined by (λ1 , λ2 , . . .) −→ (λ1 α1 , λ2 α2 , . . .), where (α1 , α2 , . . .) ∈ c0 .

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Below we collect a few immediate consequences of the definition. For the properties of compactoid sets needed in the proof, see Section 3.8. Theorem 8.1.3 Let E, F, G be locally convex spaces. Then we have the following: (i) C(E, F ) is a subspace of L(E, F ). (ii) Let T ∈ C(E, F ). Then the restriction of T to any subspace of E is compactoid. (iii) Let T ∈ C(E, F ), let D be a subspace of F for which D ⊃ T (E). Then T , considered as an operator E−→D, is compactoid. (iv) Let T ∈ C(E, F ). Then T (E) is of countable type. (v) Let T ∈ L(E, F ), S ∈ L(F, G). Then, if S or T is compactoid then so is S ◦T. (vi) If E or F is finite-dimensional then L(E, F ) = C(E, F ). (vii) If E is a normed space then an operator T : E−→F is compactoid if and only if T (BE ) is a compactoid. (viii) If E is of finite type and F is normed then L(E, F ) = C(E, F ). The same happens if E is normed and F is of finite type. (ix) Let D be a dense subspace of E, let T ∈ L(E, F ). Then T is compactoid if and only if T |D is compactoid. (x) Let E be Hausdorff. Then the identity map E−→E is compactoid if and only if E is finite-dimensional. (xi) Let D be a closed subspace of E. Then the quotient map π : E−→E/D is compactoid if and only if E/D is finite-dimensional. An operator T : E/D−→F is compactoid if and only if T ◦ π is compactoid. Proof. (i). First we prove continuity of a T ∈ C(E, F ). Let V be a zero neighbourhood in F . There is a zero neighbourhood U in E such that T (U ) is a compactoid, hence bounded. Thus, λ T (U ) ⊂ V for some λ ∈ K × , and U ⊂ T −1 (T (U )) ⊂ λ T −1 (V ). Therefore, T −1 (V ) contains the zero neighbourhood λ−1 U , hence is one itself. Clearly the zero operator is compactoid and C(E, F ) is closed for scalar multiplication. Let T1 , T2 ∈ C(E, F ). To prove that T1 + T2 ∈ C(E, F ), let U1 , U2 be zero neighbourhoods in E for which T1 (U1 ) and T2 (U2 ) are compactoids. Then U1 ∩ U2 is a zero neighbourhood in E and (T1 + T2 )(U1 ∩ U2 ) ⊂ T1 (U1 ) + T2 (U2 ), a compactoid. Thus, T1 + T2 is compactoid. The proofs of (ii) and (iii) are obvious. (iv). There is a zero neighbourhood U in E such that T (U ) is a compactoid. Since U is absorbing we have E = [U ], hence T (E) = [T (U )]. Now apply 4.2.2.

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(v). Let S be compactoid. There is a zero neighbourhood U in F such that S(U ) is a compactoid. Then S ◦ T maps the zero neighbourhood T −1 (U ) into S(U ), so S ◦ T is compactoid. Now assume T is compactoid, let V be a zero neighbourhood in E such that T (V ) is a compactoid. Since S maps compactoids into compactoids we obtain that S ◦ T is compactoid. (vi). Easy by using 3.8.4(ix). (vii). Obvious from the fact that each zero neighbourhood in E contains a non-zero scalar multiple of the unit ball. (viii). First assume that E is of finite type and F is normed. Let T ∈ L(E, F ). The formula p(x) = T (x)

(x ∈ E)

defines a continuous seminorm on E. Hence Ep is finite-dimensional and Tp : Ep −→F , πp (x) → T (x) is a well-defined continuous operator, which is compactoid by (vi) and then so is T = Tp ◦ πp by (v). Now assume that E is normed and F is of finite type. Let T ∈ L(E, F ). Then T (BE ) is bounded, hence a compactoid, in F . By (vii) T is compactoid. (ix). If T is compactoid then so is T |D by (ii). Conversely, let T |D be compactoid. There is an open zero neighbourhood U in D such that T (U ) is a compactoid. Then U = V ∩ D, where V is an open zero neighbourhood in E. Density of D ensures V ⊂ U . Then T (V ) ⊂ T (U ) ⊂ T (U ), a compactoid. Thus, T is compactoid. (x). One half follows from (vi). Conversely, let the identity be compactoid. Then, some zero neighbourhood is a compactoid. Now apply 3.8.5 to conclude that E is finite-dimensional. (xi). We first prove the second assertion. If T : E/D−→F is compactoid then so is T ◦ π by (v). Conversely, let T ◦ π be compactoid. There is a zero neighbourhood U in E such that T (π (U )) is a compactoid. Since by 3.4.5 π(U ) is a zero neighbourhood in E/D we see that T is compactoid. To prove the first assertion, apply the above to the identity T : E/D−→E/D to conclude that π is compactoid if and only if T is compactoid. Now E/D is Hausdorff (3.4.8), and the conclusion follows applying (x). From now on in this chapter E, F denote locally convex spaces. For our purpose we are particularly interested in compactoid operators into c0 . We start with two simple lemmas on c0 . Let us denote the unit vectors of c0 by e1 , e2 , . . . .

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Lemma 8.1.4 For a subset X of c0 the following are equivalent: (α) X is a compactoid. (β) There exists (α1 , α2 , . . .) ∈ c0 such that X ⊂ aco{α1 e1 , α2 e2 , . . .}. Proof. (β) =⇒ (α) is a direct consequence of 3.8.24. To prove the more interesting implication (α) =⇒ (β) we first apply 3.8.25 to conclude that X ⊂ aco {x1 , x2 , . . .}, where x1 , x2 , . . . ∈ c0 , limn xn ∞ = 0. For each m ∈ N, let πm be the coordinate function (λ1 , λ2 , . . .) → λm . Since xn → 0 we have for each m ∈ N that maxn∈N |πm (xn )| exists; choose an αm ∈ K such that |αm | = maxn∈N |πm (xn )|. We proceed to show that limm αm = 0. In fact, let ε > 0. There is an N ∈ N such that xN+1 ∞ , xN+2 ∞ , . . . are all < ε. There is an M ∈ N such that |πm (xn )| < ε for all n ∈ {1, . . . , N } and m ≥ M. Thus, for all m ≥ M we have |πm (xn )| < ε for all n ∈ N, implying |αm | ≤ ε for m ≥ M. Next, we prove that X ⊂ aco {α1 e1 , α2 e2 , . . .}. For this it suffices to show that, for each n ∈ N, xn ∈ aco {α1 e1 , α2 e2 , . . .}. Let xn = ∞ m=1 βm em be its expansion with respect to the base e1 , e2 , . . .. Then, for each m, |βm | = |πm (xn )| ≤ |αm |, so that xn ∈ aco {β1 e1 , β2 e2 , . . .} ⊂ aco {α1 e1 , α2 e2 , . . .}. Lemma 8.1.5 (Product Lemma) Let (α1 , α2 , . . .) ∈ c0 . Then there are (λ1 , λ2 , . . .) ∈ c0 and (µ1 , µ2 , . . .) ∈ c0 such that λi = 0 and αi = λi µi for each i ∈ N. Proof. Let ρ ∈ K, 0 < |ρ| < 1. Let i ∈ N. If αi = 0 choose λi := ρ i . If αi = 0 √ there is a unique ni ∈ Z such that |ρ|ni +1 < |αi | ≤ |ρ|ni , and choose λi := ρ ni . Then µi := αi /λi is well-defined and αi = λi µi for all i ∈ N. We have √ λi = 0 for each i and, since |λi | ≤ max(|ρ|i , |ρ|−1 |αi |), λi → 0. For each √ i ∈ N, |µi | = |αi /λi | ≤ |αi | so that also µi → 0. Also, the notion of local convergence is needed. Definition 8.1.6 A sequence f1 , f2 , . . . in E is said to converge locally to 0 if there exists a continuous seminorm p on E and (α1 , α2 , . . .) ∈ c0 such that |fn | ≤ |αn | p for all n ∈ N. Some immediate properties of local convergence are listed below. Theorem 8.1.7 Let f1 , f2 , . . . ∈ E . Then we have the following: (i) If {f1 , f2 , . . .} is equicontinuous and (α1 , α2 , . . .) ∈ c0 then αn fn → 0 locally. (ii) If fn → 0 locally then {f1 , f2 , . . .} is equicontinuous and fn → 0 in Eb .

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(iii) fn → 0 locally if and only if there exist (λ1 , λ2 , . . .) ∈ c0 and g1 , g2 , . . . ∈ E such that fn = λn gn for all n and gn → 0 locally. (iv) If h1 , h2 , . . . ∈ F and hn → 0 locally then hn ◦ T → 0 locally for each T ∈ L(E, F ). Proof. (i). By equicontinuity p := supn∈N |fn | is a continuous seminorm and we have |αn fn | ≤ |αn | p for all n. (ii). Let a continuous seminorm p and (α1 , α2 , . . .) ∈ c0 be such that |fn | ≤ |αn | p for all n. Then |fn | ≤ (maxi∈N |αi |) p and equicontinuity follows. Now let B ⊂ E be bounded. Then C := sup{p(x) : x ∈ B} < ∞. For each x ∈ B we have |fn (x)| ≤ |αn | C; we see that fn → 0 uniformly on B. (iii). The “if” is obvious. To prove the “only if”, let p be a continuous seminorm and (α1 , α2 , . . .) ∈ c0 such that |fn | ≤ |αn | p for all n. By the Product Lemma 8.1.5 there are (λ1 , λ2 , . . .) and (µ1 , µ2 , . . .) in c0 such that λn = 0 and αn = λn µn for all n. −1 |fn | ≤ λ−1 |αn | p = |µn | p so that Put gn := λ−1 n fn . Then |gn | = λn n gn → 0 locally. (iv). Immediate. Remark 8.1.8 We will see in 9.8.2(iii) that the converse of (ii) does not hold. Now we characterize the compactoid operators E−→c0 . Theorem 8.1.9 Let T be an operator E−→c0 . Then we have the following: (i) T ∈ L(E, c0 ) if and only if T has the form x −→ (f1 (x), f2 (x), . . .),

(8.1)

where {f1 , f2 , . . .} ⊂ E is equicontinuous and fn → 0 in Eσ . (ii) T ∈ C(E, c0 ) if and only if T has the form (8.1), where fn ∈ E for all n and fn → 0 locally. Proof. The proof of (i) is straightforward. To prove (ii), let T ∈ C(E, c0 ). There is a zero neighbourhood U such that T (U ) is a compactoid in c0 . From 8.1.4 we derive the existence of an (α1 , α2 , . . .) ∈ c0 such that T (U ) ⊂ {(λ1 , λ2 , . . .) ∈ c0 : |λn | ≤ |αn | for all n}. Without loss, assume αn = 0 for each n. Let p be a continuous seminorm such that {x ∈ E : p(x) ≤ 1} ⊂ U . Thus, p(x) ≤ 1 implies |fn (x)| ≤ |αn | for all n. We see that {α1−1 f1 , α2−1 f2 , . . .} is equicontinuous so that, by 8.1.7(i), fn = αn αn−1 fn → 0 locally. Conversely, assume that T has the form x → (f1 (x), f2 (x), . . .) where fn → 0 locally. There are a continuous seminorm p and an (α1 , α2 , . . .) ∈ c0 such that |fn | ≤ |αn | p for all n. We see that the zero neighbourhood {x ∈ E : p(x) ≤ 1}

8.1 Compactoid operators

307

is mapped into {(λ1 , λ2 , . . .) ∈ c0 : |λn | ≤ |αn | for all n}, a compactoid. Hence T ∈ C(E, c0 ). We prove extension theorems for operators E−→c0 . Theorem 8.1.10 Let K be spherically complete or let E be of countable type. Let D ⊂ E be a subspace. Then we have the following: (i) Each T ∈ L(D, c0 ) can be extended to a T˜ ∈ L(E, ∞ ). (ii) Each T ∈ C(D, c0 ) can be extended to a T˜ ∈ C(E, c0 ). Proof. (i). By 8.1.9(i) T has the form x → (f1 (x), f2 (x), . . .) (x ∈ D), where fn ∈ D , |fn | ≤ p for all n and for some continuous seminorm p on D. Let p1 be a continuous seminorm on E that extends p (3.4.3). By the Hahn–Banach Theorem 4.1.1 or 4.2.4 the fn extend to f˜n ∈ E for which |f˜n | ≤ 2 p1 . Put, for x ∈ E, T˜ (x) := (f˜1 (x), f˜2 (x), . . .). We see that T˜ maps E into ∞ and satisfies the requirements. (ii). By 8.1.9(ii) T has the form x → (f1 (x), f2 (x), . . .) (x ∈ D), where fn ∈ D , |fn | ≤ |αn | p for all n and for some (α1 , α2 , . . .) ∈ c0 and continuous seminorm p on D. As in (i), by the Hahn-Banach Theorem 4.1.1 or 4.2.4, the fn extend to f˜n ∈ E for which |f˜n | ≤ 2 |αn | p1 for all n, where p1 is a continuous seminorm on E that extends p. We see that f˜n → 0 locally so that, by 8.1.9(ii), the map T˜ , with T˜ (x) = (f˜1 (x), f˜2 (x), . . .)

(x ∈ E)

satisfies the requirements. If the valuation of K is discrete we can strengthen (i) above by using:

Lemma 8.1.11 Let the valuation of K be discrete. Then the identity c0 −→c0 can be extended to an orthogonal projection ∞ −→c0 . Proof. The unit vectors e1 , e2 , . . . of c0 form an orthonormal system in ∞ that can be extended to a maximal orthonormal system {ei : i ∈ I }, with N ⊂ I . By 2.5.4 this system is an orthonormal base in the sense of the comments following 2.5.1. Clearly the map λi ei −→ λi ei i∈I

is the required projection.

i∈N

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Corollary 8.1.12 Let the valuation of K be discrete, let D be a subspace of E. Then every T ∈ L(D, c0 ) can be extended to a T˜ ∈ L(E, c0 ). Proof. By 8.1.10(i) there is an extension T1 ∈ L(E, ∞ ) and from 8.1.11 we obtain an orthogonal projection P : ∞ −→c0 . It suffices to set T˜ := P ◦ T1 .

8.2 Intermezzo: a curious property of ∞ The natural question on what happens to 8.1.11 and 8.1.12 when K has a dense valuation will lead to curious results on ∞ that “logically” belong to Chapter 2 but, since compactoid operators are involved, will be proved here. Theorem 8.2.1 Let the valuation of K be dense. Then there is no continuous linear surjection ∞ −→c0 . Proof. Suppose there exists a surjective T ∈ L( ∞ , c0 ); we derive a contradiction. By the Open Mapping Theorem 2.1.17 the injective map ∞ /Ker T −→c0 associated to T is a bijective homeomorphism, so that the adjoint T : c0 −→( ∞ ) is a homeomorphic injection. Since c0 = ∞ (2.5.11) is not of countable type by 2.5.15, certainly ( ∞ ) is not of countable type. By 5.5.5 K must be spherically complete. It is easily seen that then ∞ is spherically complete and hence, by 4.1.11, so is ∞ /Ker T . Since this space is isomorphic to c0 , we get that c0 , equipped with a norm . that is equivalent to the canonical supremum one, must be spherically complete. By 2.3.25, (c0 , . ) has an orthogonal base e1 , e2 , . . . . Choose λ1 , λ2 , . . . ∈ K × with 1 . 2 Then x1 := λ1 e1 , x2 := λ2 e2 , . . . also form an orthogonal base of (c0 , . ). Consider the balls in (c0 , . ) defined by 1 > λ1 e1 > λ2 e2 > · · · >

Bn := Bx1 +···+xn (xn+1 )

(n ∈ N).

We have B1 ⊃ B2 ⊃ · · · and by spherical completeness there is an x ∈ n∈N Bn , let x=

∞

µn xn

n=1

be its expansion. Then µn xn → 0, so µn → 0. On the other hand, from x − (x1 + · · · + xn ) ≤ xn+1 < xn

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309

we obtain successively |µ1 − 1| < 1, |µ2 − 1| < 1, . . ., so that |µn | = 1 for all n, conflicting µn → 0. This leads to the following result contrasting with 2.5.4. Corollary 8.2.2 Let the valuation of K be dense. Then ∞ does not have a base. Proof. Suppose {ei : i ∈ I } is a base. Then by 2.3.11 it is a t-orthogonal base for some t ∈ (0, 1]. Without loss, assume N ⊂ I and |ρ| ≤ ei ≤ 1 for all i (ρ ∈ K, 0 < |ρ| < 1). Then λi ei −→ (λ1 , λ2 , . . .) i∈I

is a continuous linear surjection ∞ −→c0 , conflicting 8.2.1.

Corollary 8.2.3 Let the valuation of K be dense. Then each T ∈ L( ∞ , c0 ) is compactoid. Proof. Let B be the closed unit ball of ∞ and suppose that T (B) is not a compactoid. Then by 3.9.6 there exists, for some t ∈ (0, 1], a (bounded) torthogonal sequence y1 , y2 , . . . in T (B) such that yn ≥ ε > 0 for some ε and all n. Then D := [y1 , y2 , . . .] = { ∞ n=1 λn yn : λn ∈ K, λn → 0} is closed in c0 , hence complemented (2.3.13), let P : c0 −→D be a continuous surjective projection. Then T˜ := P ◦ T maps ∞ into D; we prove that T˜ is surjective which will yield a contradiction by 8.2.1 since D is linearly homeomorphic to c0 (2.3.9). Let y ∈ D, y = ∞ n=1 λn yn , where λn → 0. There are x1 , x2 , . . . ∈ B such that yn = T (xn ) = P (T (xn )) = T˜ (xn ) for each n. Then, by completeness, ∞ ∞ ˜ x := ∞ n=1 λn xn exists in so that T (x) = n=1 λn yn = y.

8.3 Compactifying operators Of secondary interest are the so-called compactifying operators, which we will now discuss briefly. Definition 8.3.1 An operator E−→F is called compactifying if it is continuous and maps bounded sets of E onto compactoid sets of F . The collection of all compactifying operators E−→F is denoted by CF (E, F ). First we prove: Theorem 8.3.2 Compactoid operators are compactifying.

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Proof. Let T ∈ C(E, F ), where E, F are locally convex spaces. Then T ∈ L(E, F ). Let B ⊂ E be bounded; we prove that T (B) is a compactoid. There is a zero neighbourhood U in E such that T (U ) is a compactoid. Boundedness of B yields a λ ∈ K such that B ⊂ λ U . Then T (B) ⊂ λ T (U ) so that T (B) is a compactoid. Remark 8.3.3 The converse is not true, as the following example shows. Example 8.3.4 Let T : K (I ) −→K (I ) be the identity, where I is infinite. Then T is compactifying but not compactoid. Proof. Bounded sets in K (I ) are finite-dimensional (3.6.6(a)), so T is compactifying. By 8.1.3(x) T cannot be compactoid. Remark 8.3.5 Later we will find an E and a T ∈ L(E, c0 ) that is compactifying but not compactoid, see 9.8.2(iv). The continuity condition in 8.3.1 is not superfluous. Example 8.3.6 The identity T : (c0 , σ (c0 , c0∗ ))−→(c0 , . ∞ ) maps bounded sets into compactoids but is not continuous. Proof. Let τ be the strongest locally convex topology on c0 . We observed in 4.4.8(b) that τ is polar, and in 3.6.6(a) that τ -bounded sets are finitedimensional. Since σ (c0 , c0∗ ) is the weak topology of τ they have the same bounded sets (5.4.5), so obviously T takes σ (c0 , c0∗ )-bounded sets into compactoids. On the other hand, T cannot be continuous since (c0 , σ (c0 , c0∗ )) is of finite type whereas (c0 , . ∞ ) is not. Remark 8.3.7 Let us compare the properties of C(E, F ) and CF (E, F ) in the spirit of 8.1.3. Clearly CF (E, F ) is a subspace of L(E, F ). The natural translations of (ii), (iii), (v) and (viii) of 8.1.3 are true. As for (vii) observe that if E is normed then C(E, F ) = CF (E, F ). The same conclusion holds if E or F is finite-dimensional (vi). That (iv), (x) and (xi) are not true for “compactoid” replaced by “compactifying” follows directly from 8.3.4. We do not know whether (ix) of 8.1.3 holds for “compactifying”. But we do have the following partial answer. Theorem 8.3.8 Let E be metrizable, D a dense subspace of E and T ∈ L(E, F ). Then T is compactifying if and only if T |D is compactifying. Proof. We only have to prove the “if”. Let d be an invariant ultrametric on E defining its topology, let B ⊂ E be bounded. To see that T (B) is a compactoid we prove that each countable subset of T (B) is a compactoid (3.8.12). So take x1 , x2 , . . . ∈ B. For each n there is a yn ∈ D

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such that d(xn , yn ) ≤ n1 . Then limn (xn − yn ) = 0, so {x1 − y1 , x2 − y2 , . . .} is bounded. Hence {y1 , y2 , . . .} ⊂ D is bounded. Now since T |D is compactifying, A1 := {T (y1 ), T (y2 ), . . .} is a compactoid. Also, limn (T (xn ) − T (yn )) = 0, so A2 := {T (x1 ) − T (y1 ), T (x2 ) − T (y2 ), . . .} is a compactoid, then so is {T (x1 ), T (x2 ), . . .} ⊂ A1 + A2 . Like in 8.1.9 we now describe the compactifying operators E−→c0 . Theorem 8.3.9 An operator T : E−→c0 is compactifying if and only if it has the form x −→ (f1 (x), f2 (x), . . .), where {f1 , f2 , . . .} ⊂ E is equicontinuous and fn → 0 in Eb . Proof. Let T be compactifying. By continuity and 8.1.9(i) T has the form of (8.1) with {f1 , f2 , . . .} ⊂ E equicontinuous; we prove fn → 0 in Eb . Let B ⊂ E be bounded. Then T (B) is a compactoid in c0 , so by 8.1.4 there is an (α1 , α2 , . . .) ∈ c0 such that T (B) ⊂ {(λ1 , λ2 , . . .) ∈ c0 : |λn | ≤ |αn | for all n}. We see that fn (B) ⊂ BK (0, |αn |) for each n; it follows that fn → 0 uniformly on B. Conversely, let T have the above form. Then, by 8.1.9(i), T ∈ L(E, c0 ). Let B ⊂ E be bounded. Then since fn → 0 uniformly on B, there is an (α1 , α2 , . . .) ∈ c0 such that fn (B) ⊂ BK (0, |αn |) for each n. We see that T (B) ⊂ {(λ1 , λ2 , . . .) ∈ c0 : |λn | ≤ |αn | for all n}, which is a compactoid. Remark 8.3.10 One may wonder whether the equicontinuity condition above is superfluous. Of course it can be dropped if E is polarly barrelled (7.1.7) or normed (straightforward verification), but not in general. In fact, the noncontinuous operator of 8.3.6 has the form x → (f1 (x), f2 (x), . . .) where f1 , f2 , . . . are the (continuous) coordinate functions, fn → 0 uniformly on bounded (finite-dimensional) sets.

8.4 (Semi-)Montel spaces The following definitions are somewhat unusual but have the advantage of revealing a certain “duality”. Definition 8.4.1 (i) E is called semi-Montel if, for each normed space F , each T ∈ L(F, E) is compactoid.

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(ii) E is called nuclear if, for each normed space F , each T ∈ L(E, F ) is compactoid. Definition 8.4.2 A reflexive semi-Montel space is called Montel. Remark 8.4.3 There is a slight difference between the “classical” and our definition of semi-Montelness and nuclearity. Some comparisons are given in the chapter notes. We first study (semi-)Montel spaces. In the next section nuclearity will enter the picture. Theorem 8.4.4 (i) The only normable semi-Montel spaces are the finite-dimensional ones. (ii) Spaces of finite type are semi-Montel. Proof. (i). Clearly finite-dimensional spaces are (semi-)Montel. Conversely, if E is an infinite-dimensional normed space then the identity E−→E is not compactoid (8.1.3(x)), so E cannot be semi-Montel. The proof of (ii) follows directly from 8.1.3(viii). More examples of spaces that are and that are not (semi-)Montel will be given along the chapter. We prove some characterizations of semi-Montel spaces, where (δ) below is commonly used as definition. Recall that c00 = (c00 , . ∞ ) and that, for a continuous seminorm p on E, Ep is the space E/Ker p, normed by p defined by p(πp (x)) = p(x)

(x ∈ E),

where πp is the canonical map E−→Ep . Theorem 8.4.5 The following are equivalent: (α) (β) (γ ) (δ) (ε) (ζ )

E is semi-Montel. Each T ∈ L(c00 , E) is compactoid. No infinite-dimensional subspace of E is normable. Each bounded subset of E is a compactoid. For each normed space F , each T ∈ L(E, F ) is compactifying. For each continuous seminorm p on E the map πp : E−→Ep is compactifying.

Proof. (α) =⇒ (β), (α) =⇒ (γ ) and (δ) =⇒ (ε) =⇒ (ζ ) are trivial. To prove (β) =⇒ (δ) and (γ ) =⇒ (δ), suppose (δ) does not hold; we show that E

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contains a subspace isomorphic to c00 . By 3.9.16 there is a bounded “orthogonal” sequence e1 , e2 , . . . in E, not tending to 0. Thus, we may assume that there exist a continuous seminorm p on E and an r > 0 such that e1 , e2 , . . . are orthogonal with respect to p and p(en ) ≥ r > 0 for all n. The formula T ((λ1 , λ2 , . . .)) =

∞

(8.2)

λn en

n=1

defines an injective linear map T : c00 −→E. We prove that T is a homeomorphic embedding. To show continuity, let q be a continuous seminorm on E. Then C := supn∈N q(en ) < ∞. From (8.2) we get q(T ((λ1 , λ2 , . . .))) ≤ maxn∈N |λn | q(en ) ≤ C maxn∈N |λn |, and continuity is proved. To prove continuity of T −1 we apply “orthogonality” and get ∞ ∞ −1 λn en ) = max |λn | p(en ) ≥ r max |λn | = r T ( λn en )∞ , p( n=1

n∈N

n∈N

n=1

and continuity of T −1 follows. To conclude we prove (ζ ) =⇒ (α). So, assume (ζ ), let F be a normed space and T ∈ L(F, E). Then T (BF ) is bounded. By assumption, for each continuous seminorm p on E, πp (T (BF )) is a compactoid in Ep . By 3.8.16 T (BF ) is a compactoid, i.e., T is compactoid and we have (α). Remarks 8.4.6 (a) If E is sequentially complete the formula of (8.2) makes sense for (λ1 , λ2 , . . .) ∈ c0 . Having this is mind it is easy to see that, for sequentially complete E, we may replace in (β) the space c00 by c0 . If sequential completeness fails the above replacement cannot be made. In fact, E := c00 satisfies that every T ∈ L(c0 , E) is compactoid (by the Baire Category Theorem, dim c0 /Ker T < ∞, so also dim T (c0 ) < ∞), hence T is compactifying. But E is not semi-Montel (8.4.4(i)). (b) Semi-Montel spaces need not be polar. (In fact, let E be the non-polar space of 4.5.6. Each bounded set B in E is certainly σ (E, E ∗ )-bounded, i.e., weakly bounded with respect to the strongest locally convex topology ν in E, which is polar (4.4.8(b)). So B is ν-bounded (5.4.5), hence finitedimensional (3.6.6(a)). It follows that E is semi-Montel.) However, with an eye on duality theory we will be mainly interested in polar semi-Montel spaces.

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Problem Do there exist “pathological” semi-Montel spaces E, for example E Hausdorff, E = {0}? Theorem 8.4.7 Let E be Hausdorff, polar, and semi-Montel. Then we have the following: (i) Each weakly convergent sequence in E is convergent. (ii) E is weakly quasicomplete if and only if E is quasicomplete. Proof. By 7.4.11(iii) and since weakly bounded sets are bounded (5.4.5) we will be done as soon as we can prove that a bounded net (xi )i∈I converging weakly to x ∈ E, converges to x. But this follows from 5.2.12 and the fact that {x} ∪ {xi : i ∈ I } is bounded, hence a compactoid by assumption and 8.4.5(α)=⇒(δ). For polar spaces we have a criterion for semi-Montelness in terms of the dual. Theorem 8.4.8 Let E be a polar space. Then the following are equivalent: (α) E is semi-Montel. (β) Every equicontinuous set in E is a compactoid in Eb . (γ ) The topology of E is defined by the seminorms x −→ sup{|f (x)| : f ∈ S}

(x ∈ E),

(8.3)

where S ⊂ E is equicontinuous and compactoid in Eb . For the proof we need some preliminary machinery. First, we present a new criterion for compactoidity in polar spaces. Theorem 8.4.9 Let A be a bounded absolutely convex subset of a polar space E. Then the following are equivalent: (α) A is a compactoid. (β) For each zero neighbourhood U in E there exists a closed finitecodimensional subspace H of E such that A ∩ H ⊂ U . Proof. (β) =⇒ (α) (For this implication we do not use polarity of E.) Let U be a zero neighbourhood; we prove that A ⊂ U + aco G for some finite set G ⊂ E. Let λ ∈ K, |λ| > 1, let H be a closed finite-codimensional subspace of E such that A ∩ H ⊂ λ−1 U . Let π : E−→E/H be the canonical map. Then π(A) is a bounded absolutely convex set in the finite-dimensional space E/H and we have that π(A) ⊂ aco G1 for some finite set G1 ⊂ λ π(A) = π (λ A) (3.9.8). Choose a finite set G ⊂ λ A with π(G) = G1 . Then π (A) ⊂ π (aco G). Let x ∈ A. There is a y ∈ aco G such that π (x) = π (y), i.e., x − y ∈ H . But

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also x − y ∈ A − λ A = λ A so that x − y ∈ H ∩ λ A = λ H ∩ λ A = λ (A ∩ H ) ⊂ λ λ−1 U = U , and we see that x ∈ aco G + U . (α) =⇒ (β). Suppose (β) were not true; we derive a contradiction. There is a zero neighbourhood U such that A ∩ H ⊂ U for each closed finitecodimensional subspace H . Thus, for each such H we can find an xH ∈ A ∩ H , xH ∈ U . Now H → xH forms in a natural way a net in the compactoid A, and converges obviously weakly to 0. Then by 5.2.12 also xH → 0 in the initial topology, conflicting xH ∈ U for all H . Secondly, we introduce a new topology on E that is weaker than b(E , E) but stronger than σ (E , E). Definition 8.4.10 The topology c(E , E) of uniform convergence on compactoids is the locally convex topology induced by the seminorms f −→ max{|f (x)| : x ∈ X}

(f ∈ E ),

where X runs through the collection of all compactoid subsets of E. The following lemma is easy to prove. Lemma 8.4.11 The topologies σ (E , E) and c(E , E) coincide on equicontinuous subsets of E . Proof. Let p be a continuous seminorm on E, let S := {f ∈ E : |f | ≤ p}; it suffices to show that, on S, weak star convergence implies c(E , E)convergence. So let (fi )i∈I be a net in S converging pointwise to f ∈ S. Let X ⊂ E be a compactoid, let ε > 0. There is a finite set G ⊂ E such that X ⊂ {x ∈ E : p(x) < ε} + aco G. There is an i0 ∈ I such that, for i ≥ i0 , |f (y) − fi (y)| < ε for all y ∈ G, hence for all y ∈ aco G. Then, for i ≥ i0 , |f (y) − fi (y)| < ε for all y ∈ {x ∈ E : p(x) < ε} + aco G, hence for all y ∈ X. Now we have the material to prove 8.4.8. Proof of 8.4.8. (α) =⇒ (β). Suppose E is semi-Montel. An equicontinuous set in E is σ (E , E)-bounded, hence a σ (E , E)-compactoid, so a c(E , E)compactoid by the previous lemma and 3.8.13. Now use that, by (α) and 8.4.5(α)=⇒(δ), b(E , E) = c(E , E). (For this part we did not need polarity of E). (β) =⇒ (γ ). Since E is polar its topology is generated by the seminorms defined as in (8.3), where S ⊂ E is equicontinuous. Then apply (β). (γ ) =⇒ (α). By 8.4.5(δ)=⇒(α) we have to prove that each absolutely convex bounded set B in E is a compactoid. Let U be a zero neighbourhood in E;

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we find a closed finite-codimensional subspace H ⊂ E such that B ∩ H ⊂ U (and by 8.4.9 we are done). We may assume that U = {x ∈ E : |f (x)| ≤ 1 for all f ∈ S}, where S ⊂ E is a compactoid in Eb . Since B ◦ is a zero neighbourhood in Eb , there is a finite-dimensional space D ⊂ E such that S ⊂ B ◦ + D. Then U ⊃ B ◦◦ ∩ H , with H := {x ∈ E : f (x) = 0 for all f ∈ D}. We see that B ∩ H ⊂ U. Remark 8.4.12 Polar semi-Montel spaces need not be of countable type (see also 8.4.6(b)). In fact, every space with the strongest locally convex topology is polar, and is semi-Montel as bounded sets are finite-dimensional (3.6.6(a)), so it suffices to consider K (I ) where I is uncountable. Observe that this space is polar, Hausdorff, complete and barrelled, see 3.4.22, 4.4.8(b) and 7.1.16(i). It is even Montel, see 8.4.27(a). For spaces of countable type we can add some more characterizations to (α)–(ζ ) of 8.4.5. Theorem 8.4.13 Let E be of countable type. Then the following are equivalent: (α) E is semi-Montel. (β) If {f1 , f2 , . . .} ⊂ E is equicontinuous and fn → 0 in Eσ then fn → 0 in Eb . (γ ) The topology of E is induced by the seminorms x −→ max |fn (x)| n∈N

(x ∈ E),

where {f1 , f2 , . . .} ⊂ E is equicontinuous and fn → 0 in Eb . (δ) Each T ∈ L(E, c0 ) is compactifying. Proof. (β) ⇐⇒ (δ) is a direct consequence of 8.1.9(i) and 8.3.9. (This equivalence is true for any E.) (α) =⇒ (β). Let {f1 , f2 , . . .} ⊂ E be equicontinuous and such that fn → 0 in Eσ . By 8.4.8 {f1 , f2 , . . .} is a compactoid in Eb . Also, σ (E , E) is closely related to b(E , E). So by 3.8.23, b(E , E) = σ (E , E) on {f1 , f2 , . . .}. Therefore, fn → 0 in Eb . (δ) =⇒ (γ ). First observe that by equicontinuity the seminorms given in (γ ) are continuous. Thus, we will be done if we can prove that each continuous seminorm p on E is dominated by one of the form given in (γ ). To this end consider the space Ep , which is of countable type by assumption, so it can linearly and homeomorphically be embedded into c0 (2.3.9). This yields a continuous operator T : E−→Ep −→c0 for which there are C1 , C2 > 0 such that

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C1 p(x) ≤ T (x)∞ ≤ C2 p(x) for all x ∈ E. By (δ) and 8.3.9, T has the form x → (f1 (x), f2 (x), . . .), with {f1 , f2 , . . .} ⊂ E is equicontinuous and fn → 0 in Eb . We see that, for all x ∈ E, p(x) ≤ C1−1 T (x)∞ = C1−1 maxn∈N |fn (x)| and (γ ) is proved. (γ ) =⇒ (α). Clearly, if {f1 , f2 , . . .} ⊂ E is such that fn → 0 in Eb then {f1 , f2 , . . .} is a compactoid in Eb . So by assumption we obtain that the topology of E is defined by the seminorms considered in (8.3), and by 8.4.8 E is semi-Montel. Corollary 8.4.14 Let (E, τ ) be of countable type. Then (E, τ ) is semi-Montel if and only if τ is the weakest topology making all T ∈ CF ((E, τ ), c0 ) continuous. Proof. Apply 8.3.9 and 8.4.13.

Remark 8.4.15 The example given in the second part of 8.4.6(a) shows that in (δ) of 8.4.13 we cannot replace c0 by c00 . On the other hand, in (δ) we may replace c0 by any (fixed) infinite-dimensional Banach space F , because any of such F contains a subspace isomorphic to c0 (2.3.9). Next we give some consequences of 8.4.8. Corollary 8.4.16 Let E be polar. Then we have the following: (i) If Eb is semi-Montel then E is semi-Montel. (ii) If E is metrizable or polarly barrelled and E is semi-Montel then so is Eb . Proof. We apply 8.4.8. (i). If every bounded set in Eb is a compactoid, then certainly equicontinuous sets are. (ii). Let B ⊂ Eb be bounded. Then by 7.6.4 or 7.1.7, B is equicontinuous, hence a compactoid in Eb . Remark 8.4.17 The conclusion of (ii) is not true in general. Indeed, E := (c0 , σ (c0 , ∞ )) is semi-Montel but Eb = ( ∞ , . ∞ ) is not semi-Montel. Corollary 8.4.16 leads to the following result on Montel spaces (i.e., reflexive semi-Montel spaces, see 8.4.2). Corollary 8.4.18 The following are equivalent: (α) E is Montel. (β) E is reflexive and Eb is Montel. (γ ) E is semi-Montel, polar, polarly barrelled and quasicomplete.

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Proof. (α) =⇒ (β). E is reflexive, hence so is Eb (7.4.12) and Eb is semiMontel (8.4.16(ii)). (β) =⇒ (γ ). E is semi-Montel (8.4.16(i)); 7.4.11(iii) and 7.4.13 furnish the other properties of (γ ). (γ ) =⇒ (α). Reflexivity of E follows from 7.4.19. Together with semiMontelness this yields Montelness. Corollary 8.4.19 The strong dual of a Montel space is Montel. Corollary 8.4.20 A polar semi-Montel Fr´echet space is Montel. Remark 8.4.21 We will see in 8.6.1 that metrizable semi-Montel spaces are automatically of countable type, hence polar. Corollary 8.4.22 Let K be spherically complete. Then E is reflexive if and only if E is Montel. Proof. Apply 7.4.19 and 8.4.18(γ ) =⇒ (α).

Remark 8.4.23 If K is not spherically complete, c0 is a reflexive space (7.4.3) that is not semi-Montel (8.4.4(i)). We conclude this section by proving hereditary properties for (semi-) Montelness. They are somewhat less powerful than the ones for nuclearity (8.5.7). First we give a result related to subspaces of countable type. Theorem 8.4.24 The following are equivalent: (α) E is semi-Montel. (β) Each subspace (of countable type) is semi-Montel. (γ ) Each countably generated subspace is semi-Montel. Proof. From 8.4.5(α) ⇐⇒ (γ ) it is clear that (α) =⇒ (β) holds. (β) =⇒ (γ ) is obvious. For (γ ) =⇒ (α) just observe that for a set X in E, if any countable subset of X is bounded (compactoid) then so is X (3.6.3(iv), 3.8.12), and apply 8.4.5(α) ⇐⇒ (δ). Theorem 8.4.25 (i) Subspaces of semi-Montel spaces are semi-Montel. (ii) Products and projective limits of semi-Montel spaces are semi-Montel. (iii) Locally convex direct sums of semi-Montel spaces are semi-Montel.

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319

Proof. (i). Is done in 8.4.24. (ii). Let {Ei : i ∈ I } be a collection of semi-Montel spaces, let E := i∈I Ei , and let F be a normed space, T ∈ L(F, E). To show that T is compactoid, let BF be the closed unit ball of F . Then since πi ◦ T ∈ L(F, Ei ) = C(F, Ei ) for each i ∈ I (where πi : E−→Ei is the canonical projection) we have that πi (T (BF )) is a compactoid in Ei for each i. By 3.8.14, T (BF ) is a compactoid in E. Now also the statement on projective limits follows by using (i). (iii). Here we use (α) ⇐⇒ (δ) of 8.4.5. Let {Ei : i ∈ I } be a collection of semi-Montel spaces, put E := i∈I Ei . Let B be a bounded set in E, let πi : E−→Ei be the canonical projections. Then πi (B) is bounded, hence a compactoid in Ei for each i and πi (B) ⊂ {0} for all but finitely many i (3.6.4(ii)). Applying 3.8.17 we obtain compactoidity of B in E. In the same spirit, for Montelness we obtain the following. Theorem 8.4.26 (i) Closed, polarly barrelled subspaces of Montel spaces are Montel. (ii) Products and locally convex direct sums of Montel spaces are Montel. Proof. (i). Combine 8.4.18(α) ⇐⇒ (γ ) and 8.4.25(i). (ii). Apply 7.4.23 and 8.4.25(ii) and (iii).

Remarks 8.4.27 (a) It follows from 8.4.26(ii) that every vector space equipped with the strongest locally convex topology is Montel. (b) From 8.4.26(ii) we also deduce that every complete locally convex space of finite type is Montel. In fact, by 5.3.11 such a space is isomorphic to some K I . However, there exist quasicomplete spaces that are of finite type (hence semi-Montel, 8.4.4(ii)) but that are not Montel. Indeed, E := (c0 , σ (c0 , c0∗ )) is a quasicomplete space of finite type (5.4.7). Suppose E is polarly barrelled; we derive a contradiction (then E is not Montel by 8.4.18). For every pointwise bounded subset S of c0∗ there exist g1 , . . . , gn ∈ c0∗ such that |f (x)| ≤ max1≤i≤n |gi (x)| for all f ∈ S, x ∈ c0 . So 1≤i≤n Ker gi ⊂ Ker f for all f ∈ S. By elementary algebra we obtain that S ⊂ [g1 , . . . , gn ]. In particular, every bounded subset of ∞ ( (c0 ) , 2.5.11) is finitedimensional, a contradiction. (c) We do not have an example of a semi-Montel space whose completion is not semi-Montel, but it is our conjecture that such examples exist. However, by using 8.3.8 and (α) ⇐⇒ (ε) of 8.4.5 we obtain that if D is a dense semi-Montel subspace of a metrizable E then E itself is semi-Montel.

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In 9.2.10(ii) we will see that a similar result holds for locally convex spaces with an “orthogonal” base. (d) In 9.8.2(v) we will present a Fr´echet Montel space of countable type with a quotient (in fact, c0 ) that is not semi-Montel. At the same time this will lead to a closed subspace of a Montel space that is not Montel. (e) In 11.2.13−11.2.17 we will show that (semi-)Montelness is preserved by taking inductive limits only under additional assumptions. (f) We do not know whether projective limits of projective sequences of Montel spaces are Montel, but obviously the projective limit of a projective sequence of Fr´echet Montel spaces is Montel.

8.5 Nuclear spaces We now treat nuclearity in the same spirit as (semi)-Montelness in the previous section. Recall (8.4.1) that a locally convex space E is called nuclear if, for each normed space F , each T ∈ L(E, F ) is compactoid. First, we formulate a few characterizations, where 8.5.1(β) below is commonly used as definition of nuclearity. Notice that, contrary to the semi-Montel case, nuclear spaces are of countable type, hence polar. (See 8.4.12 and 8.5.1(δ) below.) Theorem 8.5.1 The following are equivalent: (α) E is nuclear. (β) For each continuous seminorm p there is a continuous seminorm q ≥ p such that the canonical map Eq −→Ep is compactoid. (γ ) For each continuous seminorm p the canonical map πp : E−→Ep is compactoid. (δ) E is of countable type and each T ∈ L(E, c0 ) is compactoid. (ε) E is of countable type, and if {f1 , f2 , . . .} ⊂ E is equicontinuous and fn → 0 in Eσ then fn → 0 locally. (ζ ) The topology of E is induced by the seminorms x −→ max |fn (x)| n∈N

(x ∈ E),

where {f1 , f2 , . . .} ⊂ E and fn → 0 locally. Proof. The equivalence (β) ⇐⇒ (γ ) is straightforward, (α) =⇒ (γ ) follows directly from the definition of nuclear space, and (δ) ⇐⇒ (ε) from 8.1.9.

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(γ ) =⇒ (δ). First we see that E is of countable type. Let p be continuous seminorm on E. By assumption πp : E−→Ep is compactoid, so by 8.1.3(iv) πp (E) = Ep is of countable type. Now let us prove that each T ∈ L(E, c0 ) is compactoid. For that, note that p(x) := T (x) (x ∈ E) defines a continuous seminorm on E, so by assumption πp : E−→Ep is compactoid. Also, Tp : Ep −→c0 , πp (x) → T (x) is a well-defined isometry of Ep into c0 . Hence T = Tp ◦ πp is compactoid (8.1.3(v)). (δ) =⇒ (ζ ). Clearly the seminorms described in (ζ ) are continuous by equicontinuity of {f1 , f2 , . . .} (8.1.7(ii)). Now let p be a continuous seminorm on E. By (δ) the space Ep is of countable type, so there is a linear homeomorphic embedding S : Ep −→c0 (2.3.9). The composition S ◦ πp is in L(E, c0 ) hence in C(E, c0 ) by (δ). So, by 8.1.9(ii) it has the form x → (f1 (x), f2 (x), . . .) where fn ∈ E , fn → 0 locally. Then S(πp (x))∞ = max |fn (x)|

(x ∈ E).

n∈N

As S is a homeomorphism there exists a C > 0 such that S(πp (x))∞ ≥ C p(x) (x ∈ E) and we see that p(x) ≤ C −1 max |fn (x)| n∈N

(x ∈ E),

and (ζ ) follows. (ζ ) =⇒ (α). Let F be a normed space, let T ∈ L(E, F ), let us prove that T is compactoid. By (ζ ) there exist f1 , f2 , . . . ∈ E , fn → 0 locally such that T (x) ≤ p(x) := maxn∈N |fn (x)| for all x ∈ E. Clearly Ep is isoh

metrically embedded into c0 through the map πp (x) −→ (f1 (x), f2 (x), . . .). By 8.1.7(iii) we can write fn = λn gn where λn ∈ K, λn → 0 and gn → 0 locally. Let q(x) := maxn∈N |gn (x)| (x ∈ E). Then q is a continuous seminorm on E, so U := {x ∈ E : q(x) ≤ 1} is a zero neighbourhood in E. Since h(πp (U )) ⊂ {(α1 , α2 , . . .) ∈ c0 : |αn | ≤ |λn | for all n} we have that h(πp (U )) is a compactoid in c0 (8.1.4), hence πp (U ) is a compactoid in Ep . Further, Tp : Ep −→F , πp (x) → T (x) is a well-defined continuous operator. So finally we obtain that T (U ) = Tp (πp (U )) is a compactoid and we have compactoidity of T . We have the following important conclusion (from 8.4.5(ζ ) and 8.5.1(γ )). Corollary 8.5.2 Nuclear spaces are semi-Montel. Specializing, we arrive at the next result (from 8.4.20 and 8.5.2). Corollary 8.5.3 Nuclear Fr´echet spaces are Montel.

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Remarks 8.5.4 (a) In 9.8.2(vi) we will give an example of a Fr´echet Montel space of countable type that is not nuclear. (b) The fact that nuclear spaces are automatically of countable type (see (δ) and (ε)) makes the discussion of nuclearity less complicated than the one presented in Section 8.4 for semi-Montel spaces. (c) Properties (δ)–(ζ ) can be viewed as the counterpart of 8.4.13. (d) We have seen in 8.4.5(γ ) that E is semi-Montel if and only if no infinitedimensional subspace of E is normable. With the duality between semiMontelness and nuclearity (8.4.1) in mind, one could think that perhaps E is nuclear if and only if no infinite-dimensional quotient of E is normable, see the chapter notes for some discussion. The following results are the nuclear versions of 8.4.14 and 8.4.4 respectively. Corollary 8.5.5 (E, τ ) is nuclear if and only if τ is the weakest topology on E making all T ∈ C((E, τ ), c0 ) continuous. Proof. Apply 8.1.9(ii) and 8.5.1.

Theorem 8.5.6 (i) The only normed nuclear spaces are the finite-dimensional ones. (ii) Spaces of finite type are nuclear. Proof. The proof of (i) follows from 8.4.4(i) and 8.5.2, and (ii) follows from 8.1.3(viii) and the definition of nuclear space. Next we prove hereditary properties. Notice that they are far-reaching (compare with the ones for semi-Montel spaces, see 8.4.25 and 8.4.27). Theorem 8.5.7 (i) Subspaces of nuclear spaces are nuclear. (ii) Let D be a dense subspace of E. Then E is nuclear if and only if D is nuclear. In particular, the completion of a Hausdorff nuclear space is nuclear. (iii) Products and projective limits of nuclear spaces are nuclear. (iv) Countable locally convex direct sums of nuclear spaces are nuclear. (v) Quotients of nuclear spaces are nuclear. (vi) The inductive limit of an inductive system of nuclear spaces is nuclear. Proof. (i). Let D be a subspace of a nuclear space E, let F be a normed space and let T ∈ L(D, F ); we prove that T is compactoid. The formula q(x) := T (x) defines a continuous seminorm on D that by 3.4.3 extends

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to a continuous seminorm p on E. By assumption πp : E−→Ep is compactoid, hence so is its restriction πp |D : D−→πp (D). Also, Tp : πp (D)−→F , πp (y) → T (y) (y ∈ D) is a well-defined continuous operator. Therefore, T = Tp ◦ πp |D is compactoid (8.1.3(v)). (ii). One half follows from (i), so let D be nuclear, let F be a normed space and let T ∈ L(E, F ). Then T |D is compactoid, hence so is T by 8.1.3(ix), and the nuclearity of E follows. (iv). Let E1 , E2 , . . . be nuclear, set E = i∈N Ei , let F be a normed space and let T ∈ L(E, F ); we shall prove that T is compactoid. Now T ◦ φi ∈ L(Ei , F ) = C(Ei , F ) for each i ∈ N (where φi : Ei −→E is the canonical injection), so there is an absolutely convex zero neighbourhood Ui in Ei such that (T ◦ φi )(Ui ) is a compactoid in F . By using scalar multiplication we may assume that diam (T ◦ φi )(Ui ) → 0, so that C := ∞ i=1 (T ◦ φi )(Ui ) is a ∞ compactoid in F . Now U := i=1 φi (Ui ) is a zero neighbourhood in E with T (U ) ⊂ C, and compactoidity of T follows. (v). Let D be a closed subspace of a nuclear space E, let π : E−→E/D be the canonical map. Let F be a normed space, let T ∈ L(E/D, F ). Then T ◦ π ∈ L(E, F ) = C(E, F ). Now apply 8.1.3(xi) to conclude that T is compactoid. (vi). Follows from (iv) and (v). (iii). Let {Ei : i ∈ I } be a family of nuclear spaces, let E := i∈I Ei , let T ∈ L(E, F ) where F is a mormed space. T −1 (BF ) is a zero neighbourhood in E, hence it contains i∈I \G Ei for some finite set G ⊂ I . The image of i∈I \G Ei under T is a subspace, in BF , so it must be {0}, and we have a factorization E GG GG GG G π GG #

/F ; ww w w w ww ww i∈G Ei T

T1

where π is the obvious projection and T1 ∈ L( i∈G Ei , F ). Since i∈G Ei = i∈G Ei we have that T1 is compactoid by (iv). Then so is T = T1 ◦ π (8.1.3(v)). Now the statement on projective limits follows by using (i). We have the following Pull Back Principle for bounded sets (compare with 3.8.33). Theorem 8.5.8 If E is a Fr´echet nuclear space and D is a closed subspace of E then every bounded subset of E/D is the image under the quotient map E−→E/D of a bounded subset of E.

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Proof. E/D is Fr´echet (3.5.9) and nuclear (8.5.7(v)). Hence by 8.4.5 and 8.5.2 every bounded subset of E/D is a compactoid. Now the conclusion follows from 3.8.33. Remarks 8.5.9 (a) If we drop nuclearity the above conclusion is not longer true, as we will show in 9.8.2(i). (b) One may wonder whether a locally convex space is nuclear if each countably generated subspace is nuclear (compare with 8.4.24). The answer is no. In fact, take K (I ) where I is uncountable. Each countably generated subspace is either finite-dimensional or isomorphic to K (N) , so is nuclear by 8.5.7(iv). But K (I ) itself is not of countable type, hence not nuclear by 8.5.1.

8.6 Semi-Montelness, nuclearity and metrizability In this section we study mainly metrizable semi-Montel spaces and its relation with nuclearity of the dual. First we shall prove the following non-Archimedean counterpart of [103], 11.6.2. Theorem 8.6.1 Every metrizable semi-Montel space is of countable type. The classical proof (see e.g. [103]) carries over to the non-Archimedean setting if K is locally compact but fails for general K. Therefore, we shall use quite different techniques to arrive at the proof; in fact, we shall apply the theory of t-frames developed in Section 3.10. We need the following lemma. Lemma 8.6.2 Each uncountable subset of a metrizable locally convex space contains a bounded infinite subset. Proof. Let p1 ≤ p2 ≤ · · · be seminorms generating the topology (3.5.2). Let X be an uncountable subset of the space. From {x ∈ X : p1 (x) ≤ n} X= n∈N

we infer that X has an uncountable subset X1 on which p1 is bounded. By repeating this argument, replacing X by X1 and p1 by p2 we arrive at an uncountable X2 ⊂ X1 on which p2 is bounded. Inductively we obtain in this way uncountable sets X ⊃ X1 ⊃ X2 ⊃ · · · such that pn is bounded on Xn for each n. Now since all Xn are infinite we can select mutually distinct x1 , x2 , . . . such that xn ∈ Xn for all n. Then Y := {x1 , x2 , . . .} is infinite and clearly pn is bounded in Y for each n, i.e., Y is bounded.

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325

Proof of 8.6.1. Let (E, τ ) be a metrizable semi-Montel space. Let p1 ≤ p2 ≤ · · · be seminorms generating τ . It suffices to prove that the normed space E1 := (Ep1 , p 1 ) is of countable type (recall that p1 (πp1 (x)) = p1 (x), x ∈ E, where πp1 : E−→E1 is the canonical surjection). To this end, let t ∈ (0, 1), let X be a maximal t-frame in E1 and suppose X is uncountable; we derive a contradiction. (Then we are done by 3.10.5.) Since 1 X= {x ∈ X : p1 (x) ≥ }, n n∈N we see that X contains an uncountable Y such that p1 (x) ≥ r > 0 for all x ∈ Y and some r > 0. Let Z ⊂ E be a set of representatives of Y modulo πp1 . Then by 8.6.2 Z contains a bounded infinite set Z1 . As E is semi-Montel, Z1 is a compactoid in E (8.4.5(α)=⇒(δ)), so Y1 := πp1 (Z1 ) is an infinite compactoid in E1 . Let x1 , x2 , . . . be a (p 1 -bounded) sequence in Y1 with xn = xm whenever n = m. Then x1 , x2 , . . . is contained in the t-frame X so that it is a t-frame sequence. Hence p1 (xn ) → 0 by 3.10.7, but on the other hand, since Y1 ⊂ Y ⊂ X, we have p1 (xn ) ≥ r for all n, a contradiction. Next we prove: Theorem 8.6.3 The strong dual of a metrizable semi-Montel space is nuclear. Proof. Let E be a metrizable semi-Montel space, let p be a continuous seminorm on Eb . By 8.5.1(ζ ) =⇒ (α) it suffices to show that p ≤ q, where q is a continuous seminorm on Eb of the form f −→ max |n (f )| , n∈N

with n ∈ E , n → 0 locally. To this end, first observe that p is dominated by a seminorm p1 of the form f → sup{|f (x)| : x ∈ B} for some bounded set B ⊂ E. By assumption and 8.4.5(α)=⇒(δ) B is a compactoid, so by 3.8.25 there exists a sequence x1 , x2 , . . . in E tending to 0 such that B ⊂ aco {x1 , x2 , . . .}, so that q(f ) : = sup{|f (x)| : x ∈ aco {x1 , x2 , . . .}} = max |f (xn )| = max |jE (xn )(f )| n∈N

n∈N

defines a continuous seminorm q ≥ p1 ≥ p on Eb . It remains to be shown that jE (xn ) → 0 locally. There exist λ1 , λ2 , . . . ∈ K × such that |λn | → ∞ and λn xn → 0 (7.6.3). Then the formula r(f ) = maxn∈N |f (λn xn )| defines a −1 continuous seminorm on Eb and since (λ−1 1 , λ2 , . . .) ∈ c0 and |jE (xn )(f )| =

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r(f ) for all n and all f ∈ E , we have jE (xn ) → 0 |f (xn )| ≤ λ−1 n locally. Our next purpose is to prove the following. Theorem 8.6.4 Let E be a metrizable semi-Montel space. Then Eb is Montel, and E is reflexive (Montel) if and only if E is Fr´echet. A key point for the proof is 8.6.7, for which we also need two preliminary results. First, we extend 3.2.4 to metrizable spaces of countable type. Theorem 8.6.5 Let E be a metrizable space of countable type. Then, for each absolutely convex set A ⊂ E there is a countable X ⊂ A such that A ⊂ aco X. Proof. By 3.5.2 there are seminorms p1 ≤ p2 ≤ · · · all of countable type, that generate the topology of E. Let n ∈ N, let πn : E−→Epn be the canonical map. Since Epn is a normed space of countable type and πn (A) is absolutely convex, by 3.2.4 there is a countable Xn ⊂ A such that aco πn (Xn ) is dense in πn (A), in other words, aco Xn is pn -dense in A. Then X := n∈N Xn ⊂ A is countable and aco X is pn -dense in A for every n. Hence A ⊂ aco X. Second, we prove a theorem on bounded subsets of metrizable spaces of countable type, extending 3.8.28. Theorem 8.6.6 Let D be a dense subspace of a metrizable space E of countable type. Then for each bounded set A ⊂ E there is a bounded set B ⊂ D such that A ⊂ B. Proof. We may assume that A is absolutely convex. By 8.6.5 there is a countable subset X ⊂ A such that A ⊂ aco X. It suffices to prove the existence of a bounded Y ⊂ D such that X ⊂ Y . (Then B := aco Y meets the requirements). Let p1 ≤ p2 ≤ · · · be seminorms generating the topology (3.5.2). Let X := {x1 , x2 , . . .}. There exists a sequence x11 , x12 , . . . in D converging to x1 and such that p1 (x1n ) ≤ p1 (x1 ) + 1 for all n. There exists a sequence x21 , x22 , . . . in D converging to x2 and such that p1 (x2n ) ≤ p1 (x2 ) + 1, p2 (x2n ) ≤ p2 (x2 ) + 1 for all n. Continuing inductively we arrive at a countable subset Y := {xmn : m, n ∈ N} of D for which clearly X ⊂ Y . A straightforward device shows that Y is bounded. Now, we have the material for the promised Theorem 8.6.7. Theorem 8.6.7 Let E be a metrizable semi-Montel space. Then its completion is also semi-Montel, and the adjoint (E ∧ )b −→Eb of the inclusion E−→E ∧ is a bijective linear homeomorphism.

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327

Proof. The first statement has been proved in 8.4.27(c). The adjoint map f → f |E is clearly a bijection and continuous. Openness follows from 8.6.6 after noting that, by 8.6.1, E is of countable type. With 8.6.7 in hand, we can prove 8.6.4. Proof of 8.6.4. Recall that every Montel space is reflexive. We first prove the second statement. If E is Fr´echet then E is Hausdorff, polar (from 8.6.1), barrelled and (quasi)complete, and Montelness follows from 8.4.18. Conversely, if E is reflexive it is weakly quasicomplete (7.4.13), hence quasicomplete by 7.4.11(iii) and therefore Fr´echet. To prove the first statement observe that by 8.6.7 E ∧ is semi-Montel. Then note that by 8.4.19 and what we just proved, (E ∧ )b is Montel and that Eb and (E ∧ )b are isomorphic by 8.6.7. The previous theory of this section now leads to the following characterization of metrizable semi-Montel spaces. Theorem 8.6.8 For a polar metrizable space E the following are equivalent: (α) (β) (γ ) (δ) (ε) (ζ ) (η)

E is semi-Montel. Eb is semi-Montel. Eb is Montel. Each bounded subset of Eb is a metrizable compactoid. Eb is nuclear. E and Eb are of countable type. Eb is of countable type and every weakly convergent sequence in E is convergent. (θ ) E is of countable type and each bounded sequence in Eb converging in σ (E , E) converges in Eb . (ι) E ∧ is Montel.

If, in addition, E is Fr´echet the above statements are equivalent to each one of (α) and (ζ ) below. (α) E is Montel. (ζ ) E is of countable type and each σ (E , E)-convergent sequence in E is convergent in Eb . Proof. First recall that by metrizability of E, for subsets of E , “bounded in Eb ” is the same as “equicontinuous” (7.3.4 and 7.6.4). (α) =⇒ (γ ). This is 8.6.4 (α) =⇒ (ε). This is 8.6.3.

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(α) =⇒ (δ). By 8.6.1, E is of countable type and by 8.4.11, σ (E , E) = b(E , E) on equicontinuous subsets of E , so applying 7.6.11(α) =⇒ (β) we obtain that bounded sets in Eb are metrizable compactoids. Since the implications (γ ) =⇒ (β), (ε) =⇒ (β), (δ) =⇒ (β) are trivial and since (β) =⇒ (α) follows from 8.4.16(i), we conclude that (α) − (ε) are equivalent. The equivalence (δ) ⇐⇒ (θ ) is provided by 7.6.11 whereas (α) ⇐⇒ (η) follows directly from 7.6.13. Next we prove (α) ⇐⇒ (ζ ). By 5.5.2, in a space of countable type the weakly convergent sequences coincide with the convergent ones. Also, metrizable semiMontel spaces are of countable type (8.6.1). Then apply 7.6.13. To prove (α) =⇒ (ι) apply 8.4.27(c) and 8.6.4. The converse follows from 8.4.25(i). Now assume that E is Fr´echet. Then (α) =⇒ (α) follows from 8.6.4 and (α) =⇒ (α) is trivial. Now assume (ζ ). Since E is (polarly) barrelled we have by 7.1.7 that a σ (E , E)-convergent sequence is automatically bounded in Eb so (ζ ) follows. Again (ζ ) =⇒ (ζ ) is trivial.

8.7 Examples of (semi-)Montel and nuclear spaces In this section we will test the examples we have met in previous chapters on (semi-)Montelness and nuclearity. In the next chapters we will encounter more examples. Since normable (semi-)Montel or nuclear spaces are finite-dimensional (8.4.4(i) or 8.5.6(i)) we focus on the true locally convex case. Recall (8.4.2) that a Montel space is a reflexive semi-Montel space. As usual, Throughout Section 8.7 X is a zero-dimensional Hausdorff topological space.

8.7.1 Spaces of continuous functions We start with the simple case (C(X), τs ) (see 3.7.1). Theorem 8.7.1 (C(X), τs ) is nuclear. It is Montel if and only if X is discrete. Proof. Nuclearity (hence semi-Montelness, 8.5.2) follows from the fact that (C(X), τs ) is of finite type and 8.5.6(ii). The rest is furnished by 7.5.6.

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329

Next, we consider the topologies of compact and strict convergence (see 3.7.3, 3.7.14). Theorem 8.7.2 The following are equivalent: (α) (β) (γ ) (δ) (ε)

(C(X), τc ) is nuclear. (C(X), τc ) is semi-Montel. (BC(X), β0 ) is nuclear. (BC(X), β0 ) is semi-Montel. Each compact subset of X is finite.

Proof. 8.5.2 takes care of (α) =⇒ (β) and (γ ) =⇒ (δ). If (ε) holds then τc = τs , so (α) follows from 8.7.1. To prove (β) =⇒ (ε) and (δ) =⇒ (ε), let Y ⊂ X be compact. By the Tietze–Urysohn extension Lemma (2.5.23) there is a linear T : C(Y )−→C(X) such that T (f )|Y = f and T (f )∞ = f ∞ for all f ∈ C(Y ). It is easily seen that T : C(Y )−→(BC(X), β0 ) and T : C(Y )−→(C(X), τc ) are homeomorphic embeddings. By semi-Montelness and 8.4.5(α) =⇒ (γ ) it follows for both cases (β) and (δ) that C(Y ) must be finite-dimensional, i.e., Y is finite. Finally we prove (ε) =⇒ (γ ). Let p be a continuous seminorm on (BC(X), β0 ). By 8.5.1(ζ ) =⇒ (α) it suffices to prove that p(f ) ≤ maxn∈N |gn (f )| (f ∈ BC(X)), where gn ∈ (BC(X), β0 ) , gn → 0 locally. We may assume that p has the form x → supx∈X |f (x)| |φ(x)| where φ : X−→K vanishes at infinity. By (ε), for each ε > 0 the set {x ∈ X : |φ(x)| ≥ ε} is finite, so that there are mutually distinct x1 , x2 , . . . ∈ X such that φ(x) = 0 if x ∈ {x1 , x2 , . . .} and (φ(x1 ), φ(x2 ), . . .) ∈ c0 . Put gn (f ) := f (xn ) φ(xn )

(n ∈ N, f ∈ BC(X)).

Then gn ∈ (BC(X), β0 ) and p(f ) = maxn∈N |gn (f )| (f ∈ BC(X)). To show that gn → 0 locally we use the Product Lemma 8.1.5 obtaining (λ1 , λ2 , . . .) ∈ c0 , (µ1 , µ2 , . . .) ∈ c0 such that λn µn = φ(xn ) for all n. Then := ∞ n=1 µn ξ{xn } vanishes at infinity, so the formula q(f ) = |f |µ maxn∈N (xn )| n | = maxn∈N |f (xn )| |(xn )| (f ∈ BC(X)) defines a continuous seminorm on (BC(X), β0 ) and |gn (f )| ≤ |λn | q(f ) for all f ∈ BC(X).

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Theorem 8.7.3 (i) (C(X), τc ) is Montel if and only if X is discrete. (ii) (BC(X), β0 ) is Montel if and only if X is finite. Proof. The proof of (ii) follows directly from 7.5.10 and (δ) ⇐⇒ (ε) of the previous theorem. Now, let us prove (i). If X is discrete then τc = τs and Montelness follows from 8.7.1. Conversely, suppose (C(X), τc ) is Montel, Then each compact subset of X is finite (8.7.2), so τc = τs and discreteness of X follows again from 8.7.1. Next in line are the spaces (Cc (X), τi ) where X is locally compact. Recall that τi = τWc , where Wc is the set of all functions X−→K that are bounded on compact sets (3.7.22). Theorem 8.7.4 Let X be locally compact. Then the following are equivalent. (α) (Cc (X), τi ) is semi-Montel. (β) (Cc (X), τi ) is Montel. (γ ) X is discrete. Proof. (α) =⇒ (γ ). Let Y ⊂ X be compact. Then the natural embedding (C(Y ), . ∞ )−→ (Cc (X), τWc ) is linear and homeomorphic. Thus, by 8.4.5(α)=⇒(γ ), C(Y ) must be finite-dimensional, implying finiteness of Y , and (γ ) follows. If (γ ) holds then (Cc (X), τi ) = K (X) is Montel by 8.4.26(ii) so we have (β). Theorem 8.7.5 Let X be locally compact. Then (Cc (X), τi ) is nuclear if and only if X is discrete and countable. Proof. Let (Cc (X), τi ) be nuclear. Then by semi-Montelness (8.5.2) and by 8.7.4, X is discrete, so (Cc (X), τi ) = K (X) . Also, (Cc (X), τi ) is of countable type by 8.5.1, hence X is countable (4.2.17(a)). Conversely, if X is discrete and countable then (Cc (X), τi ) = K (X) is nuclear by 8.5.7(iv). Remark 8.7.6 In 9.7.5 we will prove that the spaces of analytic functions A(r) and A† (r) are nuclear and Montel. We do not know of interesting conditions in order that O(A) is semi-Montel or nuclear.

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331

8.7.2 Spaces of differentiable functions From now on in this section X is a non-empty subset of K without isolated points. Theorem 8.7.7 The spaces (C n (X), τsn ) (n ∈ {0, 1, . . .}) and (C ∞ (X), τs∞ ) (see 3.7.42) are nuclear, but not Montel. Proof. The spaces are of finite type, hence nuclear (8.5.6(ii)). On the other hand, by 7.5.13 they are not reflexive. Theorem 8.7.8 (C n (X), τcn ) (n ∈ {0, 1, . . .}) (see 3.7.36) is not semi-Montel. Proof. Apply 7.5.15 and 8.4.5(α) =⇒ (δ).

Theorem 8.7.9 The space (C ∞ (X), τc∞ ) (see 3.7.36) is nuclear. Proof. According to 7.5.18, for each n ∈ {0, 1, . . .} and compact Y ⊂ X, the set {f ∈ C ∞ (X) : pYn+1 (f ) ≤ 1} is a compactoid in (C ∞ (X), pYn ). This is the same as saying that the canonical map C ∞ (X)pYn+1 −→C ∞ (X)pYn is compactoid. The statement now follows from 8.5.1(β) =⇒ (α). As for the question whether (C ∞ (X), τc∞ ) is Montel, we only can say that it is equivalent to the open problems stated before (text following 7.2.18 and 7.5.21). Theorem 8.7.10 For the space (C ∞ (X), τc∞ ), Montelness, reflexivity and polar barrrelledness are equivalent. Proof. From the previous theorem and 8.5.2 this space is semi-Montel. Then apply 7.5.20. The discussion for the space BC ∞ (X) (see 3.7.46) is simple. Theorem 8.7.11 The following are equivalent: (α) (β) (γ ) (δ)

BC ∞ (X) is semi-Montel. BC ∞ (X) is Montel. BC ∞ (X) is nuclear. X is precompact.

Proof. BC ∞ (X) is Fr´echet (3.7.47), hence complete and barrelled, and it is polar (4.4.8(a)). It follows from 8.4.18 that (α) and (β) are equivalent. Also, (γ ) =⇒ (α) follows from 8.5.2.

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(α) =⇒ (δ). By reading the proof of 7.5.23, and applying 8.4.5(α)=⇒(δ), we see that for any K (not necessarily spherically complete) semi-Montelness implies precompactness of X. (δ) =⇒ (γ ). By 4.3.18 we may assume that X is compact. Then BC ∞ (X) = ∞ (C (X), τc∞ ). Now apply 8.7.9. Also, the characterizations for C0∞ (X) (see 3.7.57) are easy. Theorem 8.7.12 Suppose balls in X are compact. Then the following are equivalent: (α) (β) (γ ) (δ)

C0∞ (X) is semi-Montel. C0∞ (X) is Montel. C0∞ (X) is nuclear. X is compact.

Proof. (α) ⇐⇒ (β) and (γ ) =⇒ (α) follow as in the proof of 8.7.11. (α) =⇒ (δ). By reading the proof of 7.5.23 and observing that V ⊂ C0∞ (X) we see that, for any K, semi-Montelness implies precompactness of X. Thus X, being a ball in X, must be compact. (δ) =⇒ (γ ). From (δ) we get C0∞ (X) = BC ∞ (X), so we can apply (δ) =⇒ (γ ) of 8.7.11. Now, for locally compact X ⊂ K, we look into the case of the spaces (Ccn (X), τWn ) and (Cc∞ (X), τW∞ ), where W := C ∞ (X) (see 3.7.53). Lemma 8.7.13 Let B be a bounded subset of (Cc∞ (X), τW∞ ). Then is precompact.

f ∈B

supp f

Proof. Suppose not. Then there are mutually disjoint balls B1 , B2 , . . . in X, all with the same radius r > 0, each of them having a non-empty intersection with f ∈B supp f . Thus, for each m ∈ N there is an fm ∈ B not vanishing on Bm ; choose xm ∈ Bm such that fm (xm ) = 0. Put λm := fm (xm )−1 ρ −m with ρ ∈ K, 0 < |ρ| < 1, and define φ : X−→K by λm if m ∈ N, x ∈ Bm φ(x) := 0 if x ∈ X \ m∈N Bm . Since all Bm have the same radius, m∈N Bm is clopen in X, so that ∞ φ is locally constant, hence in C (X). Thus, m → fm φ∞ must be bounded. In particular, m → |fm (xm ) φ(xm )| = ρ −m must be bounded, a contradiction.

8.7 Examples of (semi-)Montel and nuclear spaces

333

Theorem 8.7.14 Let X be locally compact, W := C ∞ (X). Then we have the following: (i) For n ∈ {0, 1, . . .}, (Ccn (X), τWn ) is not semi-Montel. (ii) If X is closed in K (e.g. when balls in X are compact) the space (Cc∞ (X), τW∞ ) is semi-Montel. Proof. From 3.7.53(iv) it follows that (Ccn (X), τWn ) contains an infinitedimensional normable subspace and (i) follows by 8.4.5(α)=⇒(γ ). As for (ii) we will use 8.4.5(α) ⇐⇒ (δ). Let B be a bounded set in ∞ (Cc (X), τW∞ ). By 8.7.13 and our assumption on X there is an open compact U ⊂ X such that supp f ⊂ U for all f ∈ B. It follows by 3.7.53(iv), that B is the image of a bounded set B ⊂ (C ∞ (U ), τc∞ ) under the natural embedding (C ∞ (U ), τc∞ )−→(Cc∞ (X), τW∞ ). Now (C ∞ (U ), τc∞ ) is nuclear (8.7.9), hence semi-Montel (8.5.2) so that B , hence B, is a compactoid. Remarks 8.7.15 (a) We do not know whether (Cc∞ (X), τW∞ ) is semi-Montel for arbitrary locally compact X ⊂ K. (b) We do not know reasonable conditions on locally compact X in order that (Cc∞ (X), τW∞ ) is nuclear. Of course, X should be separable (4.3.20). Theorem 8.7.16 Let X be locally compact, W := C ∞ (X). Then (Cc∞ (X), τW∞ ) is Montel if and only if X is compact. Proof. Apply 7.5.27 and 8.7.14(ii).

Finally we consider the case where W is the collection of all polynomials functions X−→K, see 3.7.55. Theorem 8.7.17 Let balls in X be compact, let W be as above. Then for n (X), τWn ) are not semi-Montel whereas S(X) = n ∈ {0, 1, . . .} the spaces (BCW ∞ ∞ (BCW (X), τW ) is nuclear and Montel. n (X), τWn ) contains Proof. In the proof of 7.5.29 we have seen that (BCW an infinite-dimensional Banach space, so it cannot be semi-Montel by 8.4.5(α)=⇒(γ ). Theorem 7.5.31, together with 8.5.1(α) ⇐⇒ (β), tells us that S(X) is nuclear, hence semi-Montel by 8.5.2. Reflexivity follows from 7.5.32. We conclude that S(X) is Montel.

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8.8 Notes In 8.1.1 we introduced the notion of compactoid operator, which can be put as follows. Let E, F be Hausdorff locally convex spaces. An operator T : E−→F is compactoid if there exists a compactoid X ⊂ F such that T −1 (X) is a zero neighbourhood. In the literature also two more restrictive concepts are being used. T is called compact if there exists a complete compactoid X ⊂ F such that T −1 (X) is a zero neighbourhood; semicompact if there exists a compactoid Banach disk X ⊂ F such that T −1 (X) is a zero neighbourhood (see the comments before 11.1.11 for the definition of a Banach disk). It is not hard to see that T compact =⇒ T semicompact =⇒ T compactoid. The three notions coincide if F is quasicomplete. Those concepts were studied in [84], [85], [99] and [131]. Lemmas 8.1.4 and 8.1.5 were proved in [73]. The proof of 8.2.1 is taken from [193], 5.19. The compactifying operators (Section 8.3) were studied in [64], [111] for spherically complete K and in [6], [121], [126] for general K. In [62] it was proved that the conclusion of 8.4.22 holds when K is locally compact and in [196] it was extended to arbitrary spherically complete fields. One may wonder whether there is a connection between non-Archimedean and classical semi-Montelness and nuclearity (see 8.4.3). As for semi-Montel spaces, in [103], 11.5, they are defined as those for which every bounded set is relatively compact. The natural non-Archimedean translation, “the closure of each bounded set is a complete compactoid”, is somewhat stronger than “each bounded set is a compactoid”; the last one is equivalent to our concept of semi-Montelness (8.4.5(δ)). For quasicomplete spaces these two notions coincide. To see that they are different in general it suffices to take a semi-Montel space a la 8.4.5(δ), which is not quasicomplete (e.g. (C(X), τs ) for non-discrete X, see 3.7.2(iii) and 8.7.1). As for nuclear spaces, let us consider (β) of 8.5.1, which is easily seen to be equivalent to (β) For each continuous seminorm p there is a continuous seminorm q ≥ p such that the canonical map Eq∧ −→Ep∧ is compactoid. This statement, interpreted classically, yields exactly the definition of a Schwartz space ([103], 21.1.1). So, in this sense our nuclear spaces form the counterparts of the classical Schwartz spaces. However, it is well-known ([193], 4.40(α) ⇐⇒ (ε)) that compactoid operators between Banach spaces are the natural substitute of the classical nuclear ones ([103], formula (*) of p. 376). Hence, by [103], 21.2.1(1) ⇐⇒ (8), our nuclear spaces are also the counterparts of the classical nuclear spaces!

8.8 Notes

335

We point out a curious difference between non-Archimedean and classical nuclearity. Recall that for a real or complex Hausdorff locally convex E, nuclearity is equivalent to the following definition given by Grothendieck in [98]: E is nuclear if and only if for every real or complex Hausdorff locally convex space F the π -topology and the ε-topology on E ⊗ F coincide. For instance, this is the definition of nuclear space in [103], 16.1.4, in which book also the definitions of the π - and the ε-topologies can be found (Sections 15.1 and 16.1). However, in 10.3.15 we will prove that for all polar spaces E, F the non-Archimedean analogues of these two topologies on E ⊗ F coincide. This shows that a definition of nuclearity in terms of tensor products is not useful in p-adic Functional Analysis. In [69] De Grande-De Kimpe collected the results on non-Archimedean nuclear spaces which were up to that moment scattered over different papers (e.g. [64], [67], [68], [187]) and under different names (for instance, they were called Schwartz spaces in [64] and [67]). Locally convex spaces E for which every continuous operator E−→c0 is compactoid (see (δ) of 8.5.1) were considered in [66] and [67], for spherically complete K, and in [141] for general K. In [71] De Grande-De Kimpe studies examples of locally convex spaces F that are projective with respect to the class of nuclear Fr´echet spaces, i.e., those F that satisfy the following lifting property: For every nuclear Fr´echet space E, for every closed subspace D of E and for every T ∈ L(F, E/D) there exists an S ∈ L(F, E) such that π ◦ S = T , where π : E−→E/D is the canonical surjection. For a nuclear space clearly no infinite-dimensional quotient can be normable (see 8.5.6(i), 8.5.7(v)). As for the converse (see also 8.5.4(d)) one has the ´ following partial positive answer, proved by Sliwa in [220]. Theorem 8.8.1 ([220], Corollary 4) Let E be a Fr´echet space of countable type. Then E is nuclear if and only if no infinite-dimensional quotient of E is normable. The general case still remains unsolved, as far as we know. The proof of 8.6.1 by using the t-frames of Section 3.10 is taken from [96]. It is known that a Fr´echet space over R or C is nuclear if and only if its strong dual is nuclear ([103], 21.5.3). In the non-Archimedean case the situation is essentially different. In fact, there exist Fr´echet Montel spaces which are not nuclear (as we will see in 9.8.2(vi)) while their strong duals are nuclear by 8.6.8. In this context we point out the following characterization of spaces with nuclear dual, proved by Katsaras in [121].

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Theorem 8.8.2 ([121], 6.7) Let E be a polar Hausdorff space. Then Eb is nuclear if and only if the following two conditions hold: (i) Each bounded subset of E is a metrizable compactoid. (ii) For each sequence x1 , x2 , . . . in E tending to 0 there are λ1 , λ2 , . . . ∈ K with limn |λn | = ∞ such that λ1 x1 , λ2 x2 , . . . tends to 0. Finally we touch upon the subject of associated topologies. Let (E, τ ) be a locally convex space. It is easy to see that the finest semi-Montel topology smaller than τ exists. But it seems more natural to consider τsm , the finest polar semi-Montel topology smaller than τ . In [91] its existence was proved as well as the fact that it is defined by the seminorms x −→ sup{|f (x)| : f ∈ S}, where S runs through the collection of equicontinuous and b(E , E)compactoid subsets of E (compare with 8.4.8(γ )). Similarly the existence of τnu , the nuclear topology associated to τ (i.e., the finest nuclear topology coarser than τ ), was proved in [73], where it also was shown that the seminorms x −→ sup{|fn (x)| : n ∈ N}, with {f1 , f2 , . . .} ⊂ E , fn → 0 locally, generates τnu (compare 8.5.1(ζ )). The case E = c0 is of special relevance. In fact, Katsaras and Benekas showed in [129] that (c0 , τnu ) is a universal nuclear space in the following sense. Theorem 8.8.3 ([129], 6.1) A locally convex space is nuclear if and only if it is isomorphic to a subspace of some power of (c0 , τnu ). In [121], Section 3, the topologies τsm and τnu were connected with the socalled bounded weak topology bw, which is the finest locally convex topology on E coinciding with σ (E, E ) on bounded sets (when E is a normed space, bw was previously studied in [201]). Conditions on E were given in [121] in order that τ ∩ bw = τsm , τ ∩ bw = τnu , respectively.

9 Spaces with an “orthogonal” base

In this chapter we study locally convex spaces E having an “orthogonal” base e1 , e2 , . . . (9.1.1). We first show that for such E, (weak) sequential completeness, quasicompleteness and completeness are equivalent (9.1.6). E may have closed subspaces and quotients without an “orthogonal” base (9.2.5). We characterize bounded and compactoid sets in E in terms of e1 , e2 , . . . (9.2.7) and show that compactoids are metrizable (9.2.9(i)). E is semi-Montel if and only if Eb has an “orthogonal” base (9.2.13). We also characterize semi-Montelness (9.2.15) and nuclearity (9.2.16) in terms of properties of the base. Every infinite-dimensional Fr´echet space contains an infinite-dimensional closed subspace with an “orthogonal” base (9.3.5). Section 9.4 is a stepping stone for the sequel; here we introduce the perfect sequence spaces and the normal topology in the spirit of the classical spaces of K¨othe ([144], 6.30). We prove that it is the class of the spaces E with an “orthogonal” base for which E is weakly sequentially complete and E is weakly* sequentially complete (9.4.10). In Section 9.5 we start with an infinite matrix B of nonnegative real numbers and associate to it a perfect sequence space 0 (B) in a natural way (9.5.2, 9.5.9); we show that the class of these 0 (B) is precisely the class of all Fr´echet spaces with an “orthogonal” base (9.5.12). This fact turns out to be very useful; it enables us in Section 9.6 to translate (semi-)Montelness and nuclearity into concrete properties of the matrix B (9.6.2, 9.6.3). We apply this in Section 9.7 to spaces of analytic functions for which we prove properties that have been postponed in previous chapters (9.7.5). We continue in Section 9.8 by selecting a matrix B with special properties so as to obtain in 9.8.1 a Fr´echet Montel space with an “orthogonal” base having a quotient isomorphic to c0 . This space makes the way to answering seven questions raised previously in the book, by means of counterexamples (9.8.2).

337

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Spaces with an “orthogonal” base

9.1 Bases in locally convex spaces The locally convex theory of bases is only interesting for infinite-dimensional spaces. Also, we center the attention to Hausdorff spaces. Therefore, Unless explicitly stated otherwise, all the locally convex spaces considered in this chapter are assumed to be infinite-dimensional and Hausdorff. We usually denote them by (E, τ ) or simply by E. The notions of (Schauder) base and (t)-orthogonal base given in 2.3.10 and 2.3.5 respectively carry over to the locally convex setting as follows. Definition 9.1.1 A sequence e1 , e2 , . . .of non-zero vectors in E is called a base (of E) if for every x ∈ E there exist unique λ1 , λ2 , . . . ∈ K such that x= ∞ λn en ; in other words, if there exist unique f1 , f2 , . . . ∈ E ∗ such n=1 ∞ that x = n=1 fn (x) en for all x ∈ E. The f1 , f2 , . . . are called coefficient functionals or coordinate maps. If, in addition, the fn are in E then e1 , e2 , . . .is called a Schauder base. If the sequence e1 , e2 , . . .is a base and also is “orthogonal” in the sense of 3.9.10 then it is called an “orthogonal” base. Clearly each “orthogonal” base is Schauder (see the comments following 2.3.10 for the normed case). We have seen in 2.3.12 that, even for normed spaces, there exist bases that are not Schauder and Schauder bases that are not “orthogonal”. The leading part in this chapter will be played by the “orthogonal” bases, whereas (Schauder) bases figure only occasionally, mostly when “orthogonality” is involved. We start with the observation that spaces with a base are (strictly) of countable type, so as a direct consequence of 5.5.4(ii) we obtain: Theorem 9.1.2 For a locally convex space with a base, sequential completeness and weak sequential completeness are equivalent properties. For spaces with an “orthogonal” base we can say much more, see 9.1.6. To this end we introduce the following general construction. Definition 9.1.3 Let (E, τ ) have a base e1 , e2 , . . .with coefficient functionals f1 , f2 , . . .. For each τ -continuous seminorm p the formula p ∗ (x) := max |fn (x)| p(en ) n∈N

∗

defines a seminorm p on E (note that |fn (x)| p(en ) → 0 for all x ∈ E and that p ∗ ≥ p). Let τ ∗ be the locally convex topology on E induced by those

9.1 Bases in locally convex spaces

339

p∗ (or, equivalently, by {p ∗ : p ∈ P}, where P is a family of seminorms on E generating τ ). The basic properties of τ ∗ are summarized in the next result. Theorem 9.1.4 Let (E, τ ) have a base e1 , e2 , . . .. Then we have the following: (i) τ ≤ τ ∗ . (ii) p(en ) = p ∗ (en ) for all n and all τ -continuous seminorms p. (iii) e1 , e2 , . . . is an “orthogonal” base of (E, τ ∗ ). More generally, τ ∗ is the smallest among all the locally convex topologies on E that are ≥ τ and for which e1 , e2 , . . . is an “orthogonal” base. (iv) e1 , e2 , . . . is an “orthogonal” base of (E, τ ) if and only if τ = τ ∗ . (v) If e1 , e2 , . . . is a Schauder base and if (E, τ ) is polarly barrelled then τ = τ ∗ and e1 , e2 , . . . is an “orthogonal” base of (E, τ ). Proof. We only give details for (v), leaving the rest to the reader. Let us assume that e1 , e2 , . . . is a Schauder base of the polarly barrelled space (E, τ ). Let p be a (solid) τ -continuous seminorm. We are done as soon as we prove that p ∗ is τ -continuous (then τ = τ ∗ and “orthogonality” follows n n from (iv)). for every n there nBy solidity, n exists a sequence λ1 , λ2 , . . . in K n such that λ1 ≤ λ2 ≤ · · · and supm λm = p(en ). Hence, for each x ∈ E one gets p ∗ (x) = supn,m λnm fn (x), where f1 , f2 , . . . are the coefficient functionals associated to e1 , e2 , . . . . Then (λnm fn )n,m is a pointwise bounded sequence in E . By polar barrelledness and 7.1.7 this sequence is τ -equicontinuous, i.e., p∗ is τ -continuous. The importance of τ ∗ lies in the following. Lemma 9.1.5 Let (E, τ ) have a base and suppose that E is weakly sequentially complete. Then (E, τ ∗ ) is complete. Proof. Let e1 , e2 , . . . be a base of (E, τ ) with coefficient functionals f1 , f2 , . . .. Let (xi )i be a Cauchy net in (E, τ ∗ ). Since f1 , f2 , . . . are τ ∗ -continuous, λn := limi fn (xi ) exists for each n. Next we prove that limn λn en = 0 weakly. So let f ∈ E . Clearly we have |λn f (en )|≤ max(|fn (xi )||f (en )| , |(fn (xi ) − λn )f (en )|) Further, since f (xi ) =

∞ n=1

fn (xi ) f (en ) we obtain

lim fn (xi ) f (en ) = 0 n

for all n ∈ N, i ∈ I . (9.1)

for all i ∈ I .

(9.2)

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The following also holds: lim sup |(fn (xi ) − λn ) f (en )| = 0. i

(9.3)

n∈N

In fact, xi − xj → 0 in τ ∗ if i, j → ∞ and, as |f |∗ is τ ∗ -continuous, |f |∗ (xi − xj ) = maxn∈N fn (xi ) − fn (xj) |f (en )| → 0 if i, j → ∞. So, for every ε > 0 there is an i0 ∈ I such that fn (xi ) − fn (xj ) |f (en )| ≤ ε for all n and all i, j ≥ i0 . Now taking j → ∞ we derive that |fn (xi ) − λn )| |f (en )| ≤ ε for all n and all i ≥ i0 , which yields (9.3). By using (9.1), (9.2) and (9.3) we obtain that, given ε > 0, |λn f (en )| ≤ ε for sufficiently large n. This gives limn λn f (en ) = 0 for all f ∈ E , i.e., limn λn en = 0 weakly. Now, weak sequential completeness implies that x := ∞ n=1 λn en exists in σ (E, E ), hence in τ by 5.5.2. Then λn = fn (x) for each n. From this last equality, together with the assumption that (xi )i is τ ∗ -Cauchy, and by following a straightforward procedure, similar to the one carried out to prove (9.3), we arrive at x = limi xi in (E, τ ∗ ). The above leads to the following corollaries, the first one of which extends 9.1.2. Corollary 9.1.6 For a locally convex space with an “orthogonal” base, weak sequential completeness, sequential completeness, quasicompleteness and completeness are equivalent properties. Proof. By 9.1.2 it suffices to prove that weak sequential completeness implies completeness, which is true thanks to 9.1.4(iv) and 9.1.5. Corollary 9.1.7 In a Fr´echet space each base is “orthogonal”. Proof. Let E be a Fr´echet space, let e1 , e2 , . . . be a base of E. By 9.1.2 E is weakly sequentially complete and by 3.5.2 and 9.1.5 we have that (E, τ ∗ ) is also a Fr´echet space. Applying the Open Mapping Theorem 3.5.10 to the identity (E, τ ∗ )−→(E, τ ) we obtain that τ = τ ∗ and from 9.1.4(iv) we get “orthogonality” of e1 , e2 , . . . .

9.2 Spaces with an “orthogonal” base In this section we discuss a few hereditary properties and describe the bounded and compactoid subsets of spaces with an “orthogonal” base. We start with an easy consequence of 3.9.13 (see 2.3.6 for the normed case).

9.2 Spaces with an “orthogonal” base

341

Theorem 9.2.1 An “orthogonal” sequence in E is an “orthogonal” base of E if and only if its linear hull is dense in E. From this we derive the following. Corollary 9.2.2 Let D be a dense subspace of E. Then each “orthogonal” base of D is an “orthogonal” base of E. In particular, the completion of a locally convex space with an “orthogonal” base has an “orthogonal” base. Remark 9.2.3 Dense subspaces of spaces with an “orthogonal” base may fail to have an “orthogonal” base, see 9.9.1. Theorem 9.2.4 (i) A countable product of spaces with an “orthogonal” base has an “orthogonal” base. (ii) A countable locally convex direct sum of spaces with an “orthogonal” base has an “orthogonal” base. Proof. We prove the case of infinite countable products and locally convex direct sums; the proof for finite products (which coincide with finite locally convex direct sums) is a simple adaptation. Let E1 , E2 , . . . be locally convex spaces with an “orthogonal” base. For each n ∈ N, let en1 , en2 , . . . be an “orthogonal” base of En , let In be an index set and let {pnin : in ∈ In } be a base of continuous seminorms on En such that en1 , en2 , . . . is orthogonal with respect to each pnin . (i). Let E := n∈N En . Then the product topology on E is induced by the seminorms qnin := pnin ◦ πn , n ∈ N, in ∈ In where, for each n, πn is the n-th coordinate map on E. Consider the set, j j ··· . (9.4) {(e1 , 0, . . .) : j ∈ N} {(0, e2 , 0, . . .) : j ∈ N} It is a countable subset of E, so after an enumeration we can treat this set as a sequence. Let us prove that it is an “orthogonal” base of E. By 9.2.1 it suffices to show that it is orthogonal with respect to each qnin and that its linear hull is dense in E. Let n ∈ N, in ∈ In . Since en1 , en2 , . . . is orthogonal with respect to pnin and, for each s, j ∈ N, one verifies 0 if s = n in j qn ((0, . . . , 0, es , 0, . . .)) = in j pn (es ) if s = n, and it follows that the sequence of (9.4) is orthogonal with respect to qnin . Also, for each x := (x1 , x2 , . . .) ∈ E, qnin (x − (0, . . . , 0, xn , 0, . . .)) = 0 and

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since the linear hull of en1 , en2 , . . . is pnin -dense in En , we have that the linear hull of (0, . . . , 0, en1 , 0, . . .), (0, . . . , 0, en2 , 0, . . .), . . . (and hence of the sequence of (9.4)) is qnin -dense in E, and we are done. (ii). Let F := n∈N En . Then the locally convex direct sum topology on F is induced by the seminorms maxn∈N |λn | qnin , with qnin as in (i) and where, for each n, λn ∈ K × . Then with a similar procedure as in (i) it can be proved that the sequence of (9.4) is an “orthogonal” base of F . We leave the details to the reader. Remark 9.2.5 Uncountable products and uncountable locally convex direct sums of locally convex spaces with an “orthogonal” base may fail to be strictly of countable type (apply 4.3.5 and 4.2.17(a) respectively), and in that case they do not have an “orthogonal” base. As for subspaces and quotients, it is known that there are, “up to linear homeomorphisms”, only three infinite-dimensional Fr´echet spaces for which all subspaces have a base, and four infinite-dimensional Fr´echet spaces for which all quotients have a base. See 9.9.2 and 9.9.3, respectively. From now on in this section (except in 9.2.12), we assume that E has an “orthogonal” base e1 , e2 , . . . with coefficient functionals f1 , f2 , . . . . We characterize bounded and compactoid sets in terms of the base. First a basic lemma. Lemma 9.2.6 For each f ∈ E , {f (en ) fn : n ∈ N} is an equicontinuous subset of E . Proof. Let p be a continuous seminorm for which e1 , e2 , . . . is orthogonal and for which |f | ≤ p. For each x ∈ E and n ∈ N we have |fn (x) f (en )| ≤ p(fn (x) en ) ≤ maxi∈N |fi (x)| p(ei ) = p(x) and equicontinuity of {f (en ) fn : n ∈ N} follows. Theorem 9.2.7 Let X be a bounded set in E. Then there exist λ1 , λ2 , . . . ∈ K such that λ1 e1 , λ2 e2 , . . . is bounded and X ⊂ aco {λ1 e1 , λ2 e2 , . . .}. If, in addition, X is a compactoid then λ1 , λ2 , . . . can be chosen such that λn en → 0. Proof. We may assume that X is absolutely convex. Let ρ ∈ K, 0 < |ρ| < 1. Let n ∈ N. Since fn (X) is a bounded and absolutely convex subset of K there exists a λn ∈ K such that fn (X) ⊂ λn BK ⊂ ρ −1 fn (X).

(9.5)

9.2 Spaces with an “orthogonal” base

343

Let us prove that X ⊂ aco {λ1 e1 , λ2 e2 , . . .}. Let x ∈ X. Then, for each n, fn (x) ∈ fn (X), so that by (9.5), |fn (x)| ≤ |λn |. Thus, x=

∞

fn (x) en and |fn (x)| ≤ |λn | for all n,

(9.6)

n=1

from which it follows that x ∈ aco {λ1 e1 , λ2 e2 , . . .}. Next, we show that λ1 e1 , λ2 e2 , . . . is bounded, i.e., that the sequence n → p(λn en ) is bounded for every continuous seminorm p. We may assume that e1 , e2 , . . . is orthogonal with respect to p. Let M := sup{p(x) : x ∈ −1 M for all n. By (9.5) we have X}; let us prove that p(λn en ) ≤ ρ −1 −1 p(fn (xn ) en ) ≤ λ = ρ f (x ) for some x ∈ X, so that p(λ n n n n −1 −1n en ) = ρ −1 ρ maxi∈N p(fi (xn ) ei ) = ρ p(xn ) ≤ ρ M and the first part is proved. Now suppose that X is a compactoid; we show that λn en → 0. For that, it suffices to see (by 5.5.2) that λn en → 0 weakly. So, let f ∈ E . The sequence n → f (en ) fn is equicontinuous (9.2.6) and converges pointwise to 0 in E . It follows from 8.4.11 that f (en ) fn → 0 uniformly on X. Also, from (9.5) we have that, for each n, there is an xn ∈ X for which f (λn en ) = ρ −1 fn (xn ) f (en ) and, since ρ −1 f (en ) fn → 0 uniformly on X, we obtain that f (λn en ) → 0 and the proof is complete. The above result for compactoids can be put in a different form. Since for the chosen λn we have λn en → 0 there is a y in the completion E ∧ of E for ∧ which y = ∞ n=1 λn en . By 9.2.2, e1 , e2 , . . . is an “orthogonal” base of E . If ∧ we denote its associated coefficient functionals on E again by f1 , f2 , . . . then we have that λn = fn (y) for all n. With this in mind, together with (9.6), we can formulate the second part of 9.2.7 as follows. (Compare with 8.1.4.) Corollary 9.2.8 A subset X of E is a compactoid if and only if there is a y ∈ E ∧ such that X ⊂ {x ∈ E : |fn (x)| ≤ |fn (y)| for all n}. We also have the following immediate corollaries. Recall that E is assumed to have an “orthogonal” base. Notice that for metrizable spaces (of countable type) the compactoid (bounded) part of (ii) below was proved in 3.8.28 (8.6.6) and that for metrizable semi-Montel spaces (iii) below was proved in 8.6.7. Corollary 9.2.9 (i) Each compactoid in E is metrizable and contained in the closed absolutely convex hull of an “orthogonal” sequence tending to 0.

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Spaces with an “orthogonal” base

(ii) For each compactoid (bounded) set Y in E ∧ there exists a compactoid E∧ (bounded) set X in E such that Y ⊂ X . (iii) The adjoint (E ∧ )b −→Eb of the inclusion E−→E ∧ is a bijective linear homeomorphism. Proof. (i). Follows directly from 3.8.24 and 9.2.7. (ii). Let Y be a compactoid (bounded) set in E ∧ . By 9.2.2, e1 , e2 , . . . is an “orthogonal” base of E ∧ , so by 9.2.7 there exist λ1 , λ2 , . . . ∈ K such that n → ∧ λn en is a null (bounded) sequence and such that Y ⊂ acoE {λ1 e1 , λ2 e2 , . . .}. Choose X := aco {λ1 e1 , λ2 e2 , . . .}. Then X is a compactoid (bounded) set in E∧ E and clearly Y ⊂ X . (iii). Can be proved in the same way as in 8.6.7. Corollary 9.2.10 (i) E is semi-Montel if and only if for each sequence λ1 , λ2 , . . . in K, boundedness of {λn en : n ∈ N} implies λn en → 0. (ii) If E is semi-Montel then so is E ∧ . Proof. We know (8.4.5) that E is semi-Montel if and only if each bounded set is a compactoid. Then the “if” of (i) follows from 9.2.7 and the “only if” from 3.9.16. Also, (ii) is a direct consequence of (i) an the fact that, by 9.2.2, e1 , e2 , . . . is an “orthogonal” base of E ∧ . Remark 9.2.11 Note that (ii) was announced in 8.4.27(c). For metrizable spaces we have a partial converse of 9.2.9(i). Theorem 9.2.12 Let E be metrizable and of countable type. Suppose each compactoid is contained in the closed absolutely convex hull of an “orthogonal” sequence tending to 0. Then E has an “orthogonal” base. Proof. By metrizability E is strictly of countable type, so there exists a sequence y1 , y2 , . . . in E with E = [y1 , y2 , . . .]. Let U1 ⊃ U2 ⊃ · · · be a neighbourhood base of 0 (3.5.2). For each n, choose λn ∈ K × with λn yn ∈ Un and put xn := λn yn . Then x1 , x2 , . . . is a sequence in E tending to 0 and X := {x1 , x2 , . . .} is a compactoid in E whose linear hull is dense in E. By assumption there is an “orthogonal” sequence e1 , e2 , . . . in E tending to 0 such that X ⊂ aco {e1 , e2 , . . .}. Hence E = [X] ⊂ [e1 , e2 , . . .], i.e., E = [e1 , e2 , . . .]. It follows from 9.2.1 that e1 , e2 , . . .is an “orthogonal” base of E. It is a natural question to ask whether the strong dual of a space with an “orthogonal” base also has one, in particular, whether the coefficient functionals form an “orthogonal” base. The case E := c0 shows that this is not always true:

9.2 Spaces with an “orthogonal” base

345

by 2.5.11, Eb ∞ which, by 2.5.15, is not even of countable type! The general answer is as follows. Theorem 9.2.13 The following are equivalent: (α) (β) (γ ) (δ)

E is semi-Montel. Eb is of countable type. Eb has an “orthogonal” base. f1 , f2 , . . . is an “orthogonal” base of Eb .

Proof. (δ) =⇒ (γ ) =⇒ (β) is trivial. (β) =⇒ (α) follows from 7.6.13, 7.6.14 and the fact that semi-Montel spaces are those for which bounded sets are compactoids (8.4.5). So we only need to prove (α) =⇒ (δ). Let f ∈ E . It can be uniquely written as f = ∞ . Since {f (en ) fn : n ∈ N} is n=1 f (en ) fn in Eσ equicontinuous (9.2.6), by 8.4.11 we have f = ∞ n=1 f (en ) fn uniformly on compactoids, that is, in Eb by (α). Thus, f1 , f2 , . . . is a base of Eb . To prove “orthogonality” it suffices by 9.2.7 to show that p(f ) = maxn∈N |f (en )| p(fn ) for any seminorm p on E of the type p(f ) = max |λn f (en )| n∈N

(f ∈ E ),

where λ1 , λ2 , . . .is a sequence in K for which λn en → 0. But that follows immediately from the observation that |λn | = p(fn ) for all n. Remark 9.2.14 From (α)–(δ) it does not follow that Eb is semi-Montel. In fact, let E be the space of 7.1.19(i), that is, E := ∞ equipped with the locally convex topology induced by the seminorms (λ1 , λ2 , . . .) −→ max |λn µn | , n∈N

where (µ1 , µ2 , . . .) runs through c0 . It is easily seen that E is semi-Montel, that the unit vectors form an “orthogonal” base of E, and that Eb is isomorphic to the Banach space c0 . In Sections 9.5 and 9.6 we will treat the class of K¨othe sequence spaces of which E is a particular case. The following criterion for semi-Montelness may seem somewhat technical, but turns out to be useful later on when constructing (counter)examples, see Sections 9.7 and 9.8. Theorem 9.2.15 E is semi-Montel if and only if (9.7) below holds: For each subsequence en1 , en2 , . . . of e1 , e2 . . . and for each continuous seminorm p for which p(eni ) = 0 for all i, there exists a continuous seminorm q(e ) q ≥ p such that { p(enni ) : i ∈ N} is unbounded. (9.7) i

346

Spaces with an “orthogonal” base

Proof. First observe that the negation of (9.7) reads as follows. There exists a subsequence en1 , en2 , . . . of e1 , e2 . . . and a continuous seminorm p with p(eni ) = 0 for all i such that, for all continuous seminorms q ≥ p, the q(e ) set { p(enni ) : i ∈ N} is bounded. i Now suppose that (9.7) does not hold; we prove that E is not semi-Montel. Let n1 , n2 , . . . and p be as above. We may assume that e1 , e2 , . . . is orthogonal with respect to p. Set D := [en1 , en2 , . . .]. Then p is a norm on D and it follows by boundedness that each continuous seminorm q ≥ p is equivalent to p on D. Hence D is an infinite-dimensional normable subspace of E and by 8.4.5 E is not semi-Montel. Conversely, assume E is not semi-Montel; we prove that (9.7) does not hold. By 9.2.10(i) there are λ1 , λ2 , . . . ∈ K × such that {λn en : n ∈ N} is bounded, but λn en 0. Thus, there exist a continuous seminorm p, an ε > 0 and a subsequence λn1 en1 , λn2 en2 , . . . such that p(λni eni ) ≥ ε for all i. Now let q be a continuous seminorm, q ≥ p. By boundedness, M := supn∈N q(λn en ) < ∞. q(e ) q(λ e ) We then find that p(enni ) = p(λnni enni ) ≤ M ε −1 for all i. i

i

i

In the same spirit we have the following, much easier, criterion for nuclearity. Theorem 9.2.16 E is nuclear if and only if (9.8) below holds. For each continuous seminorm p there exists a continuous seminorm q ≥ p and non-negative ε1 , ε2 , . . . with εn → 0 such that p(en ) ≤ εn q(en ) for all n. (9.8)

Proof. Let E be nuclear, let p be a continuous seminorm. To prove (9.8) we may assume that e1 , e2 , . . . is orthogonal with respect to p. Also, we may assume that N0 := {n ∈ N : p(en ) = 0} is an infinite set (otherwise, it is obvious that q := p and εn , defined by εn := 1 if n ∈ N0 and εn := 0 if n ∈ N0 , satisfy (9.8)). By nuclearity and 8.5.1(α) =⇒ (β) there is a continuous seminorm q ≥ p such that the canonical map Eq −→Ep is compactoid, i.e., B := {x ∈ E : q(x) ≤ 1} is a p-compactoid. Let D := [{en : n ∈ N0 }]. Then pD := p|D is a norm on D, {en : n ∈ N0 } is an “orthogonal” base of the normed space (D, pD ), and B ∩ D is a compactoid in this normed space. Let ρ ∈ K, 0 < |ρ| < 1. For each n ∈ N0 , q(en ) = 0, hence there is a µn ∈ K × with |ρ| ≤ q(µn en ) ≤ 1, so that µn en is in B ∩ D. From pD compactoidity and 9.2.7, paying also attention to (9.5) of the proof of that theorem, we derive the existence of λn ∈ K (n ∈ N0 ) such that |µn | ≤ |λn | and limn∈N0 p(λn en ) = 0. For n ∈ N, put −1 |ρ| p(λn en ) if n ∈ N0 εn := 0 otherwise.

9.3 Fr´echet spaces with an “orthogonal” base

347

Clearly εn → 0 and 0 = p(en ) = εn q(en ) for n ∈ N0 . Further, for n ∈ N0 , p(en ) = |µn |−1 p(µn en ) ≤ |µn |−1 p(λn en ) ≤ |ρ|−1 q(en ) p(λn en ) = εn q(en ), so that these q and εn satisfy (9.8). Conversely, suppose (9.8) holds. Take a continuous seminorm p and let q, εn be as in (9.8). We may assume that e1 , e2 , . . . is orthogonal with respect to q. By 8.5.1(β) =⇒ (α) it suffices to see that the canonical map Eq −→Ep is compactoid, i.e., that B := {x ∈ E : q(x) ≤ 1} is a p-compactoid. For that, let x := ∞ n=1 fn (x) en ∈ B. Then q(fn (x) en ) ≤ 1, so that p(fn (x) en ) ≤ εn for all n. Thus, limn supx∈B p(fn (x) en ) = 0, from which p-compactoidity of B follows easily.

9.3 Fr´echet spaces with an “orthogonal” base We know (2.3.7) that every infinite-dimensional Banach space of countable type has an “orthogonal” base. The question arises whether the same holds for Fr´echet spaces E of countable type. Since such E are isomorphic to a closed subspace of c0N (4.2.15) and since this last space has an “orthogonal” base by 9.2.4(i), the above question is equivalent to Given a Fr´echet space with an “orthogonal” base, does it follow that its closed subspaces have an “orthogonal” base? Unfortunately, there is an example showing that both questions have negative answers, see the chapter notes for reference. But let us signal one positive answer. Recall that our spaces are assumed to be infinite-dimensional. Theorem 9.3.1 Let E be a Fr´echet space of finite type. Then E has an “orthogonal” base. Proof. By 5.3.12(a) E is isomorphic to K N , which has an “orthogonal” base by 9.2.4(i). Also, we do have the following. Recall that by definition any “orthogonal” sequence consists of non-zero vectors. Theorem 9.3.2 Every Fr´echet space contains an “orthogonal” sequence. To get 9.3.2 first we prove a basic lemma on t-orthogonality in normed spaces. Lemma 9.3.3 Let D be a finite-dimensional subspace of a vector space F with dim F = ℵ0 and let q1 , . . . , qn be norms on F . Then, for every t ∈ (0, 1), there exists a non-zero x ∈ F such that, for all i ∈ {1, . . . n}, x is t-orthogonal to D

348

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with respect to qi (that is, qi (λ x + y) ≥ t max(qi (λ x), qi (y)) for each λ ∈ K, y ∈ D). Proof. Let i ∈ {1, . . . n} and let Fi be the completion of (F, qi ). Since Fi is a Banach space of countable type and D is a closed subspace of Fi there exists a continuous projection Pi of Fi onto D with Pi ≤ t −1 (2.3.13). It is easily seen that each x ∈ F ∩ Ker Pi is t-orthogonal to D with respect to qi . Further, if G := 1≤i≤n F ∩ Ker Pi , dim (F /G) ≤ ≤

n i=1 n

dim (F /F ∩ Ker Pi ) dim (Fi /Ker Pi ) = n dim D < ∞.

i=1

Hence G = {0} and obviously any x ∈ G is t-orthogonal to D with respect to each qi , i ∈ {1, . . . , n}. Proof of 9.3.2. Let E be Fr´echet. By 3.5.2 there exists an increasing sequence p1 ≤ p2 ≤ · · · of seminorms defining the topology of E. If dim Epj < ∞ for all j then E is of finite type and so has an “orthogonal” base by 9.3.1. Thus, let us suppose that there is a j for which the normed space Epj is infinite-dimensional. We may assume j = 1. Let πp1 (x1 ), πp1 (x2 ), . . . be a linearly independent sequence in Ep1 and put F := [x1 , x2 , . . .]. Clearly dim F = ℵ0 and qn := pn |F is a norm on F for each n. Let s1 , s2 , . . .be a sequence in (0, 1) with s := ∞ n=1 sn > 0. By 9.3.3 we can construct inductively a sequence y1 , y2 , . . .of non-zero vectors in F such that, for each n, yn+1 is sn+1 -orthogonal to [y1 , . . . , yn ] with respect to qi , for all i ∈ {1, . . . , n}. Now we prove that there exists a sequence t1 , t2 , . . . in (0, 1) such that y1 , y2 , . . . is tm -orthogonal with respect to qm , hence with respect to pm , for all m (and by 3.9.15 we have that y1 , y2 , . . . is an “orthogonal” sequence in E). For that, let m ∈ N, m > 1, λ1 , . . . , λm ∈ K. Then m m−1 q1 ( λi yi ) ≥ sm max(q1 ( λi yi ), q1 (λm ym )) ≥ · · · i=1

i=1

≥ sm sm−1 · · · s1 max q1 (λi yi ) ≥ s max q1 (λi yi ). 1≤i≤m

1≤i≤m

Further, on the finite-dimensional space [y1 , . . . , ym ], q1 and qm are equivalent norms (3.4.24) so, applying the above, there exists an rm ∈ (0, 1) such that, for

9.3 Fr´echet spaces with an “orthogonal” base

349

arbitrary λ1 , . . . , λm ∈ K, m qm ( λi yi ) ≥ rm max qm (λi yi ). 1≤i≤m

i=1

Now, let k > m and λ1 , . . . , λk ∈ K. Then qm (

k

λi y i )

i=1 m ≥ sk sk−1 · · · sm+1 max(qm ( λi yi ), qm (λm+1 ym+1 ), . . . , qm (λk yk )) i=1

≥ s rm max qm (λi yi ). 1≤i≤k

Thus, for each m, the sequence y1 , y2 , . . . is tm -orthogonal with respect to qm for tm := s rm . Remark 9.3.4 The conclusions of 9.3.1 and 9.3.2 also hold for non-complete metrizable spaces, see the chapter notes. Corollary 9.3.5 Every infinite-dimensional closed subspace of a Fr´echet space contains an infinite-dimensional closed subspace with an “orthogonal” base. Many examples of Fr´echet spaces (with an “orthogonal” base) have continuous norms, i.e., the topology is generated by an increasing sequence of norms (rather than seminorms, 3.5.2), see also Section 9.6. Definition 9.3.6 A Fr´echet space with an “orthogonal” base and a continuous norm is called a K¨othe space. We have the following decomposition theorem. Theorem 9.3.7 Let E be a Fr´echet space. Then the following are equivalent: (α) E has an “orthogonal” base. (β) E is isomorphic to either K N , F , or K N × F , where F is a countable product of K¨othe spaces. Proof. (β) =⇒ (α) is provided by 9.2.4(i). To prove (α) =⇒ (β), let e1 , e2 , . . . be an “orthogonal” base of E. By 3.5.2 there is an increasing sequence of seminorms p1 ≤ p2 ≤ · · · defining the topology of E and such that e1 , e2 , . . . is orthogonal with respect to each pi . Put p0 := 0 and for each k ∈ N, Nk := {n ∈ N : en ∈ Ker pk−1 \ Ker pk }, Dk := [en : n ∈ Nk ]. We may assume that each Nk is non-empty.

350

Spaces with an “orthogonal” base

∞ For any (x1 , x2 , . . .) ∈ k∈N Dk the series k=1 xk is convergent since pi (xk ) = 0 for all i, k, k > i. Also, by 9.2.1 {en : n ∈ Nk } is an “orthogonal” base of Dk (admitting that this base may be finite in case Nk is finite). λ e , be the natural projection Let Pk : E−→Dk , x = ∞ n∈N n=1 λn en → k n n of E onto Dk , k ∈ N. Then the map E−→ k∈N Dk , x → (P1 (x), P2 (x), . . .) is a continuous linear bijection. By the Open Mapping Theorem 3.5.10 it is a linear homeomorphism. Next, let us analyze the Dk , k ∈ N. By construction, pk |Dk is a continuous norm on Dk . Hence, if Nk is finite, Dk is isomorphic to K nk with nk := Nk (apply 2.1.15 and 3.4.24) and if Nk is infinite, Dk is a K¨othe space. Now put M := {k ∈ N : Nk is finite}, N := {k ∈ N : Nk is infinite}. We obtain that if M = ∅ then E ∼ F , where F := k∈N Dk and if N = ∅ then E ∼ k∈M K nk ∼ K N (note that E is infinite-dimensional, so M must be infinite when N = ∅). It remains to discuss the case M, N = ∅. If M is infinite then k∈M K nk ∼ K N , so E ∼ K N × F , with F as above. If M is finite, choose k1 ∈ N . Then G := k∈M K nk × Dk1 is a K¨othe space and E ∼ G × k∈N ,k=k1 Dk (= G if N = {k1 }).

9.4 Perfect sequence spaces In this section we will introduce the so-called perfect sequence spaces in a rather concrete way, following the theory of K¨othe ([144], 6.30). We will see in 9.4.10 that this class is precisely the class of all weakly sequentially complete locally convex spaces E with an “orthogonal” base for which E is weakly∗ sequentially complete. We start with an algebraic introduction. A sequence space is a subspace of the vector space K N . Sequence spaces will be denoted by . For such a its K¨othe dual is the sequence space defined by ×

:= {(µ1 , µ2 , . . .) ∈ K N : lim λn µn = 0 for all (λ1 , λ2 , . . .) ∈ n

}.

It is the non-Archimedean version of the α-dual of [144], Section 30.1. (In 9.4.5 we will see that × can be interpreted as the dual of with respect to a certain topology.) Clearly we have ⊂ ×× . is called perfect if = ×× . We will show here and in the next sections that, among the sequence spaces, the perfect ones are the most important and useful, because of their influence in the applications. Obviously they are infinite-dimensional because they contain the unit vectors of K N .

9.4 Perfect sequence spaces

351

Remark 9.4.1 Some non-perfect sequence spaces will be considered in 11.4.9. The proof of the next basic result on K¨othe duals is straightforward and left to the reader. Theorem 9.4.2 (i) × (hence every perfect sequence space) contains the unit vectors of K N . (ii) × is perfect. (iii) ×× is the smallest perfect sequence space containing . Now we equip with a locally convex topology. For y = (µ1 , µ2 , . . .) ∈ a seminorm py on is defined by py (x) := max |λn µn | n∈N

(x = (λ1 , λ2 , . . .) ∈

).

×

,

(9.9)

The family of seminorms {py : y ∈ × } determines a Hausdorff locally convex topology on . It is denoted by n( , × ) and it is called the normal (natural) topology on . Let e1 , e2 , . . . be the unit vectors of K N . We prove that these vectors form an “orthogonal” base of any containing them (e.g. when is perfect, see (i) of 9.4.2). Theorem 9.4.3 Suppose en ∈ for all n. Then the sequence e1 , e2 , . . . is an “orthogonal” base of ( , n( , × )). Proof. For every x = (λ1 , λ2 , . . .) ∈ have py (x −

m n=1

, y = (µ1 , µ2 , . . .) ∈

×

and m ∈ N we

λn en ) = max |λn µn | , n>m

which tends to 0 when m → ∞. So x can be written as x=

∞

λn en in n( ,

×

).

(9.10)

n=1

Since e1 , e2 , . . . is orthogonal with respect to py for all y ∈ × we have that the expression of x as a convergent series of the form (9.10) is unique and that e1 , e2 , . . . is an “orthogonal” base of for the normal topology. Examples of perfect sequence spaces are K (N) , K N , c0 and ∞ , whose K¨othe duals are K N , K (N) , ∞ and c0 , respectively. It is easily seen that the normal topology on K (N) (resp. on K N , on c0 ) coincides with the locally convex direct sum topology on K (N) (resp. with the product topology on K N , with the norm topology on c0 ). But for ∞ we do not have the same luck because its norm

352

Spaces with an “orthogonal” base

topology is not of countable type (2.5.15) whereas its normal topology it is, by 9.4.3. For more examples of perfect sequence spaces see Section 9.5. Remark 9.4.4 It follows from 9.4.3 that every perfect sequence space equipped with the normal topology is strictly of countable type. However, K N is the only perfect sequence space that is of finite type. In fact, that be a perfect sequence K N is of finite type is well-known. Now, let space and suppose that ( , n( , × )) is of finite type. Let us show that × = K (N) . (Then = ×× = (K (N) )× = K N , and we are done.) If there would be a y = (µ1 , µ2 , . . .) ∈ × , y ∈ K (N) then {πpy (en ) : n ∈ I }, with I := {n ∈ N : µn = 0}, would be an infinite linearly independent set in the finite-dimensional normed space py , a contradiction. The dual of ( , n( , × )) can algebraically be identified with × . In fact, for each x = (λ1 , λ2 , . . .) ∈ and y = (µ1 , µ2 , . . .) ∈ × , the formula B(x, y) :=

∞

λn µn

n=1

defines a bilinear form B :

×

×

−→K and we have:

Theorem 9.4.5 Suppose en ∈ for all n. Then the map y → B(., y) is an algebraic isomorphism × −→( , n( , × )) . Proof. Writing fy := B(., y) we see that y → fy is linear and that, for each y = (µ1 , µ2 , . . .) ∈ × , fy (x) ≤ py (x) for all x ∈ , so fy ∈ ( , n( , × )) . Applying fy to the unit vectors en we obtain fy (en ) = µn for each n. Thus, y → fy is injective. To show surjectivity, let g ∈ ( , n( , × )) , let y := (g(e ), g(e2 ), . . .). Since, by (9.10), for all x = (λ1 , λ2 , . . .) ∈ we have x = ∞1 × ), then g(x) = ∞ n=1 λn en in n( , n=1 λn g(en ), so limn λn g(en ) = 0. It follows that y ∈ × and fy = g. Perfect sequence spaces can be characterized as follows. Theorem 9.4.6 The following are equivalent: (α) is perfect. (β) en ∈ for all n and ( , n( , (γ ) en ∈ for all n and ( , n( ,

× ×

)) is complete. )) is (weakly) sequentially complete.

Proof. By 9.1.6 and 9.4.3 we have (β) ⇐⇒ (γ ). (α) =⇒ (β). That en ∈ for all n follows from 9.4.2(i). To prove completeness, let (x i )i be a Cauchy net in ( , n( , × )), where x i = (λi1 , λi2 , . . .) for

9.4 Perfect sequence spaces

353

each i ∈ I . This means that for all y = (µ1 , µ2 , . . .) ∈ × and for all ε > 0 there is an i0 ∈ I such that j (9.11) maxn∈N (λin − λn ) µn ≤ ε for all i, j ≥ i0 . Then for each n, (λin )i is a Cauchy net in K, so λn := limi λin exists. By using (9.11) and a standard argument we obtain that x := (λ1 , λ2 , . . .) ∈ ×× (= by perfectness) and that limi x i = x in ( , n( , × )). (β) =⇒ (α). Let x := (λ1 , λ2 , . . .) ∈ ×× ; we prove that x ∈ . We know ×× (see (9.10)) that x = ∞ , × ) (observe that ××× = × n=1 λn en in n( because × is perfect, see 9.4.2(ii)). So the sequence m → sm := m n=1 λn en ×× ×× × is Cauchy in ( , n( , )). But by assumption we have that sm ∈ for all m, so s1 , s2 , . . . is Cauchy in ( , n( ×× , × )| ), hence in ( , n( , × )). × By (β) there exists a y ∈ such that y = ∞ ). Thus, n=1 λn en in n( , ∞ y = n=1 λn en in ( ×× , n( ×× , × )), which is Hausdorff. Therefore, x = y∈ . It follows from 9.4.3 and 9.4.6 that every perfect sequence space equipped with the normal topology has an “orthogonal” base and is complete. These facts, together with 9.2.8 and 9.2.9(i), lead to the following properties of compactoid subsets of perfect sequence spaces. Corollary 9.4.7 Let := ( , n( , we have the following: (i) A set X ⊂ such that

×

)) be a perfect sequence space. Then

is a compactoid if and only if there exists a (δ1 , δ2 , . . .) ∈ X ⊂ {(λ1 , λ2 , . . .) ∈

: |λn | ≤ |δn | for all n}.

(ii) Every compactoid subset of is metrizable and contained in the closed absolutely convex hull of an “orthogonal” sequence in tending to 0. Remark 9.4.8 Property (i) is an extension of 8.1.4, which covers the case := c0 (recall that the normal and the norm topologies on c0 coincide). As a direct consequence of 9.4.5 and 9.4.7(i) we can give a new description of the normal topology as follows. Corollary 9.4.9 Let be a perfect sequence space. Then n( × , ) is the topology of uniform convergence on the n( , × )-compactoid subsets of . We finish the section by showing the announced connection between perfect sequence spaces and spaces with an “orthogonal” base. Recall that all the

354

Spaces with an “orthogonal” base

locally convex spaces E := (E, τ ) considered in this chapter are assumed to be infinite-dimensional and Hausdorff. Theorem 9.4.10 The following are equivalent: (α) (E, τ ) is a weakly sequentially complete space with an “orthogonal” base and E is weakly∗ sequentially complete. (β) (E, τ ) is isomorphic to a perfect sequence space equipped with the normal topology. Proof. (α) =⇒ (β). Let e1 , e2 , . . . be an “orthogonal” base of E with coefficient functionals f1 , f2 , . . .. Let := {(λ1 , λ2 , . . .) ∈ K N : lim λn en = 0}. n

By sequential completeness (9.1.6), for each (λ1 , λ2 , . . .) ∈ n → λn en is summable in E and therefore the formula T ((λ1 , λ2 , . . .)) :=

∞

the sequence

λn en

n=1

defines a map T : −→E that is clearly a linear bijection. We prove that T is actually a homeomorphism of ( , n( , × )) onto E. (Then ( , n( , × )) is weakly sequentially complete and, as contains the unit vectors of K N , by 9.4.6 it follows that is perfect and we are done.) Thus, let τ be the locally convex topology on E induced by T ; it is generated by the seminorms x −→ max |fn (x)| |µn | , n∈N

(9.12)

where (µ1 , µ2 , . . .) runs through × ; we have to prove that τ = τ . If p is a τ -continuous seminorm then |fn (x)| p(en ) → 0 for all x ∈ E. By choosing (µ1 , µ2 , . . .) ∈ K N such that p(en ) ≤ |µn | ≤ |ρ|−1 p(en ) for all n, where ρ ∈ K, 0 < |ρ| < 1, we see that (µ1 , µ2 , . . .) ∈ × and that, for each x ∈ E, p(x) ≤ maxn∈N |fn (x)| |µn |, so that p is τ -continuous. Conversely, let p be a τ -continuous seminorm of the form (9.12). Then µn fn → 0 pointwise and by assumption the pointwise sum g := ∞ n=1 µn fn is in (E, τ ) . So there exists a τ -continuous seminorm q such that |g(x)| ≤ q(x) for all x ∈ E. By applying this to x := en we find |µn | ≤ q(en ) for all n. We may assume that e1 , e2 , . . .is orthogonal with respect to q. Then, for x ∈ E, p(x) = maxn∈N |fn (x)| |µn | ≤ maxn∈N |fn (x)| q(en ) = q( ∞ n=1 fn (x) en ) = q(x) and we obtain that p is τ continuous. (β) =⇒ (α). Suppose E ∼ := ( , n( , × )) for some perfect sequence space . It suffices to see that has the properties required for E in (α). By 9.4.3, has an “orthogonal” base. Weak sequential completeness follows from

9.5 K¨othe sequence spaces

355

9.4.6. Finally, since × is perfect by 9.4.2(ii), applying 9.4.6 to ( × , n( × , )) we arrive at weak sequential completeness of ( × , n( × , )) which, by 9.4.5, means weak∗ sequential completeness of . Corollary 9.4.11 Every Fr´echet space with a base is isomorphic to a perfect sequence space equipped with the normal topology. Proof. Let E be a Fr´echet space with a base (which is “orthogonal” by 9.1.7). By 9.1.6 E is weakly sequentially complete and by the Banach–Steinhaus Theorem (7.1.3 and 7.1.8.b) E is weakly∗ sequentially complete. Then apply 9.4.10.

9.5 K¨othe sequence spaces Corollary 9.4.11 shows a connection between Fr´echet spaces with a base and sequence spaces. The purpose of this section is to make this connection more precise. In fact, we will prove (9.5.12) that the class of Fr´echet spaces with a base coincides with the class formed by the K¨othe sequence spaces that we introduce below. We start by showing that for Fr´echet spaces with a base we can use infinite matrixes to describe the topology. Theorem 9.5.1 Let E be a Fr´echet space with an “orthogonal” base e1 , e2 , . . . and coefficient functionals f1 , f2 , . . .. Then there exists an infinite matrix (bnk )k,n (k, n ∈ N) consisting of real numbers satisfying the following conditions: 0 ≤ bnk ≤ bnk+1 for all k, n, for all n there is a k such that bnk > 0, limn |fn (x)| bnk = 0 for all k ∈ N, x ∈ E, and such that the topology of E is generated by the sequence of seminorms q1 ≤ q2 ≤ · · · , where each qk (k ∈ N) is defined by qk (x) := max |fn (x)| bnk n∈N

(x ∈ E).

Proof. There is a sequence of seminorms q1 ≤ q2 ≤ · · · defining the topology of E (3.5.2) and such that qk (x) = max |fn (x)| qk (en ) n∈N

for all k ∈ N, x ∈ E.

Then putting bnk := qk (en ) we have an infinite matrix having the required properties. Conversely, we can assign spaces to certain matrixes.

356

Spaces with an “orthogonal” base

Definition 9.5.2 Let B := (bnk )k,n (k, n ∈ N) be an infinite matrix consisting of real numbers satisfying the conditions 0 ≤ bnk ≤ bnk+1 for all k, n, for each n there is a k such that bnk > 0.

(9.13)

The K¨othe sequence space associated to the matrix B is 0

(B) := {(λ1 , λ2 , . . .) ∈ K N : limn |λn | bnk = 0 for all k}.

Note that if bnk := 1 for all k, n then 0 (B) = c0 . Also, we will see in 9.5.13 that the spaces 0 (B) with bnk > 0 for all k, n, equipped with the normal topology, are the “models” of the K¨othe spaces introduced in 9.3.6. 0

Remark 9.5.3 Different matrixes B may produce the same

(B).

From now on in this section B := (bnk )k,n (k, n ∈ N) is an infinite matrix consisting of real numbers satisfying (9.13). Clearly

0

(B) =

k∈N

Ek where, for each k,

Ek := {(λ1 , λ2 , . . .) ∈ K N : lim |λn | bnk = 0}. n

(9.14)

For k ∈ N, let us equip Ek with the seminorm (λ1 , λ2 , . . .) → maxn∈N |λn | bnk (which is a norm if and only if bnk > 0 for all n). The monotonicity condition imposed in (9.13) for the matrix B implies that E1 , E2 , . . . is a projective sequence. Let τp0 be the corresponding projective topology on 0 (B). Theorem 3.4.31 provides the following description of this topology. Theorem 9.5.4 The projective topology τp0 on 0 (B) is generated by the sequence of seminorms p1 ≤ p2 ≤ · · · , where each pk (k ∈ N) is defined by pk (x) := max |λn | bnk n∈N

(x = (λ1 , λ2 , . . .) ∈

0

(B)).

(9.15)

From 9.5.1 we conclude the following. Theorem 9.5.5 Let E be a Fr´echet space with an “orthogonal” base. Then there exists a matrix B such that E is isomorphic to ( 0 (B), τp0 ). Proof. For each k, n ∈ N let bnk be as in 9.5.1 and let B := (bnk )k,n . From the conclusions of that theorem we derive that B satisfies the conditions of (9.13) and that the map E−→ 0 (B), x → (f1 (x), f2 (x), . . .) is a bijective linear homeomorphism when we equip 0 (B) with the topology induced by the seminorms pk of (9.15), that is, with the projective topology τp0 . Next we show that the converse of 9.5.5 is true.

9.5 K¨othe sequence spaces

357

Theorem 9.5.6 ( 0 (B), τp0 ) is a Fr´echet space with an “orthogonal” base formed by the unit vectors of K N . Proof. Let p1 ≤ p2 ≤ · · · be the sequence of seminorms generating τp0 and defined according to (9.15). The unit vectors e1 , e2 , . . . of K N are orthogonal with respect to each pk and also one verifies that every x = (λ1 , λ2 , . . .) ∈ 0 (B) can be written as 0 (B), τp0 ), (9.16) x= ∞ n=1 λn en in ( so that e1 , e2 , . . . is an “orthogonal” base of this space, which is Hausdorff by the second condition of (9.13) and metrizable by 3.5.2. In order to get completeness, let (x r )r be a τp0 -Cauchy sequence in 0 (B), with x r := (λr1 , λr2 , . . .) for each r. Let n ∈ N and choose k as in the second condition of (9.13), i.e., with bnk > 0. Since k → 0 when r, s → ∞, (9.17) pk (x r − x s ) = maxm∈N λrm − λsm bm r in particular λn − λsn → 0 when r, s → ∞, so that the sequence λ1n , λ2n , . . . of the n-th coordinates of the x r is Cauchy in K. Hence λn := limr λrn exists. Now, using that (9.17) holds for all k and following a standard argument one can easily derive that x r → x := (λ1 , λ2 , . . .) in ( 0 (B), τp0 ). As a direct application of 9.5.5 and 9.5.6 we arrive at the first step towards the purpose of this section. Corollary 9.5.7 Let E be a Fr´echet space. Then E has an “orthogonal” base if and only if there exists a matrix B such that E is isomorphic to ( 0 (B), τp0 ). On the other hand, we have seen in Section 9.4 that the topology usually considered in a sequence space is the normal one. Our second step (9.5.11) is to prove that, in any K¨othe sequence space 0 (B), the normal and the projective topologies coincide. For that we need to describe the K¨othe dual 0 (B)× . Definition 9.5.8 For each k ∈ N, let Nk := {n ∈ N : bnk > 0}, Fk := {(λ1 , λ2 , . . .) ∈ K N : supn∈Nk |λn | /bnk < ∞ and λn = 0 for n ∈ Nk }. (9.18) Put ∞ (B) := k∈N Fk . Note that if bnk := 1 for all k, n then ∞ (B) = ∞ . It may happen that Nk = ∅ (and then Fk = {0}). However, it follows from (9.13) that eventually Nk = ∅.

358

Spaces with an “orthogonal” base

Also, Nk may be finite and non-empty (and then Fk finite-dimensional). In fact, the formula 1 if k ≥ n k bn := 0 if k < n, defines a matrix satisfying (9.13) such that Nk = {1, . . . , k} for all k. One verifies the following. Lemma 9.5.9

0

(B) is a perfect sequence space for which

0

(B)× =

∞ (B).

Proof. Firstly we show that ∞ (B)× = 0 (B) (so 0 (B) is perfect by 9.4.2(ii)). For each k, let Ek and Fk be the sequence spaces defined in (9.14) and (9.18) respectively. Then Fk× = Ek , from which we obtain × Fk )× = Fk× = Ek = 0 (B). ∞ (B) = ( k∈N

k∈N

k∈N

It remains to prove that ∞ (B) is perfect. (Then 0 (B)× = ∞ (B)×× = ∞ (B), and we are done). Let pk (k ∈ N) be the seminorms defined in (9.15), which by 9.5.4 generate the projective topology τp0 of 0 (B). The map ∞ (B)−→(

0

(B), τp0 ) , y = (µ1 , µ2 , . . .) → fy , fy (x) := = (λ1 , λ2 , . . .) ∈

0

∞

λn µn , x

n=1

(B),

Ek×

for all k, the series appearing in the defis well-defined. Indeed, as Fk ⊂ inition of fy (x) is convergent and in this way we get an fy ∈ ( 0 (B), τp0 ) . Also, the above map is an algebraic isomorphism. We show surjectivity, the rest is straightforward. Given g ∈ ( 0 (B), τp0 ) , take k ∈ N and C > 0 such that |g(x)| ≤ C pk (x) for all x ∈ 0 (B), in particular |g(en )| ≤ C pk (en ) = C bnk for all n, where e1 , e2 , . . . are the unit vectors of K N . It follows that y := (g(e1 ), g(e2 ), . . .) ∈ Fk ⊂ ∞ (B). Also, by (9.16), every 0 x = (λ1 , λ2 , . . .) ∈ 0 (B) can be written as x = ∞ (B), τp0 ), n=1 λn en in ( ∞ so g(x) = n=1 λn g(en ) = fy (x), which means that g = fy and we get surjectivity. Since ( 0 (B), τp0 ) is a Fr´echet space (9.5.6), applying the Banach–Steinhaus Theorem (7.1.3 and 7.1.8(b)) we obtain that ∞ (B) is sequentially complete with respect to σ ( ∞ (B), 0 (B)) (= σ ( ∞ (B), ∞ (B)× )). From 9.4.6 we derive that ∞ (B) is perfect. Remark 9.5.10 The inclusion Fk ⊂ Ek× (see the proof of 9.5.9) can be strict. In fact, there are matrixes for which Fk is finite-dimensional (e.g. the one defined in the comments following 9.5.8), however Ek× is infinite-dimensional because it contains the unit vectors of K N .

9.5 K¨othe sequence spaces

359

Theorem 9.5.11 ( 0 (B), n( 0 (B), ∞ (B))) is a Fr´echet space for which the unit vectors of K N form an “orthogonal” base. The normal topology coincides with the projective one τp0 , i.e., n( 0 (B), ∞ (B)) is generated by the sequence of seminorms p1 ≤ p2 ≤ · · · defined in (9.15). Proof. It suffices to prove that n( 0 (B), ∞ (B)) = τp0 , i.e., that the topologies on 0 (B) induced by {py : y ∈ ∞ (B)} (with py as in (9.9)) and by {pk : k ∈ N} (with pk as in (9.15)) coincide. The rest follows from 9.5.4 and 9.5.6. Let k ∈ N and let Nk and Fk be as in (9.18). Let ρ ∈ K, 0 < |ρ| < 1. For each n ∈ Nk there is an integer in with |ρ|in +1 ≤ bnk ≤ |ρ|in . Put µn := ρ in +1 if n ∈ Nk ; µn := 0 otherwise. Then supn∈Nk |µn | /bnk ≤ 1, so y := (µ1 , µ2 , . . .) ∈ Fk ⊂ ∞ (B) and also pk ≤ |ρ|−1 py . Conversely, let y := (µ1 , µ2 , . . .) ∈ ∞ (B). There is a k such that y ∈ Fk , so that M := supn∈Nk |µn | /bnk < ∞ and µn = 0 for n ∈ Nk . Then py ≤ M pk . Now, putting together 9.5.7 and 9.5.11 we arrive at the result announced at the beginning of the section. Corollary 9.5.12 (Compare with 9.3.7) Let E be a Fr´echet space. Then E has an “orthogonal” base if and only if there exists a matrix B such that E is isomorphic to ( 0 (B), n( 0 (B), ∞ (B))). In the same way, the K¨othe spaces of 9.3.6 that appeared in 9.3.7 are described as follows. Corollary 9.5.13 A Fr´echet space E is a K¨othe space if and only if there exists a matrix B, with bnk > 0 for all k, n, such that E is isomorphic to ( 0 (B), n( 0 (B), ∞ (B))). Remark 9.5.14 It follows from 9.4.2(ii) and 9.5.9 that ∞ (B) is a perfect sequence space for which ∞ (B)× = 0 (B). For k ∈ N, let us equip the space Fk of (9.18) with the seminorm (λ1 , λ2 , . . .) → supn∈Nk |λn | /bnk (which is a norm if and only if Nk = N). The monotonicity condition imposed in (9.13) for the matrix B implies that F1 , F2 , . . . is an inductive sequence. Hence, we can consider on ∞ (B) = ∞ k∈N Fk the corresponding inductive topology τi . For each k, the inclusion Fk −→ ( ∞ (B), n( ∞ (B), 0 (B))) is easily seen to be continuous. Therefore, n( ∞ (B), 0 (B)) ≤ τi∞ and so this inductive topology is Hausdorff. In 9.7.2 we will show that for certain matrixes B the equality τi∞ = n( ∞ (B), 0 (B)) holds. In 11.4.6 we will see that this coincidence of topologies does not always occur.

360

Spaces with an “orthogonal” base

We finish the section by describing the class of the normable K¨othe sequence spaces. (For the definition of c0 (N, s), see the end of Subsection 2.5.2.) Theorem 9.5.15 The following are equivalent: (α) ( 0 (B), n( 0 (B), ∞ (B))) is normable. (β) There is a s : N−→(0, ∞) such that 0 (B) = c0 (N, s). (γ ) There is a s : N−→(0, ∞) such that 0 (B) = 0 (B), with B := (βnk )k,n , βnk = s(n) for all k, n. Proof. The equivalence (β) ⇐⇒ (γ ) is immediate. (α) =⇒(β). By normability there is a y = (µ1 , µ2 , . . .) ∈ ∞ (B) (= 0 (B)× , 9.5.9) such that py (defined as in (9.9)) is a norm on 0 (B) inducing the normal topology. In particular, |µn | = py (en ) = 0 for all n. Let F := {(δ1 , δ2 , . . .) ∈ K N : supn∈N |δn | / |µn | < ∞}. It is easily seen that F is a sequence space with F × = c0 (N, s), where s : N−→(0, ∞), n → |µn |. Now let us show that ∞ (B) = F . (Then, by using 9.5.9, 0 (B) = 0 (B)×× = × × ∞ (B) = F = c0 (N, s) and we have (β).) The inclusion F ⊂ ∞ (B) is clear because y ∈ ∞ (B). For the opposite inclusion, take z := (δ1 , δ2 , . . .) ∈ ∞ (B). There is a Cz > 0 such that pz ≤ Cz py , from which |δn | = pz (en ) ≤ Cz py (en ) = Cz |µn | for all n. So n → |δn | / |µn | is bounded, i.e., y ∈ F . (β) =⇒ (α). Let s : N−→(0, ∞) be as in (β). One can easily see that c0 (N, s)× = {(µ1 , µ2 , . . .) ∈ K N : supn∈N |µn | /s(n) < ∞} and that from this it follows that the normal topology on c0 (N, s) coincides with the one induced by the canonical supremum norm (λ1 , λ2 , . . .) → maxn∈N |λn | s(n). So we get (α).

9.6 Barrelledness, reflexivity, Montelness and nuclearity of sequence spaces The first goal of this section is to study when a K¨othe sequence space 0 (B), equipped with the normal topology, is barrelled, reflexive, (semi-)Montel or nuclear. In a second step we will discuss these properties for the K¨othe dual space ∞ (B). Recall that B := (bnk )k,n (k, n ∈ N) is an infinite matrix consisting of real numbers satisfying (9.13). The most interesting matrixes B are those for which bnk > 0 for all k, n.

9.6 Barrelledness, reflexivity, Montelness and nuclearity

361

From now on in this chapter B := (bnk )k,n (k, n ∈ N) is an infinite matrix consisting of real numbers satisfying 0 < bnk ≤ bnk+1 for all k, n. By 9.5.11, 0 (B) equipped with the normal topology is a Fr´echet space of countable type, hence polar. Recall that by 9.5.13 the class of all these K¨othe sequence spaces coincides with the class of all the K¨othe spaces introduced in 9.3.6. Hence, 0 (B) has a continuous norm and so it is not of finite type. As a direct application of 7.1.3 and 7.4.30 we have the following. Theorem 9.6.1 Let E := ( 0 (B), n( 0 (B), ∞ (B))). Then E is barrelled. If K is not spherically complete then E is reflexive. Next we describe Montelness (which coincides with reflexivity when K is spherically complete, see 8.4.22). 0

Theorem 9.6.2 Let E := ( equivalent:

(B), n(

0

(B),

∞ (B))).

Then the following are

(α) E is Montel. (β) E is semi-Montel. (γ ) For every k ∈ N and every subsequence n1 < n2 < · · · of {1, 2, . . .} there bh

exists h > k such that { bnki : i ∈ N} is unbounded. ni

Proof. For (α) ⇐⇒ (β) use 8.4.18 and the properties of E already stated above. (β) ⇐⇒ (γ ). By 9.5.11 the unit vectors e1 , e2 , . . . of K N form an “orthogonal” base of E and its topology is induced by the sequence of norms p1 ≤ p2 ≤ · · · defined in (9.15) for which we have that pk (en ) = bnk for all k, n. Then the equivalence of (β) and (γ ) is an immediate consequence of 9.2.15. Now we describe nuclearity. Theorem 9.6.3 Let E := ( and only if

0

(B), n(

0

(B),

∞ (B))).

for each k ∈ N there exists h > k such that limn

Then E is nuclear if bnk bnh

= 0.

(9.19)

Proof. The reasoning carried out for (β) ⇐⇒ (γ ) of 9.6.2 works here, except that now we have to apply 9.2.16 (instead of 9.2.15) to arrive at our conclusion. As we have announced at the beginning of the section the next purpose is to study under which conditions a K¨othe dual space ∞ (B), equipped with the normal topology, is barrelled, reflexive, (semi-)Montel or nuclear.

362

Spaces with an “orthogonal” base

We know (9.5.14) that ∞ (B) is a perfect sequence space for which × 0 (B). As a direct application of 9.4.3 and 9.4.6 we obtain that ∞ (B) = 0 (B)), is of countable ∞ (B), equipped with the normal topology n( ∞ (B), type (hence polar) and complete. Also, we have the following. Theorem 9.6.4 Let F := ( hence semi-Montel.

∞ (B), n(

∞ (B),

0

(B))). Then F is nuclear,

Proof. Let us prove that F is nuclear (then semi-Montelness follows from 8.5.2). The unit vectors e1 , e2 , . . . of K N form an “orthogonal” base of F (9.4.3) and the normal topology of F is induced by the family of seminorms {py : y ∈ 0 (B)} where, for each y = (µ1 , µ2 , . . .) ∈ 0 (B), py is defined by py (x) := max |λn µn | n∈N

(x = (λ1 , λ2 , . . .) ∈

∞ (B)).

By 9.2.16 we get nuclearity as soon as we prove that for each y = (µ1 , µ2 , . . .) ∈ 0 (B) there exists a z = (δ1 , δ2 , . . .) ∈ 0 (B), with |µn | ≤ |δn | for all n, and non-negative ε1 , ε2 , . . . with εn → 0, such that py (en ) ≤ εn pz (en ) for all n. So take such a y = (µ1 , µ2 , . . .). Choose α1 , α2 , . . . ∈ K with 1 ≥ |α1 | > |α2 | > · · · and limk αk = 0. Since limn |µn | bnk = 0 for all k, there exist 1 < n1 < n2 < . . . such that |µn | bnk < |αk |2 for all n ≥ nk , k ∈ N. For each n ∈ N, put δn := βn µn , εn := |βn |−1 , with 1 if 1 ≤ n < n1 βn := 1 if nk ≤ n < nk+1 . αk One verifies that |βn | ≥ 1 (so |µn | ≤ |δn |) for all n, that limn |βn | = ∞ (so limn εn = 0) and that z := (δ1 , δ2 , . . . , . . .) ∈ 0 (B). Also, py (en ) = |µn | = εn |δn | = εn pz (en ) for all n,

and we are done.

Next we show that any of the remaining properties ((polar) barrelledness, reflexivity, Montelness) of ∞ (B) is equivalent to (semi-)Montelness of 0 (B), which was already described in 9.6.2. Theorem 9.6.5 Let E be as in 9.6.2, let F := ( Then properties (α)–(γ ) of 9.6.2 are equivalent to:

∞ (B), n(

∞ (B),

0

(B))).

(δ) F is (polarly) barrelled. (ε) F is reflexive. (ζ ) F is Montel. Proof. (ε) ⇐⇒ (ζ ) is clear because F is semi-Montel (9.6.4) and (ε) =⇒ (δ) follows from 7.4.13. Note that, since F is of countable type, barrelledness and polar barrelledness coincide (7.1.9(ii)).

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363

(δ) =⇒ (ε). F is polar and complete (see the comments before 9.6.4) and, since F is semi-Montel, every closed bounded subset of F is a complete compactoid, by 8.4.5. Then apply 7.4.19. (α) ⇐⇒ (ζ ). We prove (α) =⇒ (ζ ); the same reasoning works for the converse by interchanging E and F . Montelness of E implies that every bounded subset of E is a compactoid (8.4.5). By 9.4.5 and 9.4.9 we have that F is isomorphic to the strong dual of E, so that F is Montel by 8.4.19. Remark 9.6.6 Let E and F be as in 9.6.5. We know that E is Fr´echet and that F is complete. But F is not metrizable. In fact, metrizability of F implies that F is Fr´echet, hence barrelled by 7.1.3, so E is Montel by 9.6.5. With the same arguments as in (α) =⇒ (ζ ) of this last theorem we obtain that the strong dual of E is isomorphic to F , which is metrizable. Then applying 7.6.7 we have that E is normable which, by 8.4.4(i), is an impossibility because E is semi-Montel and infinite-dimensional.

9.7 Spaces of analytic functions As we announced in the introduction of the chapter, in this section we prove the properties of the spaces of analytic functions that we postponed in the past (see 9.7.5). We will get our purpose by showing that the spaces of analytic functions are isomorphic to either certain K¨othe sequence spaces or to the corresponding K¨othe duals, both equipped with the normal topologies. When we work with these spaces it is convenient to use infinite matrixes (bnk )k,n with k ∈ N, n ∈ N ∪ {0}, thus allowing n to be 0. All the concepts and results given in Sections 9.5 and 9.6 extend naturally to these matrixes. For r ∈ (0, ∞), let A(r) be the space of analytic functions on B(0, r − ) introduced in 3.7.25. We saw there that A(r) is isomorphic to the space := {(λ0 , λ1 , . . .) ∈ K N∪{0} : limn |λn | s n = 0 for each 0 < s < r}, equipped with the locally convex topology induced by the family of norms (λ0 , λ1 , . . .) → maxn∈N∪{0} |λn | s n , with 0 < s < r. We have that = 0 (B) = {(λ0, λ1 , . . .) ∈ K N∪{0} : limn |λn | bnk = 0 for all k}, with B := kr n , k ∈ N, n ∈ N ∪ {0}. By 9.5.11, the topology of coin(bnk )k,n , bnk := k+1 cides with the normal one of 0 (B). Therefore, we arrive at the following. 0 Theorem 9.7.1 Let r ∈ (0, ∞). Then

kA(r) is isomorphic to ( (B), n( r n k k ∞ (B))), with B := (bn )k,n , bn := k+1 , k ∈ N, n ∈ N ∪ {0}.

0

(B),

Now, for r ∈ (0, ∞), let A† ( 1r ) be the space of overconvergent power series on B(0, 1r ), see 3.7.27. We showed there that A† ( 1r ) is algebraically

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isomorphic to the space := {(λ0 , λ1 , . . .) ∈ K N∪{0} : n → |λn | s n is = ∞ (B) = {(λ0 , λ1 , . . .) ∈ bounded for some s > 1r }. We have that K N∪{0} : n → |λn | /bnk is bounded for some k}, with B = (bnk )k,n as in 9.7.1. The algebraic isomorphism between A† ( 1r ) and ∞ (B) will be a homeomorphism when we equip ∞ (B) with the corresponding inductive topology (see 9.5.14). We are lucky that this B is an example of a matrix satisfying (9.19) and one verifies that for this kind of matrixes the inductive and the normal topology on ∞ (B) coincide, as we prove below. Let us recall some notations, see 9.5.14. Let B := (bnk )k,n be an infinite matrix (as it is usual, see the beginning of Section 9.6, when we work with general K¨othe sequence spaces we assume k, n ∈ N and 0 < bnk ≤ bnk+1 for all k, n). For each k, let Fk := {(λ1 , λ2 , . . .) ∈ K N : supn∈N |λn | /bnk < ∞} equipped with the norm (λ1 , λ2 , . . .) → supn∈N |λn | /bnk and let τi∞ be the inductive topology on ∞ (B) = k∈N Fk . Then we have the following. Theorem 9.7.2 Suppose the matrix B satisfies (9.19). Then τi∞ = n( ∞ (B), 0 (B)). Proof. The inequality n( ∞ (B), 0 (B)) ≤ τi∞ holds for any matrix B, see 9.5.14. Let us prove that τi∞ ≤ n( ∞ (B), 0 (B)). We first see that every x = ∞ (λ1 , λ2 , . . .) ∈ ∞ (B) can be written as x = ∞ n=1 λn en in ( ∞ (B), τi ), N where e1 , e2 , . . . are the unit vectors of K . For that observe that given such an x there is a k such that x ∈ Fk . If h > k is as in (9.19) one verifies Fk ⊂ Gh := {(λ1 , λ2 , . . .) ∈ K N : limn |λn | /bnh = 0}. Hence x ∈ Gh and so x = ∞ n=1 λn en in Gh , which is equipped with the norm inherited from Fh . Since the inclusion Gh −→ ( ∞ (B), τi∞ ) is continuous, we obtain that ∞ x= ∞ n=1 λn en in ( ∞ (B), τi ), and we are done. Now, let p be a τi∞ -continuous seminorm on ∞ (B) and let us show that p is n( ∞ (B), 0 (B))-continuous. Let J := {n ∈ N : p(en ) = 0}, let ρ ∈ K, 0 < |ρ| < 1. For each n ∈ J we choose an integer in for which |ρ|in +1 ≤ p(en ) ≤ |ρ|in . Define y := (µ1 , µ2 , . . .) ∈ K N by µn := 0 if n ∈ / J , µn := ρ in +1 otherwise. Let x = (λ1 , λ2 , . . .) ∈ ∞ (B). We have ∞ |λn | |µn | ≤ |λn | p(en ) for all n and since x = ∞ n=1 λn en in ( ∞ (B), τi ), limn p(λn en ) = 0, so that limn |λn µn | = 0. This means that y ∈ ∞ (B)× (= 0 (B)). Also, p(x) ≤ maxn∈N |λn | p(en ) ≤ |ρ|−1 maxn∈N |λn | |µn | for all x = (λ1 , λ2 , . . .) ∈ ∞ (B), hence p is n( ∞ (B), 0 (B))-continuous. The above leads to the following conclusion. Theorem 9.7.3 Let r ∈ (0, ∞). Then A† ( 1r ) is isomorphic to ( ∞ (B), kr n n( ∞ (B), 0 (B))), with B := (bnk )k,n , bnk := k+1 , k ∈ N, n ∈ N ∪ {0}.

9.7 Spaces of analytic functions

365

With similar arguments one can easily obtain the following descriptions for the space A(∞) of entire functions on K (3.7.25) and for the space A† (0) of germs of analytic functions at 0 (3.7.27). Theorem 9.7.4 The spaces A(∞) and A† (0) are respectively isomorphic to ( 0 (B), n( 0 (B), ∞ (B))) and to ( ∞ (B), n( ∞ (B), 0 (B))), with B := (bnk )k,n , bnk := k n , k ∈ N, n ∈ N ∪ {0}. We are in the position of giving the topological properties of the spaces of analytic functions. Recall that we know that A(r) (r ∈ (0, ∞]) and A† (r) (r ∈ [0, ∞)) are strictly of countable type (4.3.14) and barrelled (Subsection 7.2.1) but not of finite type (5.3.15) and also that A(r) is Fr´echet, not normable (3.7.26). Now, as an application of the previous results we deduce that they also have the following properties, announced previously. Corollary 9.7.5 (i) For each r ∈ (0, ∞], A(r) is nuclear and Montel. (ii) For each r ∈ [0, ∞), A† (r) is complete, nuclear and Montel. Proof. (i). By 9.7.1 (for the case r > 0) and 9.7.4 (for the case r = ∞), A(r) is isomorphic to a ( 0 (B), n( 0 (B), ∞ (B))), where the matrix B satisfies (9.19). It follows from 9.6.3 that A(r) is nuclear, hence semi-Montel by 8.5.2, then Montel by 9.6.2. (ii). By 9.7.3 (applied to 1/r, for the case r > 0) and 9.7.4 (for the case r = 0), A† (r) is isomorphic to a ( ∞ (B), n( ∞ (B), 0 (B))), so it is complete (see the comments before 9.6.4) and nuclear (9.6.4). Also, since the matrix B satisfies (9.19) we have that the corresponding K¨othe space 0 (B), equipped with the normal topology, is nuclear by 9.6.3, hence semi-Montel by 8.5.2. It follows from 9.6.5 that A† (r) is Montel. By using the descriptions of the spaces of analytic functions as sequence spaces (9.7.1, 9.7.3, 9.7.4) we obtain the following. Corollary 9.7.6 (i) Let r ∈ (0, ∞). Then A(r) and A† ( 1r ) are each other’s strong dual. (ii) A(∞) and A† (0) are each other’s strong dual. Proof. We only prove (i) as the proof of (ii) runs identically. A(r) and A† ( 1r ) are identified with ( 0 (B), n( 0 (B), ∞ (B))) and ( ∞ (B), n( ∞ (B), 0 (B))) respectively, where B is as in 9.7.1. These two sequence spaces are perfect and semi-Montel, and 0 (B)× = ∞ (B). Then apply 9.4.5 and 9.4.9 to them in order to get (i).

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Remarks 9.7.7 (a) Remark 9.6.6, together with 9.7.3 and 9.7.4 (the part for A† (0)), tell us that A† (r) is metrizable for no r ∈ [0, ∞). (b) The space C ∞ (Zp ), equipped with the topology τc∞ introduced in 3.7.36, and its strong dual C ∞ (Zp )b , can also be described as a K¨othe sequence space and the corresponding K¨othe dual respectively, see the chapter notes for details.

9.8 Basic counterexamples As we promised in the introduction, in this section we construct a space which will turn out to be a kind of “universal” counterexample. Example 9.8.1 There exists a Fr´echet Montel (K¨othe) space E of countable type having a quotient isomorphic to c0 . Proof. To construct E we will use the framework of K¨othe spaces, in other words, E will be a 0 (B), where B is a matrix with the right properties. Let N1 , N2 , . . . be a sequence of pairwise disjoint infinite subsets of N with i∈N Ni = N. For i ∈ N and n ∈ Ni , put bnk :=

ki ki n

if k ≤ i if k > i.

Let B := (bnk )n,k (k, n ∈ N). Clearly we have 0 < bnk ≤ bnk+1 for all k, n. Let E := ( 0 (B), n( 0 (B), ∞ (B))). By 9.5.11 E is Fr´echet and of countable type. To see that E is Montel we will apply 9.6.2. For that, let k ∈ N and let n1 < n2 < · · · be a subsequence of {1, 2, . . .}. Put N0 := {n1 , n2 , . . .} and, for each i, Mi := N0 ∩ Ni . If the set Mi is infinite for some i, then take any h > max(i, k). For n ∈ Mi we have in h bnh if k ≤ i ki

= in h k bn if k > i. k bh

Thus, no matter whether k ≤ i or k > i, { bnk : n ∈ Mi } is unbounded and we n are done. Now suppose that the set Mi is finite and non-empty for each i. Then there exist two strictly increasing sequences q1 , q2 , . . . and m1 , m2 , . . . in N such that qi ∈ Mmi for all i. Take h := k + 1. There is an i such that k + 1 ≤ mi

9.8 Basic counterexamples

367

(hence, for j ≥ i, k + 1 ≤ mj ). Thus, bqhj bqkj

=

k+1 k

mj for all j ≥ i.

bqh

Therefore, { bkj : j ≥ i} is unbounded and again we are done. qj

Next we show that E has a quotient that is isomorphic to c0 . Since E is Fr´echet, by the Open Mapping Theorem 3.5.10 it suffices to construct a continuous linear surjection T : E−→c0 . It is easy to see that there is a partition S1 , S2 , . . . of N such that Sm ∩ Ni = ∅ for all m, i ∈ N. (In fact, with the natural order on each Ni , we can take as Sm the set formed by the m-th elements of the Ni ). Now, let x := (λ1 , λ2 , . . .) ∈ E. Since bn1 = 1 for all n we obtain that limn |λn | = 0, so limm supn∈Sm |λn | = 0. Therefore, the formula T (x) := (g1 (x), g2 (x), . . .), with gm (x) := n∈Sm λn leads to a well-defined linear map T : E−→c0 . Also, max |gm (x)| ≤ max |λn | = max |λn | bn1 , m∈N

n∈N

n∈N

from which continuity of T follows. It remains to be proved that T is surjective. Let y := (µ1 , µ2 , . . .) ∈ c0 . As bni = i i for all n ∈ Ni , i ∈ N, there is an increasing sequence r1 < r2 < · · · in N with r1 = 1 such that limi supj ≥ri µj supn∈Ni bni = 0. Let sm ∈ Sm ∩ Ni for ri ≤ m < ri+1 , i ∈ N. Let k ∈ k and ri ≤ m < ri+1 then |µm | bskm ≤ |µm | bsi m ≤ N. If i > i supj ≥ri µj supn∈Ni bn . Thus, limm |µm | bskm = 0 for all k. Hence x := (λ1 , λ2 , . . .) ∈ K N defined by µm if n = sm for some m λn := 0 otherwise is an element of E for which T (x) = y.

Now we can present the counterexamples promised earlier in this book. Between brackets we indicate the places where they were announced. Counterexamples 9.8.2 (i) (3.6.6(c)) There exists a Fr´echet space E, a closed subspace D of E and a bounded subset Y of E/D that is not the image under the quotient map π : E−→E/D of any bounded subset of E. (ii) (7.4.28) Let K be spherically complete. Then there exists a reflexive Fr´echet space E of countable type and a closed subspace D of E such that the quotient E/D is not reflexive.

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(iii) (8.1.8) There exists a locally convex space E, an equicontinuous sequence f1 , f2 , . . . in E for which fn → 0 in Eb , but such that f1 , f2 , . . . does not tend locally to 0. (iv) (8.3.5) There exists a locally convex space E and an operator in L(E, c0 ) that is compactifying but not compactoid. (v) (8.4.27(d)) There exists a Fr´echet Montel space E of countable type and a closed subspace D of E such that the quotient E/D is not semi-Montel. (vi) (8.5.4(a)) There exists a Fr´echet Montel space E of countable type that is not nuclear. (vii) (8.4.27(d)) There exists a Montel space F having a closed subspace that is not Montel. In fact, one can choose for (i)–(vi) the space E of our previous example (which is Fr´echet-Montel and has an “orthogonal” base), whereas for F in (vii) we may take Eb (which is Montel, nuclear, has an “orthogonal” base, and is nonmetrizable). Proof. Notice that (vi) implies (iv) (8.4.13 and 8.5.1) and that (iv) implies (iii) (8.1.9(ii) and 8.3.9), so we will prove (i), (ii), (v), (vi) and (vii). Let E be as in 9.8.1, let D be a closed subspace of E such that E/D ∼ c0 . (i). The unit ball Y of E/D is not a compactoid (3.8.6). If Y = π (X) for some X ⊂ E then X is not a compactoid, hence not bounded as E is semi-Montel and by 8.4.5. (ii). E/D ∼ c0 is not reflexive (7.4.20). (v). E/D ∼ c0 is not semi-Montel (8.4.4(i)). (vi). Apply 8.5.6(i) and 8.5.7(v). (vii). Let F := Eb , with E as above. Then F is Montel (8.4.19), nuclear (8.6.3), has an “orthogonal” base (9.2.13) and is non-metrizable (7.6.7). The subspace D ⊥ := {f ∈ E : f (d) = 0 for all d ∈ D} is obviously closed in F . To conclude that D ⊥ is not Montel it suffices, by 8.4.18, to show that D ⊥ is not polarly barrelled. The adjoint π : (E/D)b −→F of the quotient map E−→E/D sends (E/D) bijectively onto D ⊥ . Of course, π is continuous, but it cannot be a homeomorphism as (E/D)b ∼ ∞ (2.5.11), an infinite-dimensional Banach space, and D ⊥ , being a subspace of F , is nuclear. Therefore, there exists a continuous seminorm p on (E/D)b (which we may assume to have the form h → sup{|h(c)| : c ∈ C} (h ∈ (E/D) ), for some bounded set C ⊂ E/D) such that f → p((π )−1 (f )) (f ∈ D ⊥ ) is not b(E , E)-continuous. The latter means that f → sup{|f (x)| : x ∈ π −1 (C)} (f ∈ D ⊥ ) is not b(E , E)-continuous. But

9.9 Notes

369

f → f (x) (f ∈ F ) is in F for all x ∈ E. It follows from 7.1.7 that D ⊥ is not polarly barrelled.

9.9 Notes Orthogonal bases in locally convex spaces were introduced by De Grande-De Kimpe in [65]. Other kinds of bases, among which the weak Schauder bases, were studied in [81], [104] and [106]. We have seen in 4.2.15 that every Fr´echet space of countable type is iso´ proved that a Fr´echet space morphic to a closed subspace of c0N . In [218] Sliwa E with a base is isomorphic to a complemented subspace of c0N if and only if E is isomorphic to one of the spaces c0 , c0 × K N , K N , c0N . However, he also proved the existence of a Fr´echet space F with a base such that every Fr´echet space with a base is isomorphic to a complemented subspace of F . ´ With respect to the converse of 9.2.2, see 9.2.3, Sliwa showed the following. Theorem 9.9.1 ([213], Corollary 7, Proposition 8) Let E be a metrizable space with an “orthogonal” base. Then the following are equivalent: (α) Every dense subspace of E has an “orthogonal” base. (β) Either E has a continuous norm or E is of finite type. ´ As for other stability properties Sliwa proved the next results, see 9.2.5. Theorem 9.9.2 ([214], Theorem 7, Proposition 9) Let E be an infinitedimensional Fr´echet space. Then the following are equivalent: (α) Every closed subspace of E has a base. (β) E is isomorphic to one of the spaces c0 , c0 × K N , K N . Theorem 9.9.3 ([219], Corollary 7, Proposition 9) Let E be an infinitedimensional Fr´echet space. Then the following are equivalent: (α) Every quotient of E has a base. (β) E is isomorphic to one of the spaces c0 , c0 × K N , K N , c0N . Corollaries 9.2.8 and 9.2.9(i) were given in [80] under the assumption of sequential completeness of the space. The problem whether every Fr´echet space of countable type has an “orthogonal” base (see the beginning of Section 9.3) was posed in [79], and was solved ´ by Sliwa in [212] by constructing even a nuclear Fr´echet space E without a base.

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Remark 9.9.4 Applying 9.2.12 to this E we conclude that there is a compactoid in E that is not contained in the closed absolutely convex hull of any “orthogonal” sequence tending to 0. This is in contrast to 9.2.9(i), and was announced in 3.9.17(a). The proof of 9.3.2 is taken from [211]. As we announced in 9.3.4, any metrizable space of finite type has an “orthogonal” base (a proof can be found in [79]). By ´ using this fact Sliwa obtained in [211] that any infinite-dimensional metrizable space has an “orthogonal” sequence. He also made a further study of the existence of “orthogonal” sequences having certain special properties in [216] and [217]. ´ In [215] Sliwa proved the following characterizations. Theorem 9.9.5 (i) A Fr´echet space with a Schauder base e1 , e2 , . . . is Montel if and only if e1 , e2 , . . . has no subsequence whose closed linear hull is isomorphic to c0 . (ii) A Fr´echet space of countable type is normable (reflexive, Montel, nuclear) if and only if each of its closed subspaces with a Schauder base is normable (reflexive, Montel, nuclear). A study of non-Archimedean sequence spaces was first attempted by Monna in [157], where he developed the basic set-theoretical results about perfect sequence spaces and some of their topological properties. A more general theory of non-Archimedean perfect sequence spaces was constructed by De Grande-De Kimpe in [63] where, among other things, she gave a version of 9.4.7(i) for bounded c-compact sets; which she extended in [73] for compactoids. This was obtained in a direct way, different from ours. In [65] she partially proved 9.4.10; her proof was completed in [173]. Also, in [64], [67] and [71] she showed how certain spaces of operators can be represented as some generalized sequence spaces. This she applied to obtain some lifting properties for operators. K¨othe sequence spaces were also introduced by De Grande-De Kimpe, in 1982, see [68], in which paper one can find 9.6.3 and 9.6.4, where she used only basic techniques for their proofs. For some extensions of the notion of a K¨othe sequence space, see [117], [118] and [119]. ´ In [220] Sliwa proved that a Fr´echet space of countable type has a K¨othe quotient if and only if it is not of finite type. In [221] he also proved the existence of a K¨othe Montel space E such that every Fr´echet space of countable type is isomorphic to a quotient of E.

9.9 Notes

371

K¨othe spaces play a role in theorems related to Physics, see e.g. [75] and [138]. As a case in point, the spaces A(∞) and A† (0) are needed in the definition of a p-adic Laplace Transform, [82]. The topological properties of these spaces (9.7.5), especially their reflexivity, are crucial for the definition of this transform. Also, the descriptions of the spaces of analytic functions in terms of K¨othe spaces, given in 9.7.1, 9.7.3 and 9.7.4 were proved in [68]. For another example, let K ⊃ Qp and consider the space C ∞ (Zp ), equipped with the topology τc∞ introduced in 3.7.36. As we announced in 9.7.7(b), this space as well as its strong dual C ∞ (Zp )b , can be described as a K¨othe sequence space and the corresponding K¨othe dual respectively. These descriptions were given in [83] and applied for the definition of a p-adic Fourier Transform. To arrive at these descriptions we need results on the Mahler base e0 , e1 , . . . in the Banach space C(Zp ). We refer to the notes to Chapter 2 for the definition of the Mahler base, and to [195] for the proofs of those results. The C ∞ -functions can be characterized by their Mahler coefficients as fol ∞ lows. A function f = ∞ n=0 λn en ∈ C(Zp ) belongs to C (Zp ) if and only k if limn |λn | n = 0 for all k ∈ N. Also, for each k, the canonical norm on the Banach space C k (Zp ) is equivalent to the norm f → maxn∈N∪{0} |λn | nk . It fol lows from 9.5.11 that the map ∞ n=0 λn en → (λ0 , λ1 , . . .) is a bijective linear ∞ homeomorphism of C (Zp ) onto the K¨othe sequence space {(λ0 , λ1 , . . .) ∈ K N∪{0} : limn |λn | nk = 0 for each k} (called the space of rapidly decreasing sequences), when we equip this last one with the normal topology. So we have that C ∞ (Zp ) is isomorphic to ( with B := The K¨othe dual of this ∞ (B)

(bnk )k,n , bnk 0

0

(B), n(

0

(B),

∞ (B))),

:= n , k ∈ N, n ∈ N ∪ {0}. k

(B) is

= {(λ0 , λ1 , . . .) ∈ K N∪{0} : n → |λn |

1 nk

is bounded for some k}.

One verifies that C ∞ (Zp )b (called the space of distributions) is isomorphic to ( ∞ (B), n( ∞ (B), 0 (B))). In fact, it follows from the above and from 9.4.5 applied to := 0 (B), ∞ ∞ that the map y : (µ1 , µ2 , . . .) → ϕy , ϕy ( n=0 λn en ) := n=0 λn µn , is an algebraic isomorphism ∞ (B)−→ C ∞ (Zp ) . If, by using this map, we transfer the normal topology of ∞ (B) to C ∞ (Zp ) , we obtain the topology τ on C ∞ (Zp ) of uniform convergence on the compactoid subsets of C ∞ (Zp ) (by

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applying 9.4.9). Also, C ∞ (Zp ) is nuclear (8.7.9), hence every bounded subset of C ∞ (Zp ) is a compactoid (8.4.5, 8.5.2). Thus, τ is the strong topology on C ∞ (Zp ) and we are done. Our proof of 9.8.1, including the choice of the matrix B, was inspired by ´ Example 1.9 of [215] and Theorem 2 of [220]; in the last result Sliwa even proved that any non-nuclear Fr´echet space of countable type has a quotient isomorphic to c0 .

10 Tensor products

The algebraic definition and properties of the tensor product of vector spaces over any field are recalled in Section 10.1. Assuming the scalar field to be valued allows for considering the tensor product of two absolutely convex sets (10.2.1) and for the usual tensor products p ⊗π q, p ⊗ε q of seminorms p, q on the defining spaces (10.2.3). A few technical but important facts are proved in Section 10.2 to be used in the next section, where we define the projective tensor product E ⊗π F of locally convex spaces E and F (10.3.2, 10.3.5). Notice that, for polar spaces E, F , the natural ε-topology on E ⊗ F coincides with π (10.3.15), a fact unheard of in “classical” theory. We proceed to prove that if E, F are Hausdorff (normable, metrizable, (strictly) of countable type, of finite type, polar, dual-separating, having an “orthogonal” base, nuclear) then so is E ⊗π F (10.3.11, 10.3.13, 10.3.14, 10.3.16, 10.4.4). Much more difficult is the proof of 10.4.14 stating that the projective tensor product of two metrizable semi-Montel spaces is again semiMontel. If the spaces are Fr´echet–Montel then so is their completed tensor ˆ π F (10.4.17, 10.4.19). product E ⊗ For metrizable E, F we discuss splitting of bounded (compactoid) subsets of ˆ π F into a tensor product of bounded (compactoid) sets in E, F respectively E⊗ (10.4.20, 10.4.21). As an application of the theory of this chapter we introduce in Section 10.5 vector valued continuous functions. In fact, for a non-empty zero-dimensional Hausdorff topological space X and a locally convex space E we consider the space C(X−→E) of all continuous maps X−→E, with the topology of compact convergence. In 10.5.3 we construct a natural map C(X) ⊗π E−→C(X−→E) ˆ π E−→C(X−→E) if E and extend it in 10.5.4 to an isomorphism C(X)⊗ is complete and X is a k0 -space. We discuss in 10.5.7 the isomorphism ˆ π C(Y )−→C(X × Y ). Connections between properties of C(X−→E) C(X)⊗ on one hand, and of C(X), E on the other, are given in 10.5.10−10.5.14. 373

374

Tensor products

Finally, the method of “complexification” of a real vector space is taken over in Section 10.6, where we make the observation that, for a locally convex space E and a valued field L ⊃ K, the space E ⊗π L is, in a natural way, a locally convex space over L.

10.1 The algebraic tensor product In this section we recall, for convenience, the necessary definitions and facts concerning tensor products of vector spaces. Most proofs are omitted; we trust the reader to be able to supply the details. Throughout Sections 10.1 and 10.2, E, E1 , E2 , F, F1 , F2 are vector spaces over K . In Section 10.1 we will not use the fact that K has a valuation. Definition 10.1.1 Let G be a K-vector space. A map B : E × F −→G is called bilinear if for each a ∈ E, b ∈ F the maps x → B(x, b) and y → B(a, y) are linear. Remark 10.1.2 Warning. Neither the kernel nor the image of a bilinear map need to be a subspace. Bilinear maps are, apart from a trivial case, not injective. Definition 10.1.3 A tensor product of E and F is a pair, consisting of a K-vector space E ⊗ F and a bilinear map ⊗ : E × F −→E ⊗ F such that the following Universal Property holds. For each K-vector space G and each bilinear map B : E × F −→G there is a unique linear map T : E ⊗ F −→G such that the diagram E × FF FF FF F B FFF #

⊗

G

/ E⊗F x xx xx x x T {x x

commutes. Tensor products are unique in the following sense. Theorem 10.1.4 Let (E ⊗ F, ⊗) and (E ⊗1 F, ⊗1 ) be tensor products of E and F . Then there exists a unique linear bijection T : E ⊗ F −→E ⊗1 F such that T ◦ ⊗ = ⊗1 . The next theorem furnishes existence.

10.1 The algebraic tensor product

375

Theorem 10.1.5 Let B1 := {ei : i ∈ I } and B2 := {fj : j ∈ J } be algebraic bases of E, F respectively. Let E ⊗ F := K (B1 ×B2 ) (here we violate 3.4.15 by not considering a topology on K (B1 ×B2 ) ), and let ⊗ : E × F −→E ⊗ F be defined by the formula λi ei , µj fj ) = λi µj (ei , fj ) ⊗( i

j

i,j

(here λi , µj ∈ K, all sums are finite). Then (E ⊗ F, ⊗) is a tensor product of E and F . The previous theorems enable us to speak from now on about the tensor product E ⊗ F of E and F . The image under ⊗ of (x, y) ∈ E × F will henceforth be denoted by x ⊗ y, that is, x ⊗ y := ⊗(x, y). Theorem 10.1.6 (i) Let {ei : i ∈ I } and {fj : j ∈ J } be algebraic bases of E, F respectively. Then {ei ⊗ fj : i ∈ I, j ∈ J } is an algebraic base of E ⊗ F . (ii) Im ⊗ generates E ⊗ F as an additive group. In other words, each ele ment of E ⊗ F can be written as ni=1 xi ⊗ yi for some n ∈ N, xi ∈ E, yi ∈ F . (iii) Let z ∈ E ⊗ F be written as in (ii). Let z1 , . . . , zm be an algebraic base of [y , . . . , yn ]. Then there are x1 , . . . , xm ∈ [x1 , . . . , xn ] such that z = m 1 j =1 xj ⊗ zj . (iv) Let x1 , . . . , xn ∈ E, y1 , . . . , yn ∈ F . Let y1 , . . . , yn be linearly independent and xi = 0 for all i. Then x1 ⊗ y1 , . . . , xn ⊗ yn are linearly independent. In particular, if for some x ∈ E, y ∈ F we have x ⊗ y = 0 then x = 0 or y = 0. Proof. The construction of 10.1.5, together with uniqueness, yields (i); (ii) follows from (i), and (iii) is straightforward. To prove (iv), let λ1 , . . . , λn ∈ K be such that ni=1 λi xi ⊗ yi = 0. Let j ∈ {1, . . . , n}; we prove that λj = 0. By linear independence there exists an f ∈ F ∗ such that f (yj ) = 1, f (yi ) = 0 for i = j . The bilinear map (x, y) → f (y) x induces a linear map T : E ⊗ F −→E satisfying T (x ⊗ y) = f (y) x (x ∈ E, y ∈ F ). We have 0 = T ( ni=1 λi xi ⊗ n yi ) = i=1 λi f (yi ) xi = λj xj , and since xj = 0 we have λj = 0. Remarks 10.1.7 (a) The bilinear map E × F −→F ⊗ E given by (x, y) → y ⊗ x induces a linear map ω : E ⊗ F −→F ⊗ E (called the twist map) satisfying

376

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ω(x ⊗ y) = y ⊗ x for all x ∈ E, y ∈ F . It is easily seen that ω is a bijection. Thus, in many true statements (such as (iii) and (iv) above) one may interchange the role of E and F obtaining new valid properties. We will refer to this procedure as to “using the twist map”. (b) Let b ∈ F , b = 0. By (iv) above the linear map x → x ⊗ b is an injection E−→E ⊗ F . In particular, the map x → x ⊗ 1 is a linear bijection E−→E ⊗ K. Henceforth we will identify E ⊗ K with E in the above manner. By using the twist map we obtain for a ∈ E, a = 0 a linear injection y → a ⊗ y of F into E ⊗ F . Similarly, K ⊗ E can naturally be identified with E. Next, we consider linear maps between tensor products. Theorem 10.1.8 Let T1 : E1 −→F1 , T2 : E2 −→F2 be linear maps. Then, there is a unique linear map T1 ⊗ T2 : E1 ⊗ E2 −→F1 ⊗ F2 satisfying (T1 ⊗ T2 )(x ⊗ y) = T1 (x) ⊗ T2 (y) for all x ∈ E1 , y ∈ E2 . Further, if T1 , T2 are injective (surjective) then so is T1 ⊗ T2 . Proof. All is straightforward, except perhaps injectivity. So suppose T1 , T2 are injective, let z ∈ E1 ⊗ E2 , z = 0, (T1 ⊗ T2 )(z) = 0; we arrive at a contradiction. By 10.1.6(iii), z = ni=1 xi ⊗ yi , where y1 , . . . , yn are linearly independent and all xi = 0. Then T2 (y1 ), . . . , T2 (yn ) are linearly independent and all T1 (xi ) = 0. By 10.1.6(iv) T1 (x1 ) ⊗ T2 (y1 ), . . . , T1 (xn ) ⊗ T2 (yn ) are linearly independent, conflicting the fact that their sum is 0. Remarks 10.1.9 (a) It is possible to interpret T1 ⊗ T2 as an element of the tensor product of spaces of linear maps, but we do not need this view here. (b) Let b ∈ F , b = 0, let f ∈ F ∗ be such that f (b) = 1. Then 1 ⊗ f (where 1 stands for the identity map) sends E ⊗ F onto E ⊗ K = E. We have (1 ⊗ f )(x ⊗ b) = x for all x ∈ E. A similar story goes for a ∈ E, a = 0 and a g ∈ E ∗ such that g(a) = 1, by using the twist map. (See 10.1.7.) (c) 10.1.8 is of special interest in case T1 , T2 are inclusions. It permits us to identify the tensor product E1 ⊗ E2 with the subspace [x ⊗ y ∈ F1 ⊗ F2 : x ∈ E1 , y ∈ E2 ] of F1 ⊗ F2 ; we will use this identification in the sequel without warning. We now describe Ker (T1 ⊗ T2 ). Theorem 10.1.10 Let T1 : E1 −→F1 , T2 : E2 −→F2 be linear maps. Then Ker (T1 ⊗ T2 ) = Ker T1 ⊗ E2 + E1 ⊗ Ker T2 .

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377

Proof. Put H := Ker T1 ⊗ E2 + E1 ⊗ Ker T2 . Clearly H ⊂ Ker (T1 ⊗ T2 ). To prove the converse, let D1 (D2 ) be an algebraic complement of Ker T1 (Ker T2 ) in E1 (E2 ). Let z ∈ Ker (T1 ⊗ T2 ); we prove z ∈ H . Write z = ni=1 xi ⊗ yi , where xi ∈ E1 , yi ∈ E2 . Using xi = ti + ui

(ti ∈ D1 , ui ∈ KerT1 )

yi = vi + wi (vi ∈ D2 , wi ∈ Ker T2 ) we obtain that z = ni=1 ti ⊗ vi + h for some h ∈ H . Hence (T1 ⊗ n T2 )( i=1 ti ⊗ vi ) = 0. But, by 10.1.8, T1 ⊗ T2 is injective on D1 ⊗ D2 , so that z = h ∈ H . This leads to the following. Theorem 10.1.11 Let H1 ⊂ E1 , H2 ⊂ E2 be finite-codimensional subspaces. Then H1 ⊗ E2 + E1 ⊗ H2 is finite-codimensional in E1 ⊗ E2 . Proof. Let π1 : E1 −→E1 /H1 , π2 : E2 −→E2 /H2 be the quotients maps. Then π1 ⊗ π2 : E1 ⊗ E2 −→E1 /H1 ⊗ E2 /H2 is surjective (10.1.8), so by 10.1.10 we have that H1 ⊗ E2 + E1 ⊗ H2 = Ker (π1 ⊗ π2 ) is finite-codimensional.

10.2 Algebraic tensor products, where the scalar field is valued This section is important for the sequel; it contains all fundamental technical facts needed for developing the theory of tensor products of locally convex spaces later on. Here, in Section 10.2, we study the consequences for tensor products if we let the valuation of K play its role. The fact that K is valued allows for the following. Definition 10.2.1 Let X ⊂ E, Y ⊂ F . Put X Y := {x ⊗ y : x ∈ X, y ∈ Y } and X ⊗ Y = aco (X Y ). Remarks 10.2.2 (a) It is easily seen that for subspaces X, Y of E, F respectively the definition of X ⊗ Y does not conflict previous ones.

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(b) For absolutely convex X and Y it is possible to view X ⊗ Y as the tensor product of X and Y in the category of BK -modules (see [210], Section 17.B), but we will not need this observation in this book. We now define the non-Archimedean translation of the π - and ε-tensor products of seminorms (see [103], Chapters 15, 16). Definition 10.2.3 Let p be a seminorm on E, q a seminorm on F . For z ∈ E ⊗ F put n xi ⊗ y i } (p ⊗π q)(z) := inf{ max p(xi ) q(yi ) : n ∈ N, z = 1≤i≤n

i=1

(p ⊗ε q)(z) := sup{|(f ⊗ g)(z)| : f ∈ E ∗ , g ∈ F ∗ , |f | ≤ p, |g| ≤ q}. Theorem 10.2.4 Let p, q be seminorms on E, F respectively. Then p ⊗π q and p ⊗ε q are seminorms, p ⊗ε q is polar, p ⊗ε q ≤ p ⊗π q. Also, p ⊗π q is the largest among all seminorms r on E ⊗ F with the property r(x ⊗ y) ≤ p(x) q(y) (x ∈ E, y ∈ F ).

Proof. Straightforward. To discuss the equality p ⊗π q = p ⊗ε q we need the following lemmas.

Lemma 10.2.5 Suppose E is finite-dimensional. Let p be a seminorm on E and let t ∈ (0, 1). Then there exists an algebraic base of E that is t-orthogonal with respect to p. Proof. We may assume p = 0. Ep is finite-dimensional. By 2.3.7 Ep has a t-orthogonal base {u1 , . . . , um }. Let πp : E−→Ep be the natural map and, for each i ∈ {1, . . . , m}, take xi ∈ E with πp (xi ) = ui . Let y1 , . . . , yn be an algebraic base of Ker p. Then {x1 , . . . , xm , y1 , . . . , yn } is an algebraic base of E that is t-orthogonal with respect to p. Lemma 10.2.6 Let q be a seminorm on F and let t ∈ (0, 1). Then each z ∈ E ⊗ F can be written as z=

n

xi ⊗ yi ,

i=1

where x1 , . . . , xn ∈ E, and where y1 , . . . , yn ∈ F \ {0} are t-orthogonal with respect to q. Proof. By 10.1.6(ii) we have z ∈ E ⊗ D for some finite-dimensional subspace D of F . Then apply 10.1.6(iii) for an algebraic base of D that is t-orthogonal with respect to q (see 10.2.5).

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379

Theorem 10.2.7 Let p, q be polar seminorms on E, F respectively. Then p ⊗π q is polar. In fact we even have p ⊗π q = p ⊗ε q. Proof. From 10.2.4 we get p ⊗ε q ≤ p ⊗π q. To see the opposite inequality, let t ∈ (0, 1), z ∈ E ⊗ F ; we will prove (p ⊗π q)(z) ≤ t −1 (p ⊗ε q)(z). By the previous lemma, z can be written as ni=1 xi ⊗ yi , where y1 , . . . , yn ∈ F \ {0} are t-orthogonal with respect to q. We have, by definition of p ⊗π q and polarity, (p ⊗π q)(z) ≤ max{p(xi ) q(yi ) : i ∈ {1, . . . , n}} = sup{|f (xi )| q(yi ) : f ∈ E ∗ , |f | ≤ p, i ∈ {1, . . . , n}} = sup{|f (xi )| q(yi ) : i ∈ {1, . . . , n}, f ∈ E ∗ , |f | ≤ p} n ≤ t −1 sup{q( f (xi ) yi ) : f ∈ E ∗ , |f | ≤ p} i=1 n

= t −1 sup{|g(

f (xi ) yi )| : g ∈ F ∗ ,|g| ≤ q, f ∈ E ∗ , |f | ≤ p}

i=1

=t

−1

sup{|(f ⊗ g)(z)| : f ∈ E ∗ , |f | ≤ p, g ∈ F ∗ , |g| ≤ q}

= t −1 (p ⊗ε q)(z). As we will focus our attention to p ⊗π q (rather than p ⊗ε q), From now on in this chapter we write p ⊗ q instead of p ⊗π q. We have the following relation between the “unit balls”. Theorem 10.2.8 Let p, q be seminorms on E, F respectively. (i) Let the valuation of K be dense. Then {x ∈ E : p(x) < 1} ⊗ {y ∈ F : q(y) < 1} = {z ∈ E ⊗ F : (p ⊗ q)(z) < 1}. (ii) Let the valuation of K be discrete. Then {x ∈ E : p(x) ≤ 1} ⊗ {y ∈ F : q(y) ≤ 1} = {z ∈ E ⊗ F : (p ⊗ q)(z) ≤ 1}. Proof. (i). Let (p ⊗ q)(z) < 1. There are x1 , . . . , xn ∈ E, y1 , . . . , yn ∈ F such that z = ni=1 xi ⊗ yi and max1≤i≤n p(xi ) q(yi ) < 1. For each i, choose λi ∈ K such that |λi | > q(yi ) and |λi | < p(xi )−1 if p(xi ) = 0. Then z = n −1 i=1 λi xi ⊗ λi yi ∈ {x ∈ E : p(x) < 1} ⊗ {y ∈ F : q(y) < 1}, proving one inclusion. The opposite one is trivial.

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(ii). Let ρ ∈ BK− be a uniformizing element, let (p ⊗ q)(z) ≤ 1. There are x1 , . . . , xn ∈ E, y1 , . . . , yn ∈ F such that z = ni=1 xi ⊗ yi and max1≤i≤n p(xi ) q(yi ) < |ρ|−1 . But p, q are solid by assumption (see Section 3.3) so that p(xi ) q(yi ) ∈ |K| for each i, implying max1≤i≤n p(xi ) q(yi ) ≤ 1. Now choose λi ∈ K such that |λi | = p(xi )−1 if p(xi ) = 0, |λi | > q(yi ) if p(xi ) = 0. Then z = ni=1 λi xi ⊗ λ−1 i yi ∈ {x ∈ E : p(x) ≤ 1} ⊗ {y ∈ F : q(y) ≤ 1}. Again, the opposite inclusion is trivial. The following “continuation” of 10.2.6 is technical but useful. Lemma 10.2.9 Let p, q be seminorms on E, F respectively. Let t ∈ (0, 1] and suppose y1 , . . . , yn ∈ F \ {0} are t-orthogonal with respect to q. Then for each x1 , . . . , xn ∈ E we have (p ⊗ q)(

n

xi ⊗ yi ) ≥ t max p(xi ) q(yi ). 1≤i≤n

i=1

Proof. We have to prove that max p(xj ) q(yj ) ≥ t max p(xi ) q(yi )

n

1≤j ≤m

1≤i≤n

m

whenever i=1 xi ⊗ yi = j =1 xj ⊗ yj . Put D := [x1 , . . . , xn , x1 , . . . , xm ], let s ∈ (0, 1). By 10.2.5 D has an algebraic base e1 , . . . , ek that is s-orthogonal with respect to p. We have the expansions xi = λi1 e1 + · · · + λik ek (i ∈ {1, . . . , n}) and xj = λj1 e1 + · · · + λjk ek (j ∈ {1, . . . , m}), so that kr=1 er ⊗ k m ( ni=1 λir yi ) = ni=1 xi ⊗ yi = m j =1 xj ⊗ yj = r=1 er ⊗ ( j =1 λjr yj ). By 10.1.6,(iv), after applying the twist map, we obtain, for each r ∈ {1, . . . , k}, n

λir yi =

i=1

m

λjr yj .

(10.1)

j =1

By s-orthogonality we have

max p(xj ) q(yj ) ≥ s max λjr p(er ) q(yj ) = s max p(er ) q(λjr yj )

1≤j ≤m

j,r

≥ s max p(er ) q( 1≤r≤k

j,r

m

λjr yj ),

j =1

which by (10.1) equals s max p(er ) q( 1≤r≤k

n i=1

λir yi ).

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381

By t-orthogonality this is ≥ s t max p(er ) q(λir yi ) = s t max p(λir er ) q(yi ) ≥ s t max p(xi ) q(yi ). i,r

i,r

1≤i≤n

As this holds for each s ∈ (0, 1) the lemma follows.

Most of the crucial facts needed in the sequel are listed below. Corollary 10.2.10 Let p, q be seminorms on E, F respectively. Then we have the following: (i) For x ∈ E, y ∈ F , (p ⊗ q)(x ⊗ y) = p(x) q(y). (ii) If, for some t ∈ (0, 1], y1 , . . . , yn ∈ F \ {0} are t-orthogonal with respect to q and x1 , . . . , xn ∈ E \ {0}, then x1 ⊗ y1 , . . . , xn ⊗ yn are t-orthogonal with respect to p ⊗ q. (iii) If p, q are norms then so is p ⊗ q. (iv) Let D1 ⊂ E, D2 ⊂ F be subspaces. Then (p|D1 ) ⊗ (q|D2 ) = (p ⊗ q)|(D1 ⊗ D2 ). (v) Let D1 ⊂ E, D2 ⊂ F be subspaces. If D1 is p-dense in E, D2 is q-dense in F then D1 ⊗ D2 is p ⊗ q-dense in E ⊗ F . (vi) Let e1 , e2 , . . . ∈ E \ {0} be a t-orthogonal sequence with respect to p, where t ∈ (0, 1] and let f1 , f2 , . . . ∈ F \ {0} be an s-orthogonal sequence with respect to q, where s ∈ (0, 1]. Then e1 ⊗ f1 , e1 ⊗ f2 , e2 ⊗ f1 , . . . is t s-orthogonal with respect to p ⊗ q. Proof. (i). We may assume y = 0. Then apply the previous lemma to the case n := 1, x1 := x, y1 := y, t := 1 to obtain (p ⊗ q)(x ⊗ y) ≥ p(x) q(y). The opposite inequality is covered by 10.2.4. (ii). This follows immediately from the previous lemma and (i). (iii). Let z ∈ E ⊗ F , z = 0, let t ∈ (0, 1). By 10.2.6 we can write z = n i=1 xi ⊗ yi , where y1 , . . . , yn ∈ F \ {0} are t-orthogonal with respect to q; we may assume xi = 0 for all i ∈ {1, . . . , n}. Then, by 10.2.9, (p ⊗ q)(z) ≥ t max1≤i≤n p(xi ) q(yi ) > 0 and it follows that p ⊗ q is a norm. (iv). For trivial reasons we have (p|D1 ) ⊗ (q|D2 ) ≥ (p ⊗ q)|(D1 ⊗ D2 ). To establish the opposite inequality, let z ∈ D1 ⊗ D2 , t ∈ (0, 1). By 10.2.6 we have z = ni=1 xi ⊗ yi , where y1 , . . . , yn ∈ D2 \ {0} are t-orthogonal with respect to q and x1 , . . . , xn ∈ D1 . Applying 10.2.9 we find (p ⊗ q)(z) ≥ t max1≤i≤n p(xi ) q(yi ) ≥ t ((p|D1 ) ⊗ (q|D2 ))(z), and since t ∈ (0, 1) was arbitrary, (iv) is proved. (v). Clearly, by using (i), every x ⊗ y (x ∈ E, y ∈ D2 ) lies in the p ⊗ qclosure of D1 ⊗ D2 , which is a vector space. So by 10.1.6(ii) we find that

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D1 ⊗ D2 is p ⊗ q-dense in E ⊗ D2 . A similar reasoning, using the twist map, shows that E ⊗ D2 is p ⊗ q-dense in E ⊗ F . (vi). Let n ∈ N; we prove that {ei ⊗ fj : i, j ∈ {1, . . . , n}} is t s-orthogonal with respect to p ⊗ q. Let λij ∈ K (i, j ∈ {1, . . . , n}). We have (p ⊗ q)( ni,j =1 λij ei ⊗ fj ) = (p ⊗ q)( nj=1 xj ⊗ fj ), where xj := ni=1 λij ei . From 10.2.9 we obtain that this last quantity is ≥ s max 1≤j ≤n p(xj ) q(fj ), and by t-orthogonality we have p(xj ) ≥ t max1≤i≤n λij p(ei ), so that, combining the above we find, using (i), (p ⊗ q)(

n

λij ei ⊗ fj ) ≥ t s max λij p(ei ) q(fj ) 1≤i,j ≤n

i,j =1

= t s max λij (p ⊗ q)(ei ⊗ fj ). 1≤i,j ≤n

Theorem 10.1.10 admits a version for seminorms, extending (iii) above.

Theorem 10.2.11 Let p, q be seminorms on E, F respectively. Then Ker p ⊗ q = Ker p ⊗ F + E ⊗ Ker q. Proof. Follow the proof of 10.1.10 replacing T1 , T2 by p, q respectively. In the last sentence, use that p is a norm on D1 , q is a norm on D2 so that by 10.2.10(iii) and (iv), p ⊗ q is a norm on D1 ⊗ D2 . Finally we prove the following “preservation of isometries”. Theorem 10.2.12 Let T1 : E1 −→F1 , T2 : E2 −→F2 be linear maps, let p1 , p2 , q1 , q2 be seminorms on E1 , E2 , F1 , F2 respectively, and suppose that p1 = q1 ◦ T1 , p2 = q2 ◦ T2 . Then p1 ⊗ p2 = (q1 ⊗ q2 ) ◦ (T1 ⊗ T2 ). Proof. We follow the steps (i)–(iv): (i) Let t ∈ (0, 1), z ∈ E1 ⊗ E2 . By 10.2.6 we can write z = ni=1 xi ⊗ yi , where y1 , . . . , yn ∈ E2 \ {0} are t-orthogonal with respect to p2 . (ii) By using 10.2.9 we obtain (p1 ⊗ p2 )(z) = (p1 ⊗ p2 )(

n

xi ⊗ yi ) ≥ t max p1 (xi ) p2 (yi )

i=1

1≤i≤n

= t max q1 (T1 (xi )) q2 (T2 (yi )) 1≤i≤n

n T1 (xi ) ⊗ T2 (yi )) ≥ t (q1 ⊗ q2 )( i=1

= t (q1 ⊗ q2 )(T1 ⊗ T2 )(z).

10.3 Tensor products of locally convex spaces

383

(iii) As a trivial consequence of (i) we have that the non-zero elements of T2 (y1 ), . . . , T2 (yn ) are t-orthogonal with respect to q2 . Then, by 10.2.9 we get n T1 (xi ) ⊗ T2 (yi )) (q1 ⊗ q2 )(T1 ⊗ T2 )(z) = (q1 ⊗ q2 )( i=1

≥ t max q1 (T1 (xi )) q2 (T2 (yi )) 1≤i≤n

= t max p1 (xi ) p2 (yi ) ≥ t (p1 ⊗ p2 )(z). 1≤i≤n

(iv) The theorem follows after observing that the conclusions of (ii) and (iii) hold for all t ∈ (0, 1). As an application we obtain (for notations, see 3.3.1). Theorem 10.2.13 Let p, q be seminorms on E, F respectively. Then there is a unique linear map ϕ : (E ⊗ F )p⊗q −→Ep ⊗ Fq making the diagram πp ⊗πq

/ Ep ⊗ Fq E ⊗ FM MMM q8 q qq MMM q q qq ϕ πp⊗q MMM & qqq (E ⊗ F )p⊗q commute. This ϕ is a surjective isometry with respect to the norms p ⊗ q on (E ⊗ F )p⊗q and p ⊗ q on Ep ⊗ Fq . Proof. By 10.1.10 and 10.2.11, Ker πp⊗q = Ker (πp ⊗ πq ), from which it follows that such a (unique) linear map ϕ exists and is injective. By 10.1.8, πp ⊗ πq is surjective, hence so is ϕ. It follows that ϕ is a bijection. Now apply 10.2.12 with T1 , T2 replaced by πp , πq respectively to conclude that ϕ is an isometry.

10.3 Tensor products of locally convex spaces We define natural locally convex topologies on the tensor product of locally convex spaces and derive several hereditary properties. Throughout Sections 10.3 and 10.4 E, E1 , E2 , F, F1 , F2 are locally convex spaces (over K ). The following lemma will not come as a surprise.

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Lemma 10.3.1 (Continuity Lemma) Let G be a locally convex space. A bilinear map B : E × F −→G is continuous if and only if for each continuous seminorm r on G there exist continuous seminorms p on E, q on F such that r(B(x, y)) ≤ p(x) q(y)

((x, y) ∈ E × F ).

(10.2)

Proof. Let B be continuous, let r be a continuous seminorm on G. Then U := {z ∈ G : r(z) ≤ 1} is a zero neighbourhood. Continuity of B yields the existence of (solid!) continuous seminorms p on E, q on F such that {(x, y) ∈ E × F : p(x) ≤ 1, q(y) ≤ 1} ⊂ B −1 (U ). From here a standard procedure leads to (10.2). Conversely, let (xi , yi )i∈I be a net in E × F converging to (x, y) ∈ E × F , let r be a continuous seminorm on G and ε > 0; we prove that r(B(xi , yi ) − B(x, y)) < ε for large i. By assumption there are continuous seminorms p on E, q on F for which (10.2) holds. Take ε1 > 0 with ε12 < ε, ε1 p(x) < ε, ε1 q(y) < ε. There is an i0 ∈ I such that, for i ≥ i0 , p(xi − x) < ε1 , q(yi − y) < ε1 . Then, for i ≥ i0 , r(B(xi , yi ) − B(x, y)) ≤ max(r(B(xi , yi ) − B(xi , y)), r(B(xi , y) − B(x, y))) = max(r(B(xi , yi − y)), r(B(xi − x, y))) ≤ max(p(xi ) q(yi − y), p(xi − x)q(y)) ≤ max(ε1 p(xi ), ε1 q(y)) ≤ max(ε1 p(xi − x), ε1 p(x), ε1 q(y)) < max(ε12 , ε, ε) = ε. Definition 10.3.2 The projective tensor product topology on E ⊗ F is the strongest locally convex topology π on E ⊗ F for which the bilinear map ⊗ : E × F −→E ⊗ F is continuous. The locally convex space E ⊗ F , equipped with this topology, is called the projective tensor product of E and F and denoted by E ⊗π F . Remark 10.3.3 Equivalently, we could have started in the spirit of 10.1.3 by defining “a projective tensor product of E and F ” by a pair, consisting of a locally convex space (E ⊗ F, π) and a continuous bilinear map ⊗ : E × F −→(E ⊗ F, π) such that the following Universal Property holds. For each locally convex space G and each continuous bilinear map B : E × F −→G there is a unique continuous linear map T : (E ⊗ F, π)−→G such that B = T ◦ ⊗. It is easy to see that such a “projective tensor product” is unique in a

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natural sense and that existence is furnished by 10.3.2. We leave the details to the reader. The next conclusion follows directly from the above results. Theorem 10.3.4 A seminorm r on E ⊗π F is continuous if and only if there exist continuous seminorms p on E, q on F such that r(x ⊗ y) ≤ p(x) q(y) for all x ∈ E, y ∈ F . In 10.2.4 we have seen that p ⊗ q is the largest among all seminorms r on E ⊗ F for which r(x ⊗ y) ≤ p(x) q(y) ((x, y) ∈ E × F ). Thus, we obtain the following alternative description of E ⊗π F , which will be the one that we will use in the rest of the chapter. Theorem 10.3.5 The seminorms p ⊗ q, where p and q are continuous seminorms on E and F respectively, generate the projective tensor product topology on E ⊗ F . Remarks 10.3.6 (a) If P1 and P2 are families of seminorms inducing the topology of E and F respectively then π is also induced by {p ⊗ q : p ∈ P1 , q ∈ P2 }. (b) The identifications of E ⊗ K and of K ⊗ E with E (see 10.1.7(b)) become homeomorphisms when we consider on these tensor products their projective tensor product topologies. Henceforth we will identify E ⊗π K and K ⊗π E with E. Theorem 10.3.7 If T1 ∈ L(E1 , F1 ), T2 ∈ L(E2 , F2 ) then T1 ⊗ T2 ∈ L(E1 ⊗π E2 , F1 ⊗π F2 ). Proof. Apply 10.2.12.

From 10.2.10(iv) we have the following. (See also 10.1.9(c).) Theorem 10.3.8 Let D1 ⊂ E, D2 ⊂ F be subspaces. Then the natural inclusion D1 ⊗π D2 −→E ⊗π F is a homeomorphic embedding. The following special case is worth mentioning. Theorem 10.3.9 Let b ∈ F \ {0}. If there exists a continuous seminorm q on F with q(b) = 0 then the linear map x → x ⊗ b is a homeomorphic embedding E−→E ⊗π F . If, in addition, there exists an f ∈ F with f (b) = 0 then its image is a complemented subspace of E ⊗π F . Proof. The first part follows from 10.2.10(i). For the second part, we may assume that f (b) = 1. Then 1 ⊗ f : E ⊗π F −→E ⊗π K = E is continuous, linear and the composition x → x ⊗ b → (1 ⊗ f )(x ⊗ b) is the identity on E.

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We have an easy consequence. Corollary 10.3.10 Suppose the topology of F is not the indiscrete one. Then E is isomorphic to a subspace of E ⊗π F . If, in addition, F = {0} then E is isomorphic to a complemented subspace of E ⊗π F . Now we are ready for various permanence properties. Theorem 10.3.11 Let E, F be Hausdorff. Then E ⊗π F is Hausdorff. Proof. Let z ∈ E ⊗ F , z = 0. By 10.1.6(ii) there exist finite-dimensional subspaces D1 ⊂ E, D2 ⊂ F such that z ∈ D1 ⊗ D2 . Then (3.4.24) there are norms p, q on D1 , D2 defining their topologies. These norms can be extended to continuous seminorms p˜ on E and q˜ on F respectively (3.4.3). By 10.2.10(iv) we have ˜ 2 ))(z) = (p ⊗ q)(z). ˜ ˜ 1 ) ⊗ (q|D (p˜ ⊗ q)(z) = ((p|D Since by 10.2.10(iii) p ⊗ q is a norm on D1 ⊗ D2 we obtain that (p ⊗ q)(z) = 0, so p˜ ⊗ q˜ is a continuous seminorm on E ⊗π F not vanishing at z. For the next permanence properties the following is useful. Lemma 10.3.12 Let D1 ⊂ E, D2 ⊂ F be dense subspaces. Then D1 ⊗ D2 is dense in E ⊗π F . Proof. Directly by using 10.2.10(v) for continuous seminorms p on E, q on F. Theorem 10.3.13 Let E, F be normable (metrizable, of countable type, strictly of countable type, of finite type). Then so is E ⊗π F . Proof. 1. Let E, F be normable. Then the topologies of E and F are induced by norms p on E, q on F . Thus, p ⊗ q is a norm (10.2.10(iii)) and induces the topology on E ⊗π F . 2. Let E, F be metrizable. Then E, F are Hausdorff, hence so is E ⊗π F by 10.3.11. There are countable collections of seminorms P1 on E, P2 on F that generate the topologies. Then {p ⊗ q : p ∈ P1 , q ∈ P2 } is countable and generates the topology on E ⊗π F . Now apply 3.5.2(γ )=⇒(α) to E ⊗π F . 3. Let E, F be of countable type. Let p, q be continuous seminorms on E, F respectively. There are countable sets X ⊂ E, Y ⊂ F such that [X] is p-dense in E, [Y ] is q-dense in F . Then by 10.2.10(v) [X] ⊗ [Y ] is p ⊗ q -dense in E ⊗ F . But [X] ⊗ [Y ] = [X ⊗ Y ] = [X Y ] is countably generated. We see that E ⊗π F is of countable type.

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4. Let E, F be strictly of countable type. There are countable sets X ⊂ E, Y ⊂ F such that [X] is dense in E, [Y ] is dense in F . Then (10.3.12), [X Y ] = [X] ⊗ [Y ] is countably generated and dense in E ⊗π F . 5. Let E, F be of finite type. Let p, q be continuous seminorms on E, F respectively. Then p, q are of finite type (see 5.3.1). It follows from 10.1.6(i) that Ep ⊗ Fq is finite-dimensional, hence so is (E ⊗ F )p⊗q by 10.2.13, i.e., p ⊗ q is of finite type. Theorem 10.3.14 Let E, F be polar (dual-separating). Then so is E ⊗π F . Proof. The first statement follows directly from 10.2.7, applied to polar continuous seminorms p on E, q on F . Now let E, F be dual-separating, let z ∈ E ⊗ F , z = 0. By 10.1.6(ii) there are finite-dimensional subspaces D1 ⊂ E, D2 ⊂ F such that z ∈ D1 ⊗ D2 ; let u1 , . . . , un (v1 , . . . , vm ) be an algebraic base of D1 (D2 ). Then z has an expan sion z = ni=1 m j =1 λij ui ⊗ vj . Without loss, assume λ11 = 0. The formulas 1 if i = 1 1 if j = 1 f (ui ) := g(vj ) := 0 if i ≥ 2 0 if j ≥ 2 define f ∈ D1 , g ∈ D2 (3.4.25) that can be extended, by 5.1.6, to elements (again called f, g) of E , F respectively. Then (10.3.7) f ⊗ g ∈ L(E ⊗π F, K ⊗π K) = (E ⊗π F ) , and (f ⊗ g)(z) = λ11 = 0. Remark 10.3.15 The natural non-Archimedean counterpart of the classical εtensor product of E and F (see [103], Section 16.1) is the space E ⊗ε F , which is by definition E ⊗ F , equipped with the locally convex topology induced by the seminorms p ⊗ε q (see 10.2.3), where p and q are continuous seminorms on E, F respectively. From 10.2.7, however we obtain that, contrary to the classical case, for polar spaces E and F the locally convex spaces E ⊗π F and E ⊗ε F are identical. We conclude this section with a simple observation. Theorem 10.3.16 Let E, F have “orthogonal” bases. Then so has E ⊗π F . Proof. Observe that by 10.3.11 E ⊗π F is Hausdorff. Let e1 , e2 , . . . and f1 , f2 , . . . be “orthogonal” bases of E, F respectively. Let P1 (P2 ) be a base of continuous seminorms on E (F ) with respect to which e1 , e2 , . . . (f1 , f2 , . . .) are orthogonal. By 10.2.10(vi) {ei ⊗ fj : i, j ∈ N} is orthogonal with respect to each p ⊗ q, with p ∈ P1 , q ∈ P2 . But those generate the topology of E ⊗π F . We conclude that {ei ⊗ fj : i, j ∈ N} is “orthogonal”. According to 10.3.12 the space [ei ⊗ fj : i, j ∈ N] = [ei : i ∈ N] ⊗ [fj : j ∈ N] is dense in E ⊗π F . Now apply 9.2.1 to arrive at the conclusion.

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Remark 10.3.17 In 10.4.15 and 10.4.16 we will discuss the natural converses of the above permanence properties.

10.4 Tensor products of nuclear and semi-Montel spaces We first show that tensor products of nuclear spaces are nuclear (10.4.4). Theorem 10.4.1 Let X ⊂ E, Y ⊂ F be compactoids. Then X ⊗ Y is a compactoid in E ⊗π F . Proof. Let p, q be continuous seminorms on E, F respectively, let ε > 0; we prove the existence of a finite set G ⊂ E ⊗ F such that X ⊗ Y ⊂ {z ∈ E ⊗ F : (p ⊗ q)(z) < ε} + aco G. X is bounded, so there is an rX > 0 such that p(x) ≤ rX for all x ∈ X. Analogously, there is an rY > 0 such that q(y) ≤ rY for all y ∈ Y . Let λ ∈ K, λ := 1 if the valuation of K is discrete and |λ| > 1 if the valuation of K is dense. Let r > 0 be such that r 2 < ε,

r |λ| rX < ε,

r |λ| rY < ε.

(10.3)

By compactoidity of X, Y and 3.8.9, there exist finite sets G1 ⊂ λ X, G2 ⊂ λ Y such that X ⊂ {x ∈ E : p(x) < r} + aco G1 , Y ⊂ {y ∈ F : q(y) < r} + aco G2 . Hence, X ⊗ Y ⊂ {x ∈ E : p(x) < r} ⊗ {y ∈ F : q(y) < r} + {x ∈ E : p(x) < r} ⊗ aco G2 + aco G1 ⊗ {y ∈ F : q(y) < r} + aco G1 ⊗ aco G2 . By using 10.2.10(i) and (10.3) we obtain that the first three sets of the right-hand side are contained in {z ∈ E ⊗ F : (p ⊗ q)(z) < ε}. So we get X ⊗ Y ⊂ {z ∈ E ⊗ F : (p ⊗ q)(z) < ε} + aco G1 ⊗ aco G2 , and then X ⊗ Y ⊂ {z ∈ E ⊗ F : (p ⊗ q)(z) < ε} + G1 ⊗ G2 . Therefore, G := G1 G2 meets the requirements.

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Remarks 10.4.2 (a) It is not hard to see that if X, Y are bounded then X ⊗ Y is bounded in E ⊗π F . Conversely, if neither [X] nor [Y ] have the indiscrete topology and X ⊗ Y is a compactoid (bounded), also so are X and Y . In fact, X ⊗ Y is a compactoid (bounded) in [X] ⊗π [Y ]. Then applying 10.3.9 with E, F replaced by [X], [Y ] respectively we obtain that X is a compactoid (bounded), and by using the twist map the same goes for Y . (b) Much more difficult is the following. Let Z ⊂ E ⊗π F be compactoid (bounded). Do there exist compactoid (bounded) X ⊂ E, Y ⊂ F such that Z ⊂ X ⊗ Y ? We will present an affirmative answer for an important special case in 10.4.13. Theorem 10.4.3 Let T1 : E1 −→F1 , T2 : E2 −→F2 be compactoid operators. Then so is T1 ⊗ T2 : E1 ⊗π E2 −→F1 ⊗π F2 . Proof. There are zero neighbourhoods U1 in E1 , U2 in E2 such that T1 (U1 ) and T2 (U2 ) are compactoids (see 8.1.1). By 10.2.8, U1 ⊗ U2 is a zero neighbourhood in E1 ⊗π E2 , and (T1 ⊗ T2 )(U1 ⊗ U2 ) = T1 (U1 ) ⊗ T2 (U2 ), a compactoid according to 10.4.1. It follows that T1 ⊗ T2 is compactoid. After these preparations we can prove the following. Theorem 10.4.4 The projective tensor product of two nuclear spaces is nuclear. Proof. Let E, F be nuclear. Let p, q be continuous seminorms on E, F respectively. It suffices to see that the canonical map πp⊗q : E ⊗π F −→(E ⊗ F )p⊗q is compactoid (8.5.1(γ )). By 10.2.13 the map ϕ in the commutative diagram πp ⊗πq

/ E p ⊗ Fq E ⊗π FM MMM q8 q MMM qq q q qq ϕ πp⊗q MMMM & qqq (E ⊗ F )p⊗q is a surjective isometry. Nuclearity and 10.4.3 show compactoidity of πp ⊗ πq . Then πp⊗q = ϕ −1 ◦ (πp ⊗ πq ) is also compactoid. Remark 10.4.5 For the converse of the above theorem, see 10.4.15(ii). Next we consider the more difficult question whether tensor products of semiMontel spaces are semi-Montel. We will be able to answer it positively if the spaces involved are metrizable (10.4.14). We start off with the following “splitting lemma”.

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Lemma 10.4.6 Let F be normable. Then for each bounded set B ⊂ E ⊗π F there exist bounded sets B1 ⊂ E, B2 ⊂ F such that B ⊂ B1 ⊗ B2 . Proof. Let q be a norm on F generating the topology. Let ρ ∈ K with 0 < |ρ| < 1. For each y ∈ F \ {0} there is a λ ∈ K such that |ρ| ≤ q(λ y) ≤ 1. Let t ∈ (0, 1). Each z ∈ E ⊗ F admits a representation as a finite sum z=

kz

xmz ⊗ ymz ,

m=1

where x1z , . . . , xkzz ∈ E, and y1z , . . . , ykzz ∈ F \ {0} are t-orthogonal with respect to q (10.2.6). By the above we may assume additionally that |ρ| ≤ q(ymz ) ≤ 1 for all z ∈ E ⊗ F , m ∈ {1, . . . , kz }. Now let P be the collection of continuous seminorms on E. Then (10.2.10(ii)), for each z ∈ E ⊗ F , the non-zero vectors of {xmz ⊗ ymz : m ∈ {1, . . . , kz }} are t-orthogonal with respect to p ⊗ q for all p ∈ P. By boundedness of B, for each p ∈ P, Mp := sup{(p ⊗ q)(z) : z ∈ B} < ∞. By t-orthogonality and previous assumptions we have for z ∈ B, m ∈ {1, . . . , kz } that |ρ| p(xmz ) ≤ p(xmz ) q(ymz ) = (p ⊗ q)(xmz ⊗ ymz ) ≤ t −1 (p ⊗ q)(z) ≤ t −1 Mp . We get that p(xmz ) ≤ |ρ|−1 t −1 Mp , q(ymz ) ≤ 1 for all p ∈ P, z ∈ B, m ∈ {1, . . . , kz }. Now put B1 := {xmz : z ∈ B, m ∈ {1, . . . , kz }}, B2 := {ymz : z ∈ B, m ∈ {1, . . . , kz }}. We see that B1 , B2 are bounded and that B1 B2 ⊃ {xmz ⊗ ymz : z ∈ B, m ∈ kz xmz ⊗ ymz : z ∈ B} = B. {1, . . . , kz }}, so that B1 ⊗ B2 ⊃ { m=1 Next, we introduce the strong topology on L(E, F ). Definition 10.4.7 (Compare with 7.3.1) We denote by Lb (E, F ) the space L(E, F ), equipped with the topology of uniform convergence on bounded sets, i.e., the locally convex topology induced by the seminorms T −→ sup{q(T (x)) : x ∈ B}

(T ∈ L(E, F )),

where B ⊂ E is bounded and q is a continuous seminorm on F . We need a simple fact.

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Theorem 10.4.8 If H ⊂ Lb (E, F ) is bounded and if B ⊂ E is bounded, then H (B) := {T (x) : x ∈ B, T ∈ H } is bounded in F .

Proof. Straightforward.

We introduce a key map, interpreting elements of E ⊗ F as operators, as follows. For each x ∈ E, y ∈ F the map f → f (x) y is in L(Eb , F ). By the Universal Property it leads to an operator E ⊗ F −→L(Eb , F ), described as the map z → Tz below. Definition 10.4.9 For each z = defined by the formula Tz (f ) =

n

n

i=1

xi ⊗ yi ∈ E ⊗ F , let Tz ∈ L(Eb , F ) be

f (xi ) yi

(f ∈ E ).

i=1

Each Tz is of finite rank. From 5.1.3 we have that if E is Hausdorff and polar then z → Tz is injective. Theorem 10.4.10 Let E be Hausdorff and polar. Let D be a subspace of F . Then for z ∈ E ⊗ F we have Tz ∈ L(Eb , D) ⇐⇒ z ∈ E ⊗ D. Proof. “⇐=” is obvious. To prove “=⇒”, let z = ni=1 xi ⊗ yi , where x1 , . . . , xn are linearly independent (use 10.1.6(iii) and the twist map). By assumptions there are f1 , . . . , fn ∈ E such that fj (xi ) = δij (i, j ∈ {1, . . . , n}). Then, for each j ∈ {1, . . . , n}, yj = ni=1 fj (xi ) yi = Tz (fj ) ∈ D, proving that z ∈ E ⊗ D. Theorem 10.4.11 Let E be polar and metrizable. Then the map z → Tz of 10.4.9 is a homeomorphic embedding E ⊗π F −→Lb (Eb , F ). Proof. It suffices to show that the collection P of seminorms z −→ sup{q(Tz (f )) : f ∈ B}

(z ∈ E ⊗ F ),

(10.4)

where q is a continuous seminorm on F and B is a bounded subset of Eb , induces the topology of E ⊗π F . To this end, let r ∈ P be of the form (10.4). By 7.6.4 we may assume that B is equicontinuous, i.e., that B has the form {f ∈ E : |f | ≤ p} for some continuous seminorm p on E.

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Let z ∈ E ⊗ F . For any representation z =

n i=1

xi ⊗ yi , we have

r(z) = sup{q(Tz (f )) : f ∈ E , |f | ≤ p} n f (xi ) yi ) : f ∈ E , |f | ≤ p} = sup{q( i=1

≤ sup{ max |f (xi )| q(yi ) : f ∈ E , |f | ≤ p} 1≤i≤n

≤ max p(xi ) q(yi ), 1≤i≤n

so that r(z) ≤ (p ⊗ q)(z). Conversely, let p and q be continuous seminorms on E, F respectively. Let t ∈ (0, 1). To show that p ⊗ q is dominated by a multiple of a seminorm of the form (10.4), we may assume that p is polar. Let z ∈ E ⊗ F , z = ni=1 xi ⊗ yi , where y1 , . . . , yn ∈ F \ {0} are t-orthogonal with respect to q (10.2.6). Then there is a j ∈ {1, . . . , n} for which (p ⊗ q)(z) ≤ max1≤i≤n p(xi ) q(yi ) = p(xj ) q(yj ). By polarity there exists an f ∈ E such that |f | ≤ p and f (xj ) ≥ t p(xj ). Thus, we have (p ⊗ q)(z) ≤ t −1 f (xj ) q(yj ) ≤ t −1 max |f (xi )| q(yi ) 1≤i≤n

≤ t −2 q(

n

f (xi ) yi ) = t −2 q(Tz (f ))

i=1

≤t

−2

sup{q(Tz (g)) : g ∈ E , |g| ≤ p}

= t −2 sup{q(Tz (g)) : g ∈ B}, where B := {g ∈ E : |g| ≤ p} is equicontinuous, hence bounded in Eb (7.3.4), which completes the proof. For our main result 10.4.13 we prove a final lemma. Lemma 10.4.12 Let B1 , B2 , . . . be bounded sets in a metrizable locally convex space. Then there exist λ1 , λ2 , . . . ∈ K × such that n λn Bn is bounded. Proof. Let U1 ⊃ U2 ⊃ · · · be a neighbourhood base of 0 consisting of absolutely convex sets. For each n ∈ N, choose λn ∈ K × such that λn Bn ⊂ Un . Let m ∈ N; we show that n λn Bn is contained in a multiple of Um . We have λ n Bn = λn Bn + λn Bn ⊂ λn Bn + Um + Um+1 + · · · n

n<m

⊂

n≥m

n<m

λ n Bn + U m .

n<m

λn Bn , there is a µ ∈ K such that Clearly, by boundedness of n<m n<m λn Bn ⊂ µ Um . It follows that n λn Bn is bounded. We now extend the splitting lemma 10.4.6.

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Theorem 10.4.13 Let E, F be metrizable. Suppose E is polar. Then, for each bounded set B ⊂ E ⊗π F there are bounded sets B1 ⊂ E, B2 ⊂ F such that B ⊂ B1 ⊗ B2 . Proof. Polarity and metrizability of E, 7.3.4, 7.6.1 and 7.6.4 imply that Eb has a fundamental sequence H1 ⊂ H2 ⊂ · · · of bounded sets. For each n ∈ N, put An := aco {Tz (f ) : z ∈ B, f ∈ Hn }. Then (10.4.8, 10.4.11) each An is bounded in F . By metrizability of F and 10.4.12 there are λ1 , λ2 , . . . ∈ K × such that n λn An is bounded in F . Put λ n An . A := n

Now let z ∈ B. For each f ∈ E there is an n such that f ∈ Hn , so that Tz (f ) ∈ An ⊂ [An ] ⊂ [A], showing that Tz ∈ L(Eb , [A]). Thus, from 10.4.10 we obtain B ⊂ E ⊗ [A]. We now retopologize [A] by means of the single norm pA (see 3.1.9(ii)). From now on in this proof we shall use the symbol [A] for the normed space ([A], pA ). We now prove that B is bounded in E ⊗π [A]. By applying 10.4.11 (where this time z → Tz is viewed as a map E ⊗π [A]−→Lb (Eb , [A])) this is equivalent to showing that {Tz : z ∈ B} is bounded in Lb (Eb , [A]). So, we have to see that, for each bounded set H in Eb , sup{pA (Tz (f )) : z ∈ B, f ∈ H } < ∞. In order to get this, choose an n for which H ⊂ Hn . Then {Tz (f ) : z ∈ B, f ∈ H } ⊂ {Tz (f ) : z ∈ B, f ∈ Hn } ⊂ An ⊂ λ−1 n A, so that, for each z ∈ B, f ∈ H we have pA (Tz (f )) ≤ |λn |−1 , which means that sup{pA (Tz (f )) : z ∈ B, f ∈ H } ≤ |λn |−1 < ∞. This finishes the proof that B is bounded in E ⊗π [A]. Now, by 10.4.6, there exist bounded sets B1 ⊂ E, B2 ⊂ [A] such that B ⊂ B1 ⊗ B2 . Boundedness of B2 in [A] implies that B2 is contained in λ A for some λ ∈ K. But A is bounded in F , hence so is B2 , and the theorem is proved. Corollary 10.4.14 The projective tensor product of two metrizable semiMontel spaces is (metrizable and) semi-Montel. Proof. The metrizability part is covered by 10.3.13. For semi-Montelness, firstly recall (8.4.5) that the semi-Montel spaces are those whose bounded sets are compactoids. Let E, F be metrizable and semi-Montel, let B ⊂ E ⊗π F be bounded. By 8.6.1 E is polar and then, by the previous theorem, there

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are bounded sets B1 ⊂ E, B2 ⊂ F with B ⊂ B1 ⊗ B2 . Now B1 , B2 are compactoids, hence so are B1 ⊗ B2 (by 10.4.1) and B. Next we discuss the converses of the hereditary properties of Sections 10.3 and 10.4. Theorem 10.4.15 (i) Suppose E = {0}, F = {0}. Let E ⊗π F be Hausdorff (normable, metrizable, dual-separating). Then so are E and F . (ii) Suppose neither E nor F have the indiscrete topology. Let E ⊗π F be of countable type (polar, of finite type, semi-Montel, nuclear, strictly of countable type). Then so are E and F . Proof. (i). Suppose E ⊗π F is Hausdorff. Let x ∈ E \ {0}, y ∈ F \ {0}. There is a continuous seminorm r on E ⊗π F such that r(x ⊗ y) = 0. There are continuous seminorms p on E, q on F with p ⊗ q ≥ r. Then p(x) = 0, q(y) = 0. It follows that E, F are Hausdorff. Now suppose E ⊗π F is normable (metrizable, dual-separating). By the above E, F are Hausdorff, so they do not have the indiscrete topology. Applying 10.3.10 and by using the twist map we obtain that E and F are isomorphic to subspaces of E ⊗π F , so they are normable (metrizable, dualseparating). (ii). As in (i) we have that E and F are isomorphic to subspaces of E ⊗π F . Since the first five properties are stable for subspaces (see parts (i) of 4.2.13, 4.4.16, 5.3.6, 8.4.25 and 8.5.7) we only have to consider the case where E ⊗π F is strictly of countable type. Then E ⊗π F is of countable type, hence so are E and F by the above. By assumption E and F do not have the indiscrete topology, so E = {0} and F = {0}. Applying 10.3.10 and by using the twist map we can view E and F as complemented subspaces and hence as continuous linear images of E ⊗π F . Now apply 4.2.16(i) to conclude that E, F are strictly of countable type. Remarks 10.4.16 (a) If E = {0} and F = {0} then E ⊗ F = {0} is Hausdorff, but E need not be. Now suppose that E has the indiscrete topology, i.e., the only continuous seminorm on E is the trivial one. Then, for any F , E ⊗π F also has the indiscrete topology; hence E ⊗π F is (strictly) of countable type (polar, of finite type, semi-Montel, nuclear), but F need not be. (b) Let E ⊗π F have an “orthogonal” base, E, F = {0}. We do not know whether this implies that E, F have “orthogonal” bases (compare with 10.3.16).

10.4 Tensor products of nuclear and semi-Montel spaces

395

Next, we consider some previous results in the setting of completed tensor products. Definition 10.4.17 Let E, F be Hausdorff. The completion of E ⊗π F is called ˆ πF. the completed tensor product of E and F and denoted E ⊗ Remark 10.4.18 There exist Banach spaces E, F for which E ⊗π F is not complete. For an example, take E = F := c0 . To show that c0 ⊗π c0 is not complete, let ρ ∈ K with 0 < |ρ| < 1, let e1 , e2 , . . . be the unit vectors of n c0 . For each m ∈ N, put xm := m n=1 ρ (en ⊗ en ). Then x1 , x2 , . . . is a nonconvergent Cauchy sequence in c0 ⊗π c0 . We have the following natural “completed” version of 10.4.14 Theorem 10.4.19 The completed tensor product of two Fr´echet–Montel spaces is again Fr´echet–Montel. Proof. By 10.4.14, if E, F are Fr´echet–Montel then E ⊗π F is metrizable and ˆ π F is Fr´echet, and also semi-Montel (by 8.4.27(c)), semi-Montel, so that E ⊗ hence Montel by 8.6.4. We also have the following splitting results for bounded sets and compactoids. ˆ πF Theorem 10.4.20 Let E, F be metrizable and of countable type. If B ⊂ E ⊗ is bounded then there exist bounded sets B1 ⊂ E, B2 ⊂ F such that B ⊂ ˆ π F ). B1 ⊗ B2 (the closure in E ⊗ ˆ π F is metrizable and of countable type. Proof. By 4.2.13(ii) and 10.3.13, E ⊗ Then the conclusion follows from 8.6.6 and 10.4.13. ˆ π F is a compactoid Theorem 10.4.21 Let E, F be metrizable. If A ⊂ E ⊗ then there exist null sequences (xn )n , (yn )n in E, F respectively such that ˆ π F ). A ⊂ aco {xn ⊗ yn : n ∈ N} (the closure in E ⊗ Proof. Let (pn )n , (qn )n be increasing sequences of seminorms on E, F generatˆ π F is metrizable, ing the corresponding topologies (3.5.2). Since by 10.3.13 E ⊗ we can apply 3.8.25 and 3.8.28 to obtain a null sequence (zr )r in E ⊗π F with A ⊂ aco {zr : r ∈ N} . Let λ ∈ K, |λ| > 1. Let 1 ≤ k1 < k2 < · · · be such that (pm ⊗ qm )(zr ) < |λ|−(2m+2) if r > km , m ∈ N. r For 1 ≤ r ≤ k1 , let zr = m i=1 xri ⊗ yri be a representation of zr . Assume next that kj < r ≤ kj +1 , j ∈ N. There exists a representation r < |λ|−(2j +2) for all i. We claim zr = m i=1 ari ⊗ bri with pj (ari ) qj (bri ) r that there exists a representation zr = m i=1 λri xri ⊗ yri , with λri ∈ BK , −j −j pj (xri ) ≤ |λ| , qj (yri ) ≤ |λ| for all i. Indeed, let δri , µri ∈ K be such that

396

Tensor products

|δri | ≤ pj (ari ) ≤ |λ δri |, |µri | ≤ qj (bri ) ≤ |λ µri |. If δri µri = 0, take xri := λ−j (λ δri )−1 ari , yri := λ−j (λ µri )−1 bri , λri := λ2j +2 δri µri . If δri = 0 and µri = 0 (the case δri = 0 and µri = 0 is analogous), take xri := µri λj +1 ari , yri := λ−j (λ µri )−1 bri , λri := 1. Finally, if δri = µri = 0, take xri := ari , yri := bri , λri := 1. This clearly proves our claim. Let x1 , x2 , . . . := x11 , x12 , . . . , x1m1 , x21 , x22 , . . . , x2m2 , . . . y1 , y2 , . . . := y11 , y12 , . . . , y1m1 , y21 , y22 , . . . , y2m2 , . . . Then (xn )n , (yn )n are null sequences in E, F respectively, and one verifies that, for each r ∈ N, zr ∈ aco{xn ⊗ yn : n ∈ N}. Therefore, A ⊂ aco {zr : r ∈ N} ⊂ aco {xn ⊗ yn : n ∈ N},

and we get our conclusion.

10.5 Examples of tensor products Several spaces of vector-valued continuous functions can be described as tensor products, see the chapter notes for references. In this section we focus on one of the most direct examples. We studied the space C(X) in Subsection 3.7.1. The range of f ∈ C(X) lies in K; we now will replace K by a (more general) locally convex space E. Throughout this section X, Y are non-empty zero-dimensional Hausdorff topological spaces and E is a locally convex space (over K ). Definition 10.5.1 By C(X−→E) we mean the vector space of all continuous maps X−→E. For each compact Z ⊂ X and each continuous seminorm q on E we define the seminorm q˜Z on C(X−→E) by q˜Z (f ) = max{q(f (x)) : x ∈ Z}

(f ∈ C(X−→E)).

The topology of uniform convergence on compact sets τc on C(X−→E) is the locally convex topology generated by {q˜Z : Z ⊂ X compact, q continuous seminorm on E}. Note that if E = K then τc coincides with the topology on C(X) considered in 3.7.3, i.e., the one generated by the seminorms pZ , Z ⊂ X, Z compact, where pZ (f ) = max{|f (x)| : x ∈ Z} (f ∈ C(X)).

10.5 Examples of tensor products

397

From now on in this section we will assume that C(X−→E), C(Y −→E), C(X) (= C(X−→K )) and C(Y ) (= C(Y −→K )) are equipped with the locally convex topology τc . We will compare C(X) ⊗π E and C(X−→E). Thanks to the Universal Property of the tensor product we can define the following. Definition 10.5.2 Let : C(X) ⊗ E−→C(X−→E) be the linear map given by the formula n n ( fi ⊗ yi )(x) = fi (x) yi i=1

(x ∈ X),

i=1

where f1 , . . . , fn ∈ C(X), y1 , . . . , yn ∈ E. Theorem 10.5.3 is a homeomorphic C(X−→E) with dense image.

embedding

C(X) ⊗π E−→

Proof. Direct verification shows that is injective. To see that is a homeomorphism onto its image it suffices to prove that, for a compact Z ⊂ X and a continuous seminorm q on E, we have for ϕ ∈ C(X) ⊗ E, q˜Z ((ϕ)) = (pZ ⊗ q)(ϕ). Let t ∈ (0, 1). By 10.2.6 we can write ϕ as ni=1 fi ⊗ yi , where fi ∈ C(X), and y1 , . . . , yn ∈ E \ {0} are t-orthogonal with respect to q. Then by 10.2.9, we obtain (pZ ⊗ q)(ϕ) ≥ t max pZ (fi ) q(yi ) = t max max |fi (x)| q(yi ) 1≤i≤n

1≤i≤n x∈Z

= t max max |fi (x)| q(yi ) ≥ t max q( x∈Z 1≤i≤n

x∈Z

n

fi (x) yi )

i=1

= t max q((ϕ))(x) = t q˜Z ((ϕ)). x∈Z

Conversely, we have, by using the definition of q˜Z , n q˜Z ((ϕ)) = max q( fi (x) yi ) ≥ max t max |fi (x)| q(yi ) x∈Z

i=1

x∈Z

1≤i≤n

= t max max |fi (x)| q(yi ) = t max pZ (fi ) q(yi ) 1≤i≤n x∈Z

1≤i≤n

≥ t (pZ ⊗ q)(ϕ). Combining the results we find t (pZ ⊗ q)(ϕ) ≤ q˜Z ((ϕ)) ≤ t −1 (pZ ⊗ q)(ϕ).

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As this holds for each t ∈ (0, 1) the equality follows. To prove that Im is dense in C(X−→E), let f ∈ C(X−→E), let Z ⊂ X be compact, let q be a continuous seminorm on E, and let ε > 0. We shall produce a g ∈ Im such that max q(f (x) − g(x)) < ε. x∈Z

By the usual compactness argument there is a finite partition U1 , . . . , Un of Z consisting of relatively clopen sets, and points y1 , . . . , yn ∈ E such that q(f (x) −

n

ξUi (x) yi ) < ε

i=1

for all x ∈ Z. By the Tietze–Urysohn Lemma 2.5.23 the functions ξU1 , . . . , ξUn extend to g1 , . . . , gn ∈ C(X) respectively. Put g(x) := Then g = (

n i=1

n

gi (x) yi

(x ∈ X).

i=1

gi ⊗ yi ) ∈ Im and

max q(f (x) − g(x)) = max q(f (x) − x∈Z

x∈Z

n

ξUi (x) yi ) < ε.

i=1

In the next corollary we consider spaces E that are complete (hence Hausdorff). Corollary 10.5.4 Let E be complete, let X be a k0 -space (3.7.7). Then the mapping of 10.5.2 extends to a bijective linear homeomorphism ˆ π E−→C(X−→E). C(X)⊗ Proof. The result follows from the previous theorem as soon as we have established completeness of C(X−→E). Since E is Hausdorff, so is C(X, E). Further, as X is a k0 -space and by reading the proof of (α) =⇒ (β) of 3.7.6, replacing K by E, we obtain that an f : X−→E is continuous if and only if its restriction to compact subsets of X is continuous. Now let (fi )i be a Cauchy net in C(X−→E). By completeness of E there is an f : X−→E such that fi → f uniformly on compact sets. Then f |Z is continuous for each compact Z ⊂ X, so, by the above, f ∈ C(X−→E), and we are done. By putting E := C(Y ) we obtain: ˆ π C(Y ) is isomorphic to Corollary 10.5.5 Let X, Y be k0 -spaces. Then C(X)⊗ C(X−→C(Y )).

10.5 Examples of tensor products

399

Proof. C(Y ) is complete (3.7.9(iii)). Then apply 10.5.4.

The space C(X−→C(Y )) can be made more visible. As usual, this space as well as C(X × Y ), that appear in the next theorem, are equipped with the corresponding topologies τc . Theorem 10.5.6 Let X × Y be a k0 -space. Then C(X−→C(Y )) is isomorphic to C(X × Y ). Proof. Let ϕ ∈ C(X−→C(Y )). Then the formula (ϕ)(x, y) := ϕ(x)(y)

((x, y) ∈ X × Y )

defines a map (ϕ) : X × Y −→K. Since X × Y is a k0 -space, to see that (ϕ) is in C(X × Y ) it suffices to show that (ϕ) restricted to each compact subset of X × Y is continuous. For that, let (xi )i∈I , (yi )i∈I be nets in X, Y respectively; suppose xi → x ∈ X, yi → y ∈ Y and {yi : i ∈ I } ∪ {y} lies in a compact subset of Y . We have |(ϕ)(x, y) − (ϕ)(xi , yi )| = |ϕ(x)(y) − ϕ(xi )(yi )| ≤ max(|ϕ(x)(y) − ϕ(x)(yi )| , |ϕ(x)(yi ) − ϕ(xi )(yi )| . Now ϕ ∈ C(X−→C(Y )), so if xi → x then ϕ(xi ) → ϕ(x) uniformly on compact subsets of Y . Thus, the second term of the right-hand side of the inequality tends to 0. The first term tends to 0 just because ϕ(x) ∈ C(Y ). Therefore, the above map sends C(X−→C(Y )) into C(X × Y ) and is obviously linear. What remains to be shown is that is a bijective homeomorphism. To prove surjectivity, let f ∈ C(X × Y ). We want a ϕ ∈ C(X−→C(Y )) such that (ϕ) = f . We propose to take for ϕ the map x → fx , with fx (y) := f (x, y) (x ∈ X, y ∈ Y ). We first check that this map is in C(X−→C(Y )). Clearly, for each x ∈ X, fx ∈ C(Y ). To show that ϕ is continuous, let (xi )i be a net in X converging to x ∈ X; let us see that fxi → fx uniformly on compact subsets Z of Y . A well-known property of continuous functions of two variables yields fx (y) − fx (y) = |f (xi , y) − f (x, y)| → 0 i uniformly in y ∈ Z. Next we check that (ϕ) = f . Let (x, y) ∈ X × Y . Then (ϕ)(x, y) = ϕ(x)(y) = fx (y) = f (x, y). We conclude that is surjective.

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Tensor products

To prove injectivity, let (ϕ) = 0. Then ϕ(x)(y) = 0 for all y ∈ Y , x ∈ X. Thus, ϕ = 0, and we are done. Finally we prove that is a homeomorphism. To this end we compare the collections of seminorms on C(X−→C(Y )) given by ϕ −→ max pZ2 (ϕ(x)) = max max |ϕ(x)(y)|

(Z1 ⊂ X, Z2 ⊂ Y compact),

ϕ −→ max |(ϕ)(x, y)| = max |ϕ(x)(y)|

(Z ⊂ X × Y compact).

x∈Z1

(x,y)∈Z

x∈Z1 y∈Z2

(x,y)∈Z

Obviously, both collections induce the same topology on C(X−→C(Y )), so that is a homeomorphism. As a consequence we derive the following. ˆ π C(Y ) is isomorCorollary 10.5.7 Suppose X × Y is a k0 -space. Then C(X)⊗ phic to C(X × Y ). Proof. By 10.5.5 and 10.5.6 it suffices to see that X and Y are k0 -spaces. By symmetry it is enough to prove this for X. Let πX : X × Y −→X be the coordinate map. Let f : X−→K be a map whose restriction to each compact set is continuous. Then f ◦ πX : X × Y −→K has the same property. By assumption, f ◦ πX is continuous, from which it is easily shown that f is also continuous. Remark 10.5.8 From the above proof it follows that if X × Y is a k0 -space then so are X and Y . The following natural question arises. Is the product of two k0 -spaces again a k0 -space? The answer is “no” – see the chapter notes (10.7.2). We finish this section by applying the theory about tensor products developed in this chapter to describe the topological properties of C(X−→E), which will lead to the vector-valued versions of the corresponding results given for C(X) in previous chapters. Recall that C(X−→E) and C(X) are equipped with the topology τc of uniform convergence on compacts subsets of X. As a direct application of 10.3.10 and 10.5.3 we obtain the following, using the fact that C(X) does not have the indiscrete topology. Corollary 10.5.9 (i) E is isomorphic to a subspace of C(X−→E). (ii) Suppose E does not have the indiscrete topology. Then C(X) is isomorphic to a subspace of C(X−→E). The descriptions of the topological properties of C(X−→E) are summarized in the next results.

10.5 Examples of tensor products

401

Theorem 10.5.10 Suppose E = {0}. Then we have the following: (i) C(X−→E) is normable if and only if X is compact and E is normable. (ii) C(X−→E) is metrizable if and only if X is hemicompact (see 3.7.4) and E is metrizable. (iii) C(X−→E) is (quasi)complete if and only if X is a k0 -space and E is (quasi)complete. Proof. (i) and (ii). Recall (3.7.9(i), (ii)) that X is compact (hemicompact) if and only if C(X) is normable (metrizable). Suppose C(X−→E) is normable (metrizable). By 10.5.9(i) we get that then so is E, hence E does not have the indiscrete topology. Now applying 10.5.9(ii) we obtain that C(X) is normable (metrizable). Conversely, if C(X) and E are normable (metrizable) then, by 10.3.13, C(X) ⊗π E is normable (metrizable), hence so is C(X−→E) by 10.5.3 (and by the easily seen fact that a Hausdorff space having a normable (metrizable) dense subspace is again normable (metrizable)). (iii). Recall (3.7.9(iii)) that X is a k0 -space if and only if C(X) is (quasi)complete. Suppose C(X−→E) is (quasi)complete. Let us show that then so are C(X) and E. By 10.5.9(i) E is Hausdorff. Let y ∈ E \ {0}. We will see that the map : C(X)−→C(X−→E) sending f ∈ C(X) to the function x → f (x) y in C(X, E) is a linear homeomorphic embedding with closed image (from which it follows that Im is (quasi)complete and hence so is C(X)). Obviously is well-defined, linear and injective; also, since for each compact Z ⊂ X and each continuous seminorm q on E one verifies q˜Z ((f )) = q(y) pZ (f ), we have that is a linear homeomorphism of C(X) onto its image (note that as E is Hausdorff there is a continuous seminorm on E not vanishing at y). To prove that Im is closed, let (gi )i be a net in Im converging to a g in C(X−→E). For each i, there is an fi ∈ C(X) with gi (x) = fi (x) y for all x ∈ X. Now take x ∈ X. Then g(x) = limi gi (x) = limi fi (x) y and, since E is Hausdorff, [y] is closed in E (2.1.14(i), 3.4.24), so we obtain that g(x) ∈ [y], i.e., g(x) = λx y for some λx ∈ K. It is easily seen that f : X−→K, x → λx is in C(X), so that g(x) = f (x) y for all x ∈ X, and we get that g ∈ Im . Now consider the map E−→C(X, E) given by y → gy , where gy is the constant function y. Then this map is a linear homeomorphic embedding with closed image (again using the fact that E is Hausdorff), so that E is (quasi)complete.

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Tensor products

Finally, suppose C(X) and E are (quasi)complete. Let us show that then so is C(X, E). For completeness, it follows from 10.5.4. Next, assume that X is a k0 -space and E is quasicomplete. Since E is Hausdorff, also so is C(X−→E). Now, let (fi )i be a bounded Cauchy net in C(X−→E). Then, for each x ∈ X, (fi (x))i is a bounded Cauchy net in E. So, by quasicompleteness, there is an f : X−→E such that fi → f uniformly on compacts. Then proceed as in the proof of 10.5.4 to conclude that f ∈ C(X−→E). Theorem 10.5.11 C(X−→E) is polar (dual-separating) if and only if E is polar (dual-separating). Proof. Assume C(X−→E) is polar (dual-separating). By 10.5.9(i) E is isomorphic to a subspace of C(X−→E), hence polar (dual-separating) by 4.4.16(i) (5.1.7(i)). For the converse we first treat polarity. Let E be polar. Since C(X) is polar then so is C(X) ⊗π E by 10.3.14. Thus, C(X−→E) is also polar by 4.4.16(ii) and 10.5.3. Finally, assume E is dual-separating. Let g ∈ C(X−→E), g = 0; let us find an element in C(X−→E) not vanishing at g. There is an x ∈ X with g(x) = 0 and then there is a ϕ ∈ E with ϕ(g(x)) = 0. Thus, the map C(X−→E)−→K, h → ϕ(h(x)) is in C(X−→E) and does not vanish at g. Therefore, C(X−→E) is dual-separating. Theorem 10.5.12 Suppose E does not have the indiscrete topology. Then we have the following: (i) C(X−→E) is of countable type if and only if every compact subset of X is ultrametrizable and E is of countable type. (ii) C(X−→E) is strictly of countable type if and only if there exists a continuous injection X−→K and E is strictly of countable type. (iii) C(X−→E) is of finite type (semi-Montel, nuclear) if and only if every compact subset of X is finite and E is of finite type (semi-Montel, nuclear). Proof. (i). Recall (4.3.2) that every compact subset of X is ultrametrizable if and only if C(X) is of countable type. Let C(X−→E) be of countable type. From 10.5.9 we have that C(X) and E are isomorphic to subspaces of C(X−→E), so they are of countable type by 4.2.13(i). Conversely, if C(X) and E are of countable type then, by 10.3.13, C(X) ⊗π E is of countable type, hence so is C(X−→E) by 4.2.13(ii) and 10.5.3. (ii). Recall (4.3.4) that the existence of a continuous injection X−→K is equivalent to C(X) being strictly of countable type.

10.5 Examples of tensor products

403

Let C(X−→E) be strictly of countable type. We shall construct continuous linear surjections of C(X−→E) onto C(X) and onto E (and applying 4.2.16(i) we conclude that C(X) and E are strictly of countable type). For that, take ϕ ∈ E \ {0} (such a ϕ exists because, by (i), E is of countable type and because E does not have the indiscrete topology) and take a ∈ X. Then it is straightforward to check that the linear maps C(X−→E)−→C(X), f → ϕ ◦ f , and C(X−→E)−→E, f → f (a) are continuous and surjective. Conversely, if C(X) and E are strictly of countable type then, by 10.3.13, C(X) ⊗π E is strictly countable type, hence so is C(X−→E) by 4.2.16(ii) and 10.5.3. (iii). The proofs for the cases “finite type” and “nuclear” run in the same spirit as in (i); we turn to semi-Montelness. Recall (8.7.2) that each compact subset of X is finite if and only if C(X) is semi-Montel. Let C(X−→E) be semi-Montel. From 10.5.9 we obtain that C(X) and E are isomorphic to subspaces of C(X−→E), so they are semi-Montel by 8.4.25(i). Conversely, suppose that each compact subset of X is finite and that E is semi-Montel. The assumption on X implies that the linear map C(X−→E)−→E X , f → (f (x))x∈X is a homeomorphic embedding. Since E is semi-Montel, then so are E X and C(X−→E) by 8.4.25(i), (ii). Theorem 10.5.13 Suppose E = {0}. Then C(X−→E) is Montel if and only if X is discrete and E is Montel. Proof. Let C(X−→E) be Montel. First note that then C(X−→E) is Hausdorff, hence so is E by 10.5.9(i), so certainly E does not have the indiscrete topology. Now we see that X is discrete. From 8.4.18 it follows that C(X−→E) is quasicomplete. So, by 10.5.10(iii), X is a k0 -space, i.e., C(X) is quasicomplete (3.7.9(iii)). Further, Montelness together with 10.5.12(iii) above imply that each compact subset of X is finite. Hence τc = τs on C(X) (where τs is the topology of simple convergence on C(X), see 3.7.1). Thus, (C(X), τs ) is quasicomplete and applying 3.7.2(iii) we get that X is discrete. Next we show that E is Montel. By assumption and 10.5.12(iii) we obtain that E is semi-Montel; let us prove that E is reflexive. Since X is discrete we have that C(X−→E) = E X . Then E X is reflexive and by considering a suitable coordinate map we conclude (7.4.17) that E is also reflexive. Conversely, suppose X is discrete and E is Montel. Then C(X−→E) = E X is Montel by 8.4.26(ii). Remark 10.5.14 If E = {0} then C(X−→E) = {0} for all X, and clearly the conclusions of 10.5.10 and 10.5.13 are false.

404

Tensor products

Now suppose that E has the indiscrete topology, i.e., the only continuous seminorm on E is the trivial one. Then C(X−→E) also has the indiscrete topology. Hence, for any X, we have that E and C(X−→E) are (strictly) of countable type (of finite type, semi-Montel, nuclear), which lead to the falsity of the conclusion of 10.5.12.

10.6 Non-Archimedean complexifications In this section we apply tensor products to construct the non-Archimedean version of “complexification” of a real vector space. Let us give an algebraic introduction. Let E be a K-vector space and let L ⊃ K be a field extension of K. Then L is a K-vector space, so the tensor product E ⊗ L makes sense as a K-vector space. The formula λ (x ⊗ µ) := x ⊗ λ µ

(λ, µ ∈ L, x ∈ E)

(10.5)

defines a map L × (E ⊗ L)−→E ⊗ L making E ⊗ L into a vector space over L. We use the terms “K-linear” and “K-seminorm” (“L-linear” and “Lseminorm”) to indicate linear maps and seminorms respectively on K-vector (L-vector) spaces. Now let E carry a locally convex topology and let L ⊃ K be a complete valued field extension of K. The object E ⊗π L is a locally convex space over K (as L is also a locally convex space over K, even a Banach space). Next we show that (10.5) makes E ⊗π L into a locally convex space over L. For this we have to see that the projective tensor product topology on E ⊗ L is generated by L-seminorms. Let us denote the valuation on L again by | . |, let p be a K-seminorm on E. We want to check that p ⊗ | . | is an L-seminorm, that is, we have to verify that for z ∈ E ⊗ L, z = 0, λ ∈ L× , (p ⊗ | . |)(λ z) = |λ| (p ⊗ | . |)(z). (10.6) Let t ∈ (0, 1). Let z = ni=1 xi ⊗ µi , where x1 , . . . , xn ∈ E \ {0}, and µ1 , . . . , µn ∈ L× are t-orthogonal with respect to | . | (10.2.6). Then λ z = n × does not change ti=1 xi ⊗ λ µi . Now since multiplying with λ ∈ L orthogonality, we have by 10.2.9, n xi ⊗ λ µi ) ≥ t max p(xi ) |λ µi | (p ⊗ | . |)(λ z) = (p ⊗ | . |)( i=1

1≤i≤n

= |λ| t max p(xi ) |µi | ≥ |λ| t (p ⊗ | . |)(z). 1≤i≤n

10.7 Notes

405

In a similar way it can be shown that |λ| (p ⊗ | . |)(z) ≥ t (p ⊗ | . |)(λ z). Since this holds for all t ∈ (0, 1), we have proved (10.6). Thus, we have obtained the analogue of “complexification” of a real vector space. Theorem 10.6.1 Let L be a complete valued field extension of K. Then E ⊗π L, with the scalar multiplication given by (10.5), is a locally convex space over L. Remark 10.6.2 Let X be a k0 -space. By 10.5.4 there is a bijective K-linear homeomorphism ˆ ˆ : C(X)⊗ ˆ π L−→C(X−→L), (

n

fi ⊗ µi )(x) =

i=1

n

fi (x) µi

(x ∈ X)

i=1

(where f1 , . . . , fn ∈ C(X), µ1 , . . . , µn ∈ L). From 10.6.1 it now follows that this is also an L-linear homeomorphism.

10.7 Notes Several results of this chapter (e.g. 10.2.6, 10.2.7, 10.3.15, 10.4.4, 10.5.3) appear in [86]. Lemma 10.2.9 was proved in [210], 17.3. In 19.11 of [210] one can find another proof of the fact that tensor products of nuclear spaces are nuclear (10.4.4). It was also Schneider who in the same book proved, for the first time as far as we know, that completed tensor products of Fr´echet Montel spaces are Fr´echet Montel. His proof (only for spherically complete K), on p. 129–134 of [210], has inspired us in the set up of our theory on tensor products of semi-Montel spaces developed in Section 10.4, where we generalize Schneider’s results to semi-Montel spaces and general K (10.4.14, 10.4.19). One may wonder whether the tensor product of two spaces having the strongest locally convex topology has again this topology. It is curious that the answer depends on the algebraic dimension of the spaces involved, see [184]. Versions of 10.3.15, only for spherically complete fields, were given in [100] and [187]. In Section 9.6 of [185] one can find versions of 10.5.3 as well as of 10.5.7. Our proof of 10.4.21 is taken from 2.2 of [121]. This paper provides other versions of 10.4.20, see also [210], 20.11. We announced in 10.5.8 that the product of two k0 -spaces needs not be a k0 -space. We will show this in 10.7.2, by modifying the method given in [48], VI.8, Exercise 5, XI.9, Exercise 1, for k-spaces.

406

Tensor products

Theorem 10.7.1 Let K := Qp . Let E be a K-vector space with algebraic base {ei : i ∈ I }, let τ0 be the strongest locally convex topology on E, and let C be the collection of all τ0 -compact subsets of E. Put U := {U ⊂ E : U ∩ C is relatively τ0 -clopen in C for every C ∈ C} and consider the topology τ on E generated by U as a subbase. Then (E, τ ) is a k0 -space, and there exist τ0 -clopen zero neighbourhoods {Wi : i ∈ I } in E such that W := i∈I (ei + Wi ) is a τ -clopen set with 0 ∈ W . Proof. To get the answer we follow several steps (in the last two ones we prove our conclusions): 1. If U ∈ U then E \ U ∈ U. Proof. Direct. 2. If U1 , . . . , Un ∈ U then U1 ∩ . . . ∩ Un ∈ U. Proof. Direct. 3. U = {U ⊂ E : U ∩ D is relatively τ0 -clopen in D, for every finitedimensional subspace D ⊂ E}. Proof. Suppose that, for every finitedimensional subspace D of E, U ∩ D is relatively τ0 -clopen in D. Let C ∈ C. By 3.6.6(a), C ⊂ D for some finite-dimensional subspace D ⊂ E. So U ∩ C = U ∩ D ∩ C is τ0 -clopen in C, so that U ∈ U. Conversely, let U ∈ U, let D be a finite-dimensional subspace of E and let us equip D with the restricted topology τ0 |D (which is normable by 3.4.24). Then D = n∈N Bn , where B1 , B2 , . . . are certain multiples of the unit ball of D (which are compact by local compactness of K and 2.1.15). For each n, U ∩ Bn is clopen in Bn , and Bn is clopen in D, so we obtain that U ∩ Bn is clopen in D. Hence, U ∩ D = n∈N U ∩ Bn is open in D. Also, by step 1, we can do the same for E \ U , and the converse is proved. 4. A set U ⊂ E is τ -clopen if and only if U ∈ U. Proof. Let U ∈ U. By definition of τ , U is τ -open. As, by step 1, this also holds for E \ U , we conclude that U is τ -clopen. Conversely, let U be τ -clopen. Let C ∈ C. Now U is the union of certain members of U (by step 2). Hence U ∩ C is the union of relatively τ0 -clopen sets in C, which implies that U ∩ C is τ0 -open in C. By the same token, (E \ U ) ∩ C is τ0 -open in C. Thus, U ∩ C is τ0 -clopen in C. Since this holds for all C ∈ C, it follows that U ∈ U. 5. τ is zero-dimensional. Proof. τ is generated by U, a collection of τ -clopen sets (by step 4). 6. τ ≥ τ0 . In particular, (E, τ ) is Hausdorff. Proof. τ0 is zero-dimensional, so it is generated by its clopen sets. But any τ0 -clopen set lies in U. Hence, τ0 is generated by a subcollection of U, and we are done. 7. (E, τ ) and (E, τ0 ) have the same compact sets. Proof. By step 6 we have that every τ -compact set is τ0 -compact.

10.7 Notes

407

Conversely, let C ⊂ E be τ0 -compact, i.e., C ∈ C. To show that C is τ -compact, let (Uj )j be a covering of C where all Uj are τ -open; we prove the existence of a finite subcovering. Since τ is zero-dimensional by step 5, we may assume that all Uj are τ -clopen. This means that each Uj ∈ U (by step 4). By definition, for each j , Uj ∩ C is τ0 -relatively clopen in C, which yields a covering of C by τ0 -clopen sets. By τ0 -compactness there are j1 , . . . , jn such that Uj1 ∩ C, . . . , Ujn ∩ C cover C. Then Uj1 , . . . , Ujn is the required finite subcovering. 8. τ = τ0 on τ (τ0 )-compact sets. Proof. Let C ⊂ E be τ -compact (= τ0 compact by step 7). From step 6, the identity (C, τ |C)−→(C, τ0 |C) is continuous, hence a homeomorphism, and we see that τ = τ0 on C. 9. (E, τ ) is a k0 -space. Proof. Let U ⊂ E be such that, for each τ -compact C ⊂ E, U ∩ C is relatively τ -clopen in C. By steps 7 and 8 above we have that U ∩ C is relatively τ0 -clopen in every C ∈ C. This means that U ∈ U, i.e., that U is τ -clopen by step 4. 10. There exist τ0 -clopen zero neighbourhoods {Wi : i ∈ I } in E for which W := i∈I (ei + Wi ) is a τ -clopen set with 0 ∈ W . Proof. For each i ∈ I , let Hi := [ej : j ∈ I, j = i]. Each hyperplane Hi is τ0 -closed, hence there is a τ0 -clopen zero neighbourhood Wi such that (ei + Wi ) ∩ Hi = ∅. Then clearly 0 ∈ ei + Wi for each i, so that 0 ∈ W . Let us show that W is τ -clopen, i.e., that W ∈ U (by step 4). Let D be a finite-dimensional subspace of E. Then there is a finite set J ⊂ I such that D ⊂ [ej : j ∈ J ]. Now if i ∈ J we have (ei + Wi ) ∩ D ⊂ (ei + Wi ) ∩ [ej : j ∈ J ] ⊂ (ei + Wi ) ∩ Hi = ∅. Thus, W ∩ D = j ∈J (ej + Wj ) ∩ D, which is a finite union of τ0 -clopen subsets of D, hence τ0 -clopen in D. It follows from step 3 that W ∈ U. Example 10.7.2 There exist k0 -spaces X, Y such that X × Y is not a k0 -space. Proof. Let K := Qp , H := NN and let E, F be K-vector spaces with algebraic bases {xϕ : ϕ ∈ H} and {yn : n ∈ N} respectively. Put X := (E, τ ), Y := (F, τ ), with τ as in 10.7.1, where we proved that X, Y are k0 -spaces. In a first step we produce a set W ⊂ E × F that is clopen with respect to the topology τ on E × F (defined again according to 10.7.1), but that is not closed in the product topology τ × τ on E × F . Then we will see that this leads to conclude that X × Y is not a k0 -space. Let τ0 be the strongest locally convex topology on E × F . Put W1 := {p ϕ(n) (xϕ , yn ) : ϕ ∈ H, n ∈ N}. Then W1 is an algebraic base of E × F , so

408

Tensor products

by 10.7.1 there are τ0 -clopen zero neighbourhoods W(ϕ,n) in E × F such that W := {p ϕ(n) (xϕ , yn ) + W(ϕ,n) : ϕ ∈ H, n ∈ N} is a τ -clopen set in E × F τ ×τ with 0 ∈ W . To see that W is not τ × τ -closed we prove that 0 ∈ W . For τ ×τ this, it is enough to show that 0 ∈ W1 . Let U, V be τ -open zero neighbourhoods in E, F respectively. For each ϕ ∈ H there is a λϕ ∈ K × such that {λ xϕ : |λ| ≤ λϕ } ⊂ U. In the same way, there exist λ1 , λ2 , . . . in K × such that, for each n, {λ yn : |λ| ≤ |λn |} ⊂ V . Let ϕ˜ ∈ H be the map defined by ϕ(n) ˜ = [max(n,

− log |λn | )] + 1 log p

(n ∈ N),

˜ ˜ n) ≤ λϕ˜ . Then where [ ] indicates the entire part. Take n˜ ∈ N with p −ϕ( ˜ ˜ ˜ ˜ ˜ n) ˜ n) ˜ n) ˜ n) p ϕ( ≤ λϕ˜ ) and pϕ( ≤ xϕ˜ ∈ U (because p ϕ( yn˜ ∈ V (because pϕ( ˜ ˜ n) |λn˜ |). Hence p ϕ( (xϕ˜ , yn˜ ) ∈ (U × V ) ∩ W1 . This finishes the proof of the first step. Next we prove that X × Y is not a k0 -space. Let W ⊂ E × F be as above. As W is τ -clopen, applying step 4 of the proof of 10.7.1 we have that, for each τ0 -compact set C ⊂ E × F , W ∩ C is relatively τ0 -clopen in C. Now, since τ0 is the product of the corresponding strongest locally convex topologies of E, F , we can use steps 7 and 8 of the proof of 10.7.1 to obtain that, for each τ × τ -compact set C ⊂ E × F , W ∩ C is relatively τ × τ -clopen in C. But W is not τ × τ -closed, so X × Y is not a k0 -space, and we are done. Remark 10.7.3 Subspaces of k0 -spaces need not be k0 -spaces. In fact, let X, Y be as in the above example, where we proved that X × Y is not a k0 -space. Let Z be the Banaschewski compactification of X × Y (see Subsection 2.5.3). Then X × Y is homeomorphically embedded in Z which, by compactness, is a k0 -space. More information on various-spaces of vector-valued continuous functions, involving tensor products techniques or not, can be found in [1], [86], [112], [113], [114], [115], [116], [120], [123], [124], [127], [128], [132], [134], [176], [178], [179]. Also, tensor products were used in [64] and [119] to study generalizations of the perfect sequence spaces and the K¨othe spaces of Chapter 9.

11 Inductive limits

Locally convex inductive limits over the real and complex field appear in great abundance in many disciplines of classical analysis and its applications, see [29] for a background account. It is our impression that their non-Archimedean counterparts will play a similar role, see the chapter notes for some supporting evidence on this. Inductive limits have popped in occasionally in previous chapters, often accompanied by promises to discuss some delicate matters later on. Now that we have the necessary locally convex theory available, we are able to develop some framework on inductive limits, thereby keeping our word. After recalling the definition of an inductive limit we describe the general form of its continuous seminorms (11.1.1) and zero neighbourhoods (11.1.2). By using basically classical methods we derive results on strictness and regularity, e.g. that every strict inductive sequence of Fr´echet spaces is regular (11.1.7). This is followed by a typically non-Archimedean discussion (11.1.8) on weak forms of regularity and strictness. Grothendieck’s result saying that a Hausdorff (LF)-space is regular if and only if each bounded set is contained in a Banach disk, is translated without too much effort to the non-Archimedean case (11.1.11). We conclude Section 11.1 with a description of the dual of an inductive sequence being a projective system (11.1.13) and with a few examples (11.1.15). Stability properties are the main concern of Section 11.2. After recalling results obtained previously in the book (11.2.1) we prove in a modified classical way that the Hausdorff property (11.2.2) and metrizability (11.2.8) are not stable under taking inductive limits. Metrizability being disastrous (11.2.8), we present conditions under which the Hausdorff property (11.2.4) and (weak) (quasi)completeness (11.2.5) are guaranteed. The non-stability of polarity (11.2.11) is a non-Archimedean matter. The section is completed by discussing stability of (semi-)Montelness and reflexivity (11.2.13–11.2.17). 409

410

Inductive limits

The special case in which the canonical inclusions are compactoid maps is treated in Section 11.3. Both in the classical and non-Archimedean context such “compact(oid) inductive limits” seem to be important. Indeed, they have nice properties (11.3.5). Our proofs form a curious mixture of classical methods combined with subtleties on non-Archimedean compactoids. For certain real matrixes B we reconsider in Section 11.4 the space ∞ (B) introduced in 9.5.8. We also include a new related space 0 (B). If B satisfies (9.19) both turn out to be compactoid inductive limits (11.4.2) yielding much information on spaces such as A† (r) (11.4.3). Finally, a sophisticated choice of the matrix B leads to various examples, such as a non-regular inductive limit of Banach spaces that is not quasicomplete (11.4.10(a)). In the final Section 11.5 we prove that every compactoid in a Hausdorff (LM)-space is metrizable (11.5.5). To obtain the statement we combine the notion of resolution (11.5.1) with our theory of t-frames of Section 3.10. Especially in the case of a non-spherically complete K this metrizability result is, perhaps more than in the classical situation, of importance. This is because for such scalar fields K not much is known about compactoids in general, but for metrizable compactoids a lot of subtle information is available.

11.1 Basic facts and examples Recall (3.4.32) that an inductive sequence is an increasing sequence of locally convex spaces (En , τn ) in such a way that each inclusion (En , τn ) → (En+1 , τn+1 ) is continuous (in other words, τn+1 |En ≤ τn for all n). By 3.4.34 we can view its inductive limit as the space E := n∈N En equipped with the so-called inductive topology, which is the strongest locally convex topology τ on E for which all the inclusions (En , τn )−→(E, τ ) are continuous (in other words, τ |En ≤ τn for all n). From now on in Sections 11.1−11.2 (En )n := (En , τn )n is an inductive sequence of locally convex spaces with inductive limit E := (E, τ ). The weak topologies on E and En are denoted by σ and σn , respectively. We start with two classical results, the proofs of which are left to the reader. They describe the continuous seminorms, the continuous operators and the absolutely convex zero neighbourhoods of inductive limits. The fact that nonArchimedean absolutely convex sets are BK -modules makes the description of the continuous seminorms and of the zero neighbourhoods of an inductive

11.1 Basic facts and examples

411

limit in terms of the steps slightly simpler than in the Archimedean case (see the comments in [29], p. 43). Theorem 11.1.1 (i) A seminorm p on E is τ -continuous if and only if p|En is τn -continuous for each n. (ii) For each n, let Pn be the collection of all τn -continuous seminorms on ' En . Then P := { n pn : pn ∈ Pn for each n} is the collection of all τ continuous seminorms on E, where for each x ∈ E, ( pn (x) := inf{ max pi (xi ) : m ∈ N, 1≤i≤m

n

x=

m

xi , xi ∈ Ei for each i ∈ {1, . . . , m}}.

i=1

(iii) An operator T of E into a locally convex space is τ -continuous if and only if T |En is τn -continuous for each n. Theorem 11.1.2 For an absolutely convex set U ⊂ E the following are equivalent: (α) U is a zero neighbourhood in E. (β) U ∩ En is a zero neighbourhood in En for each n. (γ ) U = n Un , where each Un is an absolutely convex zero neighbourhood in En . (δ) U = n Vn , where V1 ⊂ V2 ⊂ · · · and each Vn is an absolutely convex zero neighbourhood in En . We now introduce the classical notions of strictness and regularity and give some results related to them. Except for 11.1.8, they are the p-adic translations of well-known facts in the Archimedean setting (see [29]). Definition 11.1.3 (i) (En )n is called strict (and E is called a strict inductive limit) if τn+1 |En = τn for each n. (ii) (En )n is called regular (and E is called a regular inductive limit) if for each bounded set B in E there exists an n such that B ⊂ En and B is bounded in En . Remarks 11.1.4 (a) An inductive sequence (En )n such that for each bounded set B in E there exists an n with B ⊂ En , is called α-regular. Clearly regularity implies αregularity. But the converse is not true, see the chapter notes for references.

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Inductive limits

(b) In 11.4.10(a) we will see that there exist inductive sequences of Banach spaces that are not even α-regular. Theorem 11.1.5 If (En )n is strict then τ |En = τn for each n. Proof. Let n ∈ N, let pn be a τn -continuous seminorm on En . By strictness and 3.4.3, pn extends to a τn+1 -continuous seminorm pn+1 on En+1 and so on. The formula p(x) = pm (x) if m ∈ N, m ≥ n, x ∈ Em , defines a seminorm p on E which is τ -continuous (11.1.1(i)) and extends pn . Thus, τ |En = τn . Under some assumptions strictness implies regularity. Theorem 11.1.6 If (En )n is strict and En is closed in En+1 for each n then (En )n is regular. Proof. By 11.1.5 it suffices to show that any sequence x1 , x2 , . . . with xn ∈ En+1 \ En for each n is not τ -bounded. There is a continuous seminorm p1 on E2 for which p1 (x1 ) ≥ 1. By strictness and 3.4.3 p1 extends to a continuous seminorm q2 on E3 . As E2 is closed in E3 , applying 4.1.6 we find a continuous seminorm r2 on E3 , which is zero on E2 , such that r2 (x2 ) ≥ 2. Set p2 := max(q2 , r2 ). Then p2 extends p1 and p2 (x2 ) ≥ 2. Continuing in this way we obtain continuous seminorms pn on En+1 such that pn+1 |En+1 = pn and pn (xn ) ≥ n for each n. The formula p(x) = pn (x) if n ∈ N, x ∈ En+1 , defines a continuous (11.1.1(i)) seminorm p on E which is unbounded on {x1 , x2 , . . .} and so this set is not τ -bounded. Corollary 11.1.7 (p-adic Dieudonn´e–Schwartz Theorem) Every strict inductive sequence of Fr´echet spaces is regular. We will also need the following technical versions of 11.1.5 and 11.1.6 for the weak topologies, especially in case K is allowed to be non-spherically complete, see e.g. 11.2.5(iii) and 11.2.16. Theorem 11.1.8 Suppose σn+1 |En = σn for all n. Then σ |En = σn for all n. If, in addition, En is weakly closed in En+1 for all n, then each weakly bounded set in E is contained and weakly bounded in some En . Proof. For the first part it suffices to prove that, for all n, each functional in En has an extension in E . So, let n ∈ N, fn ∈ En (= (En , σn ) ). Our assumption

11.1 Basic facts and examples

413

σn+1 |En = σn , together with 4.2.6 and the fact that (En+1 , σn+1 ) is of countable type, lead to the existence of an fn+1 ∈ (En+1 , σn+1 ) = En+1 that extends fn . Continuing in this way we obtain fm ∈ Em such that fm+1 |Em = fm for all m ≥ n. The formula f (x) = fm (x) if m ∈ N, m ≥ n, x ∈ Em , defines an extension f ∈ E (11.1.1(iii)) of fn . Now, suppose that additionally each En is weakly closed in En+1 ; let us show that every weakly bounded subset of E is contained and weakly bounded in some En . Since σ |En = σn for all n, it is enough to see that any sequence x1 , x2 , . . . in E, with xn ∈ En+1 \En for each n, is not weakly bounded in E. As E1 is weakly closed in E2 , there exists an f1 ∈ E2 such that |f1 (x1 )| ≥ 1. Now f1 extends to some g2 ∈ E3 (see the proof of the first part). As E2 is a polar subset of E3 (5.2.8), there exists an h2 ∈ E3 , h2 = 0 on E2 , with |h2 (x2 )| ≥ |g2 (x2 )| + 2. Then f2 := g2 + h2 is in E3 , extends f1 and |f2 (x2 )| ≥ 2. Inductively we arrive at fn ∈ En+1 such that fn+1 |En+1 = fn and |fn (xn )| ≥ n for each n. The formula f (x) = fn (x) if n ∈ N, x ∈ En+1 , defines a continuous (11.1.1(iii)) linear functional on E that is not bounded on {x1 , x2 , . . .} and so this set is not weakly bounded in E. Remark 11.1.9 Later (11.4.4) we will construct an inductive sequence of Banach spaces showing that certain natural converses of 11.1.6, 11.1.7 and 11.1.8 are false. Following Grothendieck’s idea, see [98], Chapter 5, we will show (11.1.11) that, for inductive sequences whose steps are Fr´echet spaces, regularity admits a description in terms of a completeness-like property. For that we need the next Closed Graph Theorem, which extends the one given in 2.1.19(ii) for Banach spaces, and the Fr´echet version indicated in 3.5.11. Theorem 11.1.10 (Closed Graph Theorem) Let F be a barrelled space and let G be a Fr´echet space. If T : F −→G is a linear map whose graph is closed in F × G, then T is continuous. Proof. Let U be an absolutely convex zero neighbourhood in G. Set U0 := U . Since G is metrizable there exists a base of absolutely convex zero neighbourhoods U1 , U2 , . . . in G with U0 ⊃ U1 ⊃ U2 ⊃ · · · (3.5.2), and since F is barrelled we have that T −1 (Un ) is a zero neighbourhood in F for all n = 0, 1, . . . . Our conclusion follows as soon as we obtain that T −1 (U ) ⊂ T −1 (U ).

414

Inductive limits

Let x ∈ T −1 (U ). A sequence x0 , x1 , x2 , . . . in F can be selected inductively such that xn ∈ T −1 (Un ) and x ∈ x0 + · · · + xn + T −1 (Un+1 ) for all y in G, as n = 0, 1, . . . . Then the series ∞ i=0 T (xi ) converges to a certain limi T (xi ) = 0. Also, by closedness of the Un (3.3.11(ii)), y ∈ T ( ni=0 xi ) + Un for all n and, in particular, y ∈ T (x0 ) + U0 ⊂ U0 + U0 = U . The proof is complete if we show that (x, y) is in the closure of the graph of T in F × G. Let V , W be neighbourhoods of x and y in F and G respectively and p choose p ∈ N with y + Up ⊂ W . There is a u ∈ V such that u − i=0 xi ∈ T −1 (Up+1 ), hence T (u) ∈ y + Up + Up+1 ⊂ y + Up ⊂ W . Thus, (u, T (u)) ∈ G(T ) ∩ (V × W ) and we are done. We also will need the concept of Banach disk. As in the classical case, a Banach disk in a locally convex space is a bounded absolutely convex set B for which the normed space ([B], pB ) is Banach, where pB is the Minkowski function on [B] associated to B, see 3.1.9(ii). Note that if τ1 , τ2 are locally convex topologies on a vector space F and if an absolutely convex set B ⊂ F is τ1 -bounded and τ2 -bounded, then B is a Banach disk in (F, τ1 ) if and only if it is a Banach disk in (F, τ2 ). Every sequentially complete bounded absolutely convex set in a Hausdorff space is a Banach disk (the proof is classical and left to the reader, see e.g. [171], 3.2.5). On the other hand, there are Banach disks that are not sequentially complete. For an example, let B be the closed unit ball of C 1 (Zp ). Then B is a Banach disk in C(Zp ) (because it is bounded in C(Zp ), and complete in C 1 (Zp )), but B is not (sequentially) complete in C(Zp ). Now we have the necessary material to prove the announced result. Theorem 11.1.11 Let E be a Hausdorff inductive limit of an inductive sequence (En )n of Fr´echet spaces. Then (En )n is regular if and only if each bounded subset of E is included in a Banach disk. Proof. Assume that (En )n is regular and let B be a bounded subset of E. There is an n such that B is contained and bounded in En . Then B is included in the Banach disk acoτn B. Conversely, assume that each bounded subset B of E is included in a Banach disk. Let us prove that any such B is contained and bounded in some En . To this end we may suppose that B itself is a Banach disk. Let EB := ([B], pB ). EB is a Banach space and, by boundedness, the inclusion map iB : EB −→E is continuous. For n ∈ N, put Fn := iB−1 (En ). From EB = n∈N Fn and by the Baire Category Theorem, Fn is dense in EB for some n; without loss assume all Fn are dense in EB . The next step is to prove that some Fn is barrelled, we even show

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415

that some Fn is Baire. In fact, let for each n ∈ N, Fn = i∈N Vni . Then, since EB EB = n,i∈N Vni and EB is Baire, there exist n, i such that Vni contains a ball W in EB . By density of Fn , the set W ∩ Fn is non-empty. Also, W ∩ Fn is Fn open in Fn and contained in Vni . It follows that Fn is Baire. The restriction of iB to Fn , considered as a map of the barrelled space Fn into the Fr´echet space En , has closed graph (if (xi )i is a net such that xi → x in Fn and iB (xi ) = xi → y in En then we have xi → x and xi → y in E and, since E is Hausdorff, x = y). By the Closed Graph Theorem 11.1.10 this restriction map is continuous. We finally prove that iB (EB ) ⊂ En (then B is contained and bounded in some En and we are done). For that, let z ∈ EB . By density of Fn there exist z1 , z2 , . . . ∈ Fn such that zm → z in EB . By continuity of iB : EB −→E we obtain iB (zm ) → iB (z) in E. On the other hand, (zm )m is Cauchy in Fn , so by completeness, (iB (zm ))m converges in En to some y ∈ En . Hence also iB (zm ) → y in E. As E is Hausdorff, we have iB (z) = y ∈ En . As complete bounded absolutely convex sets in Hausdorff spaces are Banach disks, as a consequence of 11.1.11 we have the following. Corollary 11.1.12 Every quasicomplete inductive limit of Fr´echet spaces is regular. We now want to describe the dual of an inductive sequence. The inclusions E1 −→E2 −→ · · · induce naturally adjoint maps · · · −→E2 −→E1 which are continuous if we equip each En with the strong topology, and we obtain a projective system whose projective limit may be considered as the dual of lim En as follows. −→

)b −→ Theorem 11.1.13 The strong duals (En )b , with the maps πn : (En+1 (En )b , f → f |En , form a projective system, and ψ : Eb → lim(En )b , g → ←− (g|E1 , g|E2 , . . .) is a continuous bijective operator. If, in addition, (En )n is regular then ψ is a homeomorphism, in other words, (lim En )b ∼ lim(En )b . −→

←−

Proof. For each n, the map πn (which is the adjoint of the inclusion En −→En+1 ) is continuous (7.3.3), so ((En )b )n is a projective system. Also, it is immediate that ψ is a well-defined injective operator. Continuity follows because, for each n, the composition of ψ with the n-th coordinate map lim(En )b −→(En )b is the map Eb −→(En )b , g → g|En which is continuous ←− as it is the adjoint of the inclusion En −→E. Next, let us see surjectivity. Let h := (h1 , h2 , . . .) ∈ lim(En )b , i.e., hn+1 |En = hn for all n. Then the formula ←−

g(x) = hn (x) if x ∈ En , defines a g ∈ E (11.1.1(iii)) for which ψ(g) = h.

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Now assume regularity and let (gi )i be a net in E such that (ψ(gi ))i converges to 0 in lim(En )b . This means that, for each n, (gi |En )i converges to 0 ←−

in (En )b , i.e., gi |En → 0 uniformly on bounded subsets of En . To prove that gi → 0 in Eb , let B ⊂ E be bounded. By regularity B is contained and bounded in some En and from the above gi → 0 uniformly on B and we are done. Remarks 11.1.14 (a) Note that by 5.2.5 the maps πn of 11.1.13 are injective for all n (i.e., ((En )b )n ) is a projective sequence) if and only if each En is weakly dense in En+1 . (b) In 11.4.5 we will apply 11.1.13 to describe the duals of some important examples of sequence spaces. (c) Starting with a projective system we may get by a procedure, “dual” to the one of 11.1.13, an inductive sequence, see 11.6.1 (compare with [29], p. 57). We complete this section by interpreting a few examples of Section 3.7 as inductive limits of Banach spaces. New information about them will be given along this chapter. Examples 11.1.15 (I) Let X be a non-empty zero-dimensional Hausdorff topological space. Assume that X is locally compact and σ -compact. Then it has a fundamental sequence of compact open sets U1 ⊂ U2 ⊂ · · · . With the obvious embeddings C(U1 ) ⊂ C(U2 ) ⊂ · · · , we have that the space of continuous functions (Cc (X), τi ) of 3.7.21 is the inductive limit (Cc (X), τi ) = lim(C(Uj ), . ∞ ). −→

This inductive sequence of Banach spaces is strict (same proof as in 3.7.53(iv), for n = 0) and hence regular by 11.1.7. (II) Let X and U1 ⊂ U2 ⊂ · · · be as in the above example. Suppose additionally that X is a subset of K without isolated points. Let n ∈ {0, 1, . . .}. With the obvious embeddings C n (U1 ) ⊂ C n (U2 ) ⊂ · · · , we have that the space of differentiable functions (Ccn (X), τWn ), with W = C ∞ (X), of 3.7.53 is the inductive limit (Ccn (X), τWn ) = lim(C n (Uj ), . n ). −→

Similarly, we obtain that this inductive sequence of Banach spaces is strict and regular. Note that for n = 0, (Cc0 (X), τW0 ) = (Cc (X), τi ). Note also that if X is not compact we may take U1 U2 · · · and then the inductive sequences of (I) and (II) are strictly increasing. (III) For the spaces A† (r), r ∈ (0, ∞) of overconvergent power series on B(0, r) and A† (0) of germs of analytic functions at 0 (see 3.7.27), we

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have A† (r) = lim Fk , −→

where Fk is the Banach space {(λ0 , λ1 , . . .) ∈ K N∪{0} : supn∈N∪{0} |λn | / bnk < ∞} equipped with the canonical supremum norm, and where, for each k ∈ N, n ∈ N ∪ {0}, n k if r > 0 k r(k+1) (11.1) bn := n if r = 0. k (This follows easily from 9.7.2–9.7.4 and the comments before 9.7.2. Note that (Fk )k is strictly increasing. In 11.4.3 we will see that these inductive sequences (Fk )k are regular and non-strict.

11.2 Stability properties of inductive limits In previous chapters we have given several results showing that some classes of locally convex spaces are stable under taking inductive limits. They are summarized in the next theorem. Theorem 11.2.1 (i) (4.2.13(v), 4.2.16(v)) Inductive limits of spaces (strictly) of countable type are (strictly) of countable type. (ii) (7.1.13(ii)) Inductive limits of (polarly) barrelled spaces are (polarly) barrelled. (iii) (8.5.7(vi)) Inductive limits of nuclear spaces are nuclear. For various other classes of spaces, however, such stability is a more delicate matter as we already announced previously. (For example, in 5.3.8 we met an inductive limit of spaces of finite type that is not of finite type.) In this section we will look into this. We start by considering the Hausdorff property (announced in 3.4.33). As in the classical case (see [57]), we will prove, on the one hand that there exist inductive sequences of Hausdorff spaces with non-Hausdorff inductive limits (11.2.2), and on the other hand that the Hausdorff property is preserved under strict and under regular inductive limits (11.2.4). Example 11.2.2 There exists an inductive sequence of Hausdorff spaces whose inductive limit is not Hausdorff.

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Proof. We will construct an inductive sequence (En )n of non-zero normed spaces such that its inductive limit has the indiscrete topology, as follows. Suppose K ⊃ Qp . Let E be the vector space of polynomial functions f : Zp −→K with f (0) = 0. For n ∈ N ∪ {0} and f ∈ E, set f n := max{|f (x)| : |x| ≤ p−n }. Then, for all f ∈ E, f 0 ≥ f 1 ≥ · · · and all . n are norms on E. Putting En := (E, . n ) we consider the inductive sequence E0 −→E1 −→E2 −→ · · · , where each arrow is the identity map. As a vector space, lim En = E. Now let p be a continuous seminorm on −→ lim En ; we shall prove that p = 0. As p|E0 is continuous, there is a C > 0 −→ such that p(f ) ≤ C f 0 for all f ∈ E. Now take any g ∈ E; it suffices to prove that p(g) ≤ C. Since g(0) = 0 there is an n for which gn ≤ 1. The function h : Zp −→K, g(x) if |x| ≤ p−n x −→ 0 if p −n < |x| ≤ 1 is continuous, h(0) = 0 and h0 ≤ 1. By the Weierstrass Theorem 1.2.17 there are polynomial functions P1 , P2 , . . . : Zp −→K such that Pr → h, and then also Pr − Pr (0) → h, uniformly on Zp . So there is an s ∈ N such that Pr − Pr (0)0 ≤ 1 for all r ≥ s. Thus, putting hm := Pm+s − Pm+s (0) (m ∈ N) we get a sequence h1 , h2 , . . . in E with hm 0 ≤ 1 for all m and such that hm → h uniformly on Zp . Then limm g − hm n = 0 and so, by continuity of the canonical map En −→ lim En , we also have limm p(g − hm ) = 0, and −→ therefore limm p(hm ) = p(g). But p(hm ) ≤ Chm 0 ≤ C for all m, from which we obtain p(g) ≤ C. Remark 11.2.3 Notice that in the above example the normed spaces En are polar, hence dual-separating. Theorem 11.2.4 (i) Strict inductive limits of Hausdorff spaces are Hausdorff. (ii) Regular inductive limits of Hausdorff spaces are Hausdorff. Proof. The proof of (i) is a direct consequence of 11.1.5. To prove (ii), suppose that (En )n is regular and that all the En are Hausdorff; let us prove that E is also Hausdorff. The closure {0} of {0} in E is a bounded subspace of E and by regularity it is contained and bounded in some En . Since En is Hausdorff we obtain that {0} = {0} and we are done.

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We continue by considering stability of various completeness properties (see 3.4.33), compare also [29], p. 47. Theorem 11.2.5 (i) Suppose that (En )n is strict. If each En is complete then so is E. (ii) Suppose that (En )n is strict and that En is closed in En+1 for all n. If each En is quasicomplete then so is E. (iii) Suppose that σn+1 |En = σn and that En is weakly closed in En+1 for all n. If each En is weakly quasicomplete then so is E. Proof. (i). Let (xi )i∈I be a Cauchy net in E. As a first step we show that there is an m ∈ N such that for any i ∈ I and any absolutely convex zero neighbourhood U in E there is a j ≥ i such that xj ∈ Em + U . Assuming the contrary we derive a contradiction. Indeed, for each m ∈ N there is an i(m) ∈ I and an absolutely convex zero neighbourhood Um in E such that xj ∈ Em + Um for all j ≥ i(m). We certainly may assume U1 ⊃ U2 ⊃ · · · . Consider the set := En ∩ Un , n

which is an absolutely convex zero neighbourhood in E (11.1.2). We claim that ⊂ Em + Um for all m. It suffices to show that En ∩ Un ⊂ Em + Um for all n, m and this indeed holds because if n ≤ m then En ⊂ Em , and if n ≥ m then Un ⊂ Um . It follows that Em + ⊂ Em + Um for all m. Choose now an index i ∈ I such that xj − xi ∈ for all j ≥ i. Letting m ∈ N be such that xi ∈ Em we arrive at the contradiction that xj = xi + (xj − xi ) ∈ Em + ⊂ Em + Um for all j ≥ i. Denoting by U the set of all absolutely convex zero neighbourhoods in E we introduce the set I × U directed by the relation “(i, U ) ≤ (j, V ) if and only if i ≤ j and V ⊂ U ”. Fix an m ∈ N with the property that we have established above. For any pair (i, U ) ∈ I × U we then have an index i (U ) ≥ i and an x(i,U ) ∈ Em such that xi (U ) − x(i,U ) ∈ U . In the next step we show that (x(i,U ) )(i,U )∈I ×U in fact is a Cauchy net in Em . By strictness and 11.1.5, a base of absolutely convex zero neighbourhood in Em is formed by the sets Em ∩ U , with U ∈ U. Fix such a U and for this one fix an i ∈ I such that xk − xl ∈ U for all k, l ≥ i. Consider now any two pairs (k, V ), (l, W ) ≥ (i, U ). We have x(k,V ) − xk (V ) ∈ V and xl (W ) − x(l,W ) ∈ W . Since k (V ) ≥ k ≥ i, l (W ) ≥ l ≥ i and V + W ⊂ U + U ⊂ U we obtain x(k,V ) − x(l,W ) = (x(k,V ) − xk (V ) ) + (xk (V ) − xl (W ) ) + (xl (W ) − x(l,W ) ) ∈ V + U + W ⊂ U .

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Since Em is complete by assumption, the Cauchy net (x(i,U ) )(i,U ) converges to some vector x in Em . We conclude by showing that the original net (xi )i∈I converges to x in E. For that, let U ∈ U. We find a pair (k, V ) ∈ I × U such that 1. x(l,W ) − x ∈ U ∩ Em ⊂ U for all (l, W ) ≥ (k, V ), 2. xk1 − xk2 ∈ U for all k1 , k2 ≥ k. As a special case of 1 we have x(k,V ∩U ) − x ∈ U , and since by construction xk (V ∩U ) − x(k,V ∩U ) ∈ V ∩ U , it follows that xk (V ∩U ) − x = (xk (V ∩U ) − x(k,V ∩U ) ) + (x(k,V ∩U ) − x) ∈ U + U ⊂ U . Also, using 2 and that k (V ∩ U ) ≥ k we have xl − xk (V ∩U ) ∈ U for l ≥ k. Hence we finally obtain that, for l ≥ k, xl − x = (xl − xk (V ∩U ) ) + (xk (V ∩U ) − x) ∈ U + U ⊂ U . (ii). By 11.2.4(i) E is Hausdorff. Let (xi )i be a bounded Cauchy net in E. Since (En )n is regular (11.1.6), there exists an n such that xi ∈ En for all i and (xi )i is a bounded net in En . Also, since τ |En = τn (11.1.5), (xi )i is a τn Cauchy net in En . By quasicompleteness of En , xi −→ x for some x ∈ En . τ Then xi −→ x. (iii). By 11.1.8 we have that σ |En = σn for all n and that every weakly bounded subset of E is contained and weakly bounded in some En . Then, proceeding as in (ii) with the obvious modifications, it follows that E is weakly quasicomplete. Remark 11.2.6 We will see in 11.4.10(a) that in general completeness properties are not preserved under taking inductive limits. Next, we consider metrizability. In 3.5.6 we announced that inductive limits of metrizable spaces need not be metrizable. In 11.2.8 we will show that, even when the steps are Banach, metrizability of the inductive limit only occurs in trivial cases. (This is the non-Archimedean version of Corollary 3 of [164].) We need a basic lemma. Lemma 11.2.7 Let A1 ⊂ A2 ⊂ · · · be absolutely convex subsets of a Hausdorff barrelled space F , covering F . Let F ∧ be the completion of F and let λ1 , λ2 , . . . be a sequence in K with 1 ≤ |λ1 | < |λ2 | < · · · and |λn | → ∞. Then F ∧ = ∧ ∧ ∧ n∈N λn An , where An denotes the closure of An in F . In particular, if each An is complete then F is complete. Proof. Suppose there exists an x ∈ F ∧ \ n∈N λn A∧n . Then, for each n, ∧ ∧ λ−1 n x ∈ F \ An and by 4.1.6 there exists a continuous seminorm pn on ∧ ∧ F such that pn (λ−1 n x) = 1 and pn (y) < 1 for all y ∈ An . Let qn = pn |F , n ∈ N. Since the sequence A1 , A2 , . . . is increasing and covers F , we obtain that q1 , q2 , . . . is pointwise bounded. By barrelledness and 7.1.5, q1 , q2 , . . .

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is equicontinuous and by a standard reasoning so is p1 , p2 , . . .. Hence the formula p(x) = supn∈N pn (x) defines a continuous seminorm on F ∧ . Now ∧ −1 −1 λ−1 n x → 0 in F , hence p(λn x) → 0, implying p(λn x) < 1 for some n, a contradiction. Theorem 11.2.8 The inductive limit of a strictly increasing inductive sequence of Banach spaces is not metrizable. Proof. Assume there exists a strictly increasing inductive sequence (En )n of Banach spaces whose inductive limit E is metrizable; we derive a contradiction. One obtains (inductively) an increasing sequence A1 , A2 , . . . of absolutely convex sets (where each An is a multiple of the closed unit ball of En ), such that E = n∈N An . Choose a sequence λ1 , λ2 , . . . in K with 1 ≤ |λ1 | < |λ2 | < · · · and |λn | → ∞. Since E is barrelled (7.1.3 and 7.1.13(ii)), applying 11.2.7 we get E ∧ = n∈N λn A∧n (for a set X ⊂ E, X∧ denotes the closure of X in E ∧ ). As E ∧ is a Fr´echet space, by the Baire Category Theorem there exists a k such that A∧k is a zero neighbourhood in E ∧ . To get our contradiction it is enough to prove that A∧k = Ak , since in this case we will have Ek = E. Let {Um : m ∈ N} be a decreasing countable base of zero neighbourhoods in E (3.5.2). Then clearly {Um∧ : m ∈ N} is a base of zero neighbourhoods in E∧. Let x ∈ A∧k . There exists a sequence x1 , x2 , . . . in Ak such that x − x1 −

m

−1 ∧ ∧ λ−1 n−1 xn ∈ λm Ak ∩ Um , m ∈ N, m ≥ 2.

n=2

−1 − x1 in E ∧ . On the other hand, since λ−1 Hence ∞ n=2 λn−1 xn = x n−1 xn → 0 −1 in Ek , we have that m → m λ x converges in E (and hence in E) to a n k n=2 n−1 point in Ak . So x − x1 ∈ Ak and then x ∈ Ak + Ak ⊂ Ak . We now apply 11.2.8 to special cases. Corollary 11.2.9 (i) Let X be a locally compact, σ -compact, non-compact zero-dimensional Hausdorff topological space. Then (Cc (X), τi ) (see 11.1.15(I)) is not metrizable. If, in addition, X ⊂ K without isolated points then, for each n ∈ N, n (Cc (X), τWn ) (see 11.1.15(II)) is not metrizable. (ii) The spaces A† (r), r ∈ [0, ∞) (see 11.1.15(III)) are not metrizable.

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Proof. We have seen in 11.1.15 that the spaces involved in (i) and (ii) are inductive limits of certain strictly increasing inductive sequences of Banach spaces, so they are not metrizable by 11.2.8. Remark 11.2.10 Statement (i) was announced in 3.7.54; (ii) was proved in 9.7.7(a). Now we discuss preservation of polarity. If K is spherically complete all locally convex spaces are polar (4.4.3(i)) and then the problem is solved trivially. Also, if each step is of countable type, so is its inductive limit, therefore it is again polar (4.4.3(ii)). But in general polarity is not preserved (announced in 4.4.17), as we show in the next example. Example 11.2.11 Let K be not spherically complete. Then there exists an inductive sequence of polar Banach spaces whose inductive limit is dualseparating but not polar. Proof. Let En , n ∈ N, be the Banach spaces considered in the proof of 7.1.10 (all of them closed subspaces of ∞ equipped with the restricted norm topology). Let E be its inductive limit. E is dual-separating as the inductive topology on E is stronger than the topology induced by the norm of ∞ . Also, from 7.1.3 and 7.1.13(ii) we have that E is barrelled. Suppose E is polar; we derive a contradiction. By 11.1.1(i) the inductive topology on E is defined by the collection of the polar seminorms p on E for which p|En is continuous on En for all n. Applying 4. of the proof of 7.1.10 we obtain that E is the locally convex space constructed in that example, which is not barrelled: a contradiction. Remark 11.2.12 Let (En )n and E be as above. In 7.1.10 we considered the vector space E, but equipped with the topology inherited from the norm of ∞ , which is the strongest polar locally convex topology on E making all the inclusions En −→E continuous (see 7.1.11). This (E, ) is an example of a so-called polar inductive limit. It clearly differs from the ordinary inductive limit. See the chapter notes for some details. We finish this section by treating (semi-)Montelness. In 8.4.27(e) we announced that inductive limits of (semi-)Montel spaces are (semi-)Montel under additional assumptions. In the next results we see that regularity and strictness appear naturally as candidates. For the Archimedean counterparts see [103], 11.4.5(e), 11.5.4(e). Theorem 11.2.13 Regular inductive limits of semi-Montel spaces are semiMontel.

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Proof. Follows easily from the fact that semi-Montel spaces are those whose bounded sets are compactoid (8.4.5). Theorem 11.2.14 Let (En )n be regular and suppose that each En is reflexive (Montel). If either: (i) K is spherically complete, or (ii) E is polar and, for each n, (En )b is of countable type, then E is reflexive (Montel). Proof. First we prove the reflexivity part by using 7.4.13. Each En is Hausdorff, polar, polarly barrelled and weakly quasicomplete. From 11.2.4(ii) it follows that E is Hausdorff. E is polarly barrelled by 7.1.13(ii). If (i) holds E is automatically polar (4.4.3(i)). It remains to be shown that E is weakly quasicomplete. Let A ⊂ E be absolutely convex, weakly bounded (i.e., bounded, 5.4.5) and weakly closed. By regularity, A ⊂ En for some n and A is bounded (hence a weak compactoid, 5.4.2) in En . Since the inclusion En → E is continuous for the weak topologies, A is also weakly closed in En . By reflexivity, A is weakly complete in En . Then A is a weak compactoid and weakly complete in En , and additionally A is weakly metrizable in En in case we have (ii) (7.6.12). Since the weak topologies of E and En are Hausdorff (5.1.6), it follows that they coincide on A (apply 6.2.1 for (i) and 3.8.38 for (ii)), so that A is weakly complete in E. Therefore, E is reflexive. Finally, let each En be Montel. By the above E is reflexive and by 11.2.13 E is semi-Montel. Thus, E is Montel. Corollary 11.2.15 Regular (strict) inductive limits of Fr´echet Montel spaces are Montel. Proof. By 11.1.7 it suffices to prove the result for regular inductive limits. A Fr´echet–Montel space, as well as its strong dual, is of countable type (8.6.8). Also, by 4.2.13(v) E is of countable type, hence polar (4.4.3(ii)). Then apply 11.2.14(ii). Theorem 11.2.16 (Compare with 11.2.14) Let (En )n be strict, let En be closed in En+1 for each n. Suppose that each En is reflexive (Montel) and of countable type. Then E is reflexive (Montel). Proof. Again we prove reflexivity by using 7.4.13. Each En is Hausdorff, polar, polarly barrelled and weakly quasicomplete. Then E is Hausdorff (11.2.4(i)), polar (4.2.13(v), 4.4.3(ii)) and polarly barrelled (7.1.13(ii)). Let us see that E is weakly quasicomplete. By using 4.2.6 and strictness we get σn+1 |En = σn for all n. Also, from the closedness assumption and 5.2.2 we have that En is

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weakly closed in En+1 for all n. Then weak quasicompleteness follows from 11.2.5(iii). Next, let each En be Montel. As (En )n is regular (11.1.6), we derive from 11.2.13 that E is semi-Montel. Further, by the above E is reflexive, hence Montel. Remarks 11.2.17 (a) For spherically complete base fields K we have the following. Let each En be a reflexive Banach space. Then E is regular and Montel. Proof. By 7.4.20 each En is finite-dimensional. Then E is strict, hence regular by 11.1.7, and E has the strongest locally convex topology. Now apply 8.4.27(a) to get Montelness. However, if K is not spherically complete, we will see in 11.4.10(a) that there exist inductive sequences of reflexive Banach spaces with Hausdorff inductive limits that are neither regular nor reflexive. (b) On the other hand, for any K ⊃ Qp , there exist inductive sequences of Fr´echet nuclear (hence Montel, 8.5.2, 8.6.4) spaces (in fact, spaces of C ∞ -functions) with Hausdorff inductive limits that are neither regular nor reflexive, see [95] for details. But the following is unknown. Problem Is semi-Montelness stable under taking inductive limits?

11.3 Compactoid inductive limits In this section we treat compactoid inductive sequences of Banach spaces, which are the adequate non-Archimedean counterpart of the classical (weakly) compact and nuclear inductive sequences (see e.g. [29], Section 2). Definition 11.3.1 An inductive sequence (En )n of Banach spaces is called compactoid if for each n there exists an m ≥ n such that the inclusion En −→Em is compactoid. In this case we say that its inductive limit is compactoid. Remarks 11.3.2 (a) Observe that from 3.8.6 we have that a compactoid inductive sequence of Banach spaces (En )n is strict if and only if each En is finite-dimensional. In particular, the inductive limits of 11.1.15(I) and (II) cannot be compactoid. On the other hand, in 11.4.3 we will show that the inductive limits of 11.1.15(III) are compactoid.

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(b) Compactoid inductive sequences can be defined for arbitrary locally convex steps. However, it turns out that these general “compactoid” inductive sequences are equivalent, in some natural way, to the ones with Banach steps treated in this section, see 11.6.2. In 11.3.5 we will describe the topological properties of the compactoid inductive limits. First two basic lemmas. Lemma 11.3.3 Let F, G be locally convex spaces, F Fr´echet, G Hausdorff. Let T : F −→G be a continuous injective operator. Let A be a closed absolutely convex compactoid in F . Then for each x ∈ F \ A there exists an absolutely convex zero neighbourhood U in F such that T x ∈ T (A) + T (U ). Proof. By injectivity T x ∈ / T (A). By metrizability and 3.8.38, the restriction of T to the complete compactoid A is a homeomorphism, hence T (A) is closed. So there exists a clopen absolutely convex zero neighbourhood V in G such that Tx ∈ / T (A) + V . Now choose U := T −1 (V ). Then T (U ) ⊂ V , so T (U ) ⊂ V and T x ∈ / T (A) + T (U ). Lemma 11.3.4 Let E be a barrelled space which is covered by an increasing sequence A1 , A2 , . . . of closed absolutely convex subsets. Then for every bounded set B in E there exist λ ∈ K × and no ∈ N such that B ⊂ λ Ano . In particular, if λ1 , λ2 , . . . is a sequence in K with 1 ≤ |λ1 | < |λ2 | < · · · and |λn | → ∞ and each An is bounded in E, then λ1 A1 , λ2 A2 , . . . is a fundamental sequence of bounded sets in E. Proof. Suppose the conclusion does not hold for some bounded set B in E; we derive a contradiction. Let λ1 , λ2 , . . . ∈ K × be as above. Then, for each n, the set B is not contained in λn An ; choose xn ∈ B \ λn An . Thus, λ−1 n xn ∈ E \ An for each n, and by 4.1.6 there exists a continuous seminorm pn on E such that pn (λ−1 n xn ) = 1 and pn (y) < 1 for all y ∈ An . The sequence p1 , p2 , . . . is pointwise bounded. By barrelledness of E and 7.1.5 the formula p(x) = supn∈N pn (x) defines a continuous seminorm on E. By boundedness of B we −1 have λ−1 n xn → 0, which implies p(λn xn ) → 0, a contradiction. Theorem 11.3.5 Let (En )n be a compactoid inductive sequence of Banach spaces, let E be its inductive limit. Then we have the following: (i) E is Hausdorff. (ii) E is barrelled. (iii) E has an increasing fundamental sequence of bounded sets which are absolutely convex, complete, metrizable and compactoid. In particular,

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(iv) (v) (vi) (vii)

(viii) (ix)

Inductive limits

every closed bounded subset of E is complete, metrizable and compactoid. E is complete. (En )n is regular. Every compactoid in E is contained and compactoid in some En . E satisfies the Mackey Convergence Condition (i.e., for each sequence x1 , x2 , . . . tending to 0 in E there exists a sequence µ1 , µ2 , . . . in K with limm |µm | = ∞ such that µ1 x1 , µ2 x2 , . . . tends to 0 in E). Eb is Fr´echet. E and Eb are strictly of countable type, Montel (hence reflexive) and nuclear. In particular, E is the strong dual of the Fr´echet space Eb .

Proof. Let τ be the inductive topology on E and, for each n, let τn be the topology on En . It is not a restriction to suppose that all the inclusions En −→En+1 are compactoid. (i). It suffices to show that for every x ∈ E1 , x = 0, there exists a zero neighbourhood W in E with x ∈ W . Choose µ, α1 , α2 , . . . ∈ K as follows. If the valuation of K is discrete, let µ = α1 = α2 = · · · := 1; if the valuation of K is dense, let 0 < |µ| < 1 and |αn | > 1 for all n such that |µ| n∈N |αn | ≤ 1. Applying 11.3.3 for F := E1 , G := E2 , T := the inclusion E1 −→E2 , A := {0} and taking into account that this T is compactoid, we find an absolutely convex zero neighbourhood U1 in E1 that is a compactoid in E2 and such that τ2 µx ∈ U1 . Now, applying 11.3.3 for F := E2 , G := E3 , T := the inclusion τ2 E2 −→E3 , A := U1 and taking into account that again this T is compactoid, we find an absolutely convex zero neighbourhood U2 in E2 that is a compactoid τ2 τ3 in E3 and such that µx ∈ U1 + U2 . By 3.8.38 it follows that τ3 = τ2 on τ2 τ2 U1 and so U1 is a compactoid (3.8.13) and complete in E3 . Also, from τ2 τ3 3.8.29,(ii) we have that (U1 + U2 )e is a complete compactoid in E3 . Further, τ2 τ3 τ2 τ3 µ x ∈ U1 + U2 implies α1 µ x ∈ (U1 + U2 )e . Continuing in the same spirit and applying 3.1.6(v) we obtain for each n an absolutely convex zero neighbourhood Un in En such that τ2

τ3

αn · · · α1 µ x ∈ (U1 + U2 + · · · + Un+1

τn+2 e

)

which, by the choice of the scalars µ and αn , implies x ∈ U1 + U2 + · · · + Un for all n. Then W := n Un is a zero neighbourhood in E (11.1.2) with x ∈ W . (ii). Is a direct consequence of 7.1.3 and 7.1.13(ii). (iii). For each n, let Bn be the closed unit ball of En . There exists a sequence ν1 , ν2 , . . . in K × such that E = n∈N νn Bn and νn Bn ⊂ νn+1 Bn+1 for all n. τn+1 is By compactoidity of the inclusion En −→En+1 , we have that An := νn Bn

11.3 Compactoid inductive limits

427

a compactoid (and clearly a complete and metrizable) set in En+1 . Also, since τ | En+1 ≤ τn+1 and by (i) τ is a Hausdorff topology on En+1 , we can apply 3.8.38 to obtain that τ = τn+1 on An , and so An is a compactoid in E (3.8.13), which is complete and metrizable. Hence, A1 , A2 , . . . is an increasing sequence of absolutely convex, complete, metrizable and compactoid sets in E covering E. As E is barrelled by (ii), from 11.3.4 we conclude that if λ1 , λ2 , . . . is a sequence in K with 1 ≤ |λ1 | < |λ2 | < · · · and |λn | → ∞, then λ1 A1 , λ2 A2 , . . . is an increasing fundamental sequence of bounded sets in E satisfying the required conditions. (iv). Follows directly from the above properties and 11.2.7. (v) and (vi). Let A1 , A2 , . . . and λ1 , λ2 , . . . be as in the proof of (iii). Let B be a bounded set in E. By using the properties of the sets An proved in (iii), there is an n such that B ⊂ λn An , so B is contained and compactoid (hence bounded) in En+1 . (vii). Again we use the sets A1 , A2 , . . . and the scalars λ1 , λ2 , . . . of the proof of (iii) and the properties proved there for them. Let x1 , x2 , . . . be τ a sequence in E with xm → 0. The set {x1 , x2 , . . .} is τ -bounded, so there is an n such that xm ∈ λn An for all m. Since τ = τn+1 on An we conclude that τn+1 {x1 , x2 , . . .} ⊂ En+1 and xm → 0. As En+1 is a Banach space it is clear that it satisfies the Mackey Convergence Condition, hence there exists a sequence τn+1 µ1 , µ2 , . . . in K with limm |µm | = ∞ and such that µm xm → 0, so also τ µm xm → 0. (viii). Since (En )n is regular by (v), it follows from 11.1.13 that Eb is isomorphic to the projective limit of the Banach spaces (En )b . From 3.5.7 we deduce that Eb is Fr´echet. (ix). First we prove that E is strictly of countable type. For each n, let Fn := (En , τn+1 |En ). By 4.2.2 and compactoidity of the inclusion En −→En+1 we have that each Fn is a normed space (strictly) of countable type. Also, it is easily seen that (Fn )n is an inductive sequence with inductive limit E, so from 4.2.16(v) we obtain that E is strictly of countable type. Next we see that E and Eb are Montel (hence reflexive). By the above and 4.4.3(ii) E is polar. E is Hausdorff by (i) and barrelled by (ii). Also, by (iii) we have that every closed bounded set in E is a complete compactoid, so that E is semi-Montel (8.4.5) and quasicomplete. Thus, E is Montel (8.4.18) and then so is Eb (8.4.19). E is nuclear. In fact, by reflexivity it suffices to see that E is nuclear, which is true by 8.6.8, because Eb is metrizable (viii) and Montel. Now, applying (iii) and 7.6.13 we obtain that Eb is of countable type, hence strictly of countable type by metrizability (viii).

428

Inductive limits

Finally to show that Eb is nuclear we need (8.5.1) that Eb is of countable type (which is already known) and that every T ∈ L(Eb , c0 ) is compactoid. To prove this last property, let T : Eb −→c0 be a continuous operator. For every m, the map f → (T (f ))m (f ∈ E ) is in E , hence by reflexivity there is an xm ∈ E such that (T (f ))m = f (xm ). As T maps into c0 we have that xm → 0 weakly τ in E, so xm → 0 (5.5.2(ii)). Since by (vii) E has the Mackey Convergence Condition, there exists a sequence µ1 , µ2 , . . . in K × with limm |µm | = ∞ but limm µm xm = 0 in E, from which B := {µ1 x1 , µ2 x2 , . . .} is τ -bounded. Therefore, B ◦ is a zero neighbourhood in Eb with T (B ◦ ) ⊂ {(α1 , α2 , . . .) ∈ c0 : |αm | ≤ |µm |−1 for all m}. The latter set is a compactoid in c0 (8.1.4) and then so is T (B ◦ ). Remarks 11.3.6 (a) Theorem 11.3.5(iii), (v) and (vi) tell us that in compactoid inductive limits the bounded and the compactoid sets coincide and are characterized in terms of the steps. (b) In 11.2.8 we saw that an inductive limit of Banach spaces is metrizable only in very few cases. If the inductive sequence is additionally compactoid then, as one can guess, the situation is even worse. In fact, one verifies the following. Let (En )n be an inductive sequence of Banach spaces, let E be its inductive limit. Then E is metrizable (normable) and (En )n is compactoid ⇐⇒ E is finite-dimensional.

Proof. Suppose E is metrizable and (En )n is compactoid. Applying 11.2.8 we may assume that En = En+1 (= E) for all n. By the Open Mapping Theorem 2.1.17 these equalities between steps are topological. Thus, (En )n is strict and by 11.3.2(a) each En , hence E, is finite-dimensional. The opposite implication is easy.

11.4 Inductive topologies on sequence spaces As we said in the notes to Chapter 9, sequence spaces associated to infinite matrixes play a role in the development of a p-adic theory related to Physics. Many of these spaces are endowed in a natural way with inductive topologies. Here we are going to apply our previous results to derive some topological properties of them. In this way we complete the theory about K¨othe sequence spaces already developed in Sections 9.5–9.7 and in particular we now keep the promises made there.

11.4 Inductive topologies on sequence spaces

429

As in Sections 9.6 and 9.7 we work with the matrixes that most frequently appear in the applications, that is, In this section B := (bnk )k,n (k, n ∈ N) is an infinite matrix consisting of real numbers satisfying 0 < bnk ≤ bnk+1 for all k, n. First, we recall the sequence space ∞ (B) introduced in 9.5.8. We have Fk , where for each k, ∞ (B) = k∈N

Fk = {(λ1 , λ2 , . . .) ∈ K N : sup |λn | /bnk < ∞}. n∈N

We know (9.5.14) that ∞ (B) is a perfect sequence space whose K¨othe dual, × othe sequence space 0 (B) introduced in 9.5.2, i.e., ∞ (B) , is the K¨ 0

(B) = {(λ1 , λ2 , . . .) ∈ K N : lim |λn | bnk = 0 for all k}. n

∞ ∞ (B) is equipped with the inductive topology τi

with respect to the inductive sequence of Banach spaces (Fk )k , where each Fk has the norm (λ1 , λ2 , . . .) → supn∈N |λn | /bnk . Now we introduce a new sequence space. Definition 11.4.1 Let 0 (B)

:=

Gk , where for each k,

k∈N

Gk := {(λ1 , λ2 , . . .) ∈ K N : lim |λn | /bnk = 0}. n

It is easily seen that ∞

×

0 (B)

=

∞

(B), with

(B) := {(λ1 , λ2 , . . .) ∈ K N : sup |λn | bnk < ∞ for all k}. n∈N

In 11.4.9(i) we will see that

0 (B)

is not always perfect.

For each k, Gk is a Banach space of countable type with the norm inherited from Fk . Also, the monotonicity condition imposed for the matrix B implies that G1 , G2 , . . . is an inductive sequence of Banach spaces. 0 (B) is equipped with the corresponding inductive topology τi0 . Next we describe when the above inductive sequences are compactoid. Theorem 11.4.2 The following are equivalent: (α) (Fk )k is compactoid. (β) (Gk )k is compactoid.

430

Inductive limits

(γ ) The matrix B satisfies (9.19). (δ) ( ∞ (B), τi∞ ) = ( 0 (B), τi0 ). (ε) ∞ (B) = 0 (B) algebraically. Proof. (α) =⇒ (β) is immediate since, for each h > k, the inclusion Gk −→Gh is the restriction to Gk of the corresponding one Fk −→Fh . (β) =⇒ (γ ). Let k ∈ N and let h > k be such that the inclusion Gk −→Gh is compactoid. Then the closed unit ball Bk of Gk is a compactoid in Gh with respect with the norm topology on Gh , which coincides with the normal one (see the proof of (β)=⇒(α) of 9.5.15). By 9.4.7(i) there exists a (δ1 , δ2 , . . .) ∈ Gh such that Bk ⊂ {(µ1 , µ2 , . . .) ∈ Gh : |µn | ≤ |δn | for all n}. Let ρ ∈ K, 0 < |ρ| < 1. For each n let jn be an integer with |ρ|jn +1 ≤ bnk ≤ |ρ|jn and call λn := ρ jn +1 . We have that λn en ∈ Bk for all n, and so |λn | ≤ |δn |. Also, since bnk /bnh ≤ |ρ|−1 |λn | /bnh ≤ |ρ|−1 |δn | /bnh , we obtain that limn bnk /bnh = 0. (γ ) =⇒ (α). Let k ∈ N. Let h > k be as in (9.19). We will prove that the inclusion Fk −→Fh is compactoid. Let ρ and jn , λn (n ∈ N) be as in the proof of (β) =⇒ (γ ). Then (λ1 , λ2 , . . .) ∈ Gh and the closed unit ball, Ck , of Fk is contained in {(µ1 , µ2 , . . .) ∈ Gh : |µn | ≤ |ρ|−1 |λn | for all n}. By 9.4.7(i) we obtain that Ck is a compactoid in Gh with respect to the normal topology. Thus, Ck is a norm-compactoid in Gh , hence in Fh , and we have compactoidity of the inclusion Fk −→Fh . (γ ) =⇒ (δ). Observe that if k and h are as in (9.19) then Fk is contained in Gh and the inclusion Fk −→Gh is continuous. Also, each Gk is isometrically included in Fk . So that the inductive limits of (Fk )k and (Gk )k coincide and we get (δ). Clearly (δ) =⇒ (ε). (ε) =⇒ (γ ). Let k ∈ N and let again ρ, jn , λn (n ∈ N) be as in the proof of (β) =⇒ (γ ). Then (λ1 , λ2 , . . .) ∈ Fk and by (ε) there exists an h > k such that (λ1 , λ2 , . . .) ∈ Gh , which implies that limn bnk /bnh = 0. Remark 11.4.3 The spaces of analytic functions A† (r), r ∈ [0, ∞) of 11.1.15(III) are particular cases of spaces ( ∞ (B), τi∞ ) for the matrixes B = (bnk )k,n defined in (11.1), which satisfy (9.19). Thus, by 11.4.2 they are compactoid inductive limits (announced in 11.3.2(a)), so that they satisfy all the properties of 11.3.5. This completes the study about these spaces carried out in the previous chapters.

11.4 Inductive topologies on sequence spaces

431

Note also that compactoidity of these inductive limits implies that they are regular (11.3.5(v)) and non-strict (11.3.2(a)), as we announced in 11.1.15(III). We extend this in the next example, promised in 11.1.9. Example 11.4.4 Let B be any matrix satisfying (9.19) such that for the induced inductive sequence (Gk )k of Banach spaces of countable type we have Gk = Gk+1 for all k. Then the following holds: (i) (Gk )k is regular. (ii) Each weakly bounded set in ( 0 (B), τi0 ) is contained and weakly bounded in some Gk . (iii) (Gk )k is not strict. (iv) Gk is (weakly) closed in Gk+1 for no k. (v) σk+1 |Gk = σk for no k, where σk denotes the weak topology on Gk , k ∈ N. Proof. The inductive sequence (Gk )k is compactoid (11.4.2), hence regular (11.3.5(v)) and non-strict (11.3.2(a)), so we have (i) and (iii). To prove (ii), note that since each Gk is of countable type, so is the inductive limit ( 0 (B), τi0 ) (4.2.13(v)). Thus, these spaces are polar (4.4.3(ii)), and their bounded sets coincide with the corresponding weakly bounded ones (5.4.5). Then apply (i). Next, suppose (iv) is not true. Then there is a k for which Gk is closed (i.e., weakly closed, 5.2.2) in Gk+1 . Since Gk is also dense in Gk+1 (because the unit vectors of K N form a Schauder base of Gk+1 ) we obtain Gk = Gk+1 , a contradiction. Finally, suppose (v) fails, i.e., σk+1 |Gk = σk for some k. Let us see that then τk+1 |Gk = τk , where τk and τk+1 are the norm topologies on Gk and Gk+1 respectively (which would imply that Gk is complete, hence closed, in Gk+1 , yielding a contradiction with (iv)). For that, let x1 , x2 , . . . be a sequence in Gk τk+1 σk+1 σk with xm −→ 0. Then xm −→ 0 and by assumption xm −→ 0. Since Gk is of τk countable type, it follows from 5.5.2(ii) that xm −→ 0 and we are done. Remark 11.4.5 Now we show that 11.1.13 can be applied to describe the duals of the sequence spaces considered in 11.4.2; it was announced in 11.1.14(b). Let B be a matrix satisfying (9.19), see 11.4.3 for some important examples of such B. By 11.4.2 we have that ( ∞ (B), τi∞ ) = ( 0 (B), τi0 ). Then clearly the K¨othe duals of ∞ (B) and of 0 (B) also coincide, that is, 0 (B) = ∞ (B). Let τp0 be the projective topology on 0 (B) with respect to the projective sequence of Banach spaces (Ek )k (see the comments preceding 9.5.4).

432

Inductive limits

Let τp∞ be the projective topology on ∞ (B) with respect to the projective sequence of Banach spaces (Hk )k where, for each k, Hk := {(λ1 , λ2 , . . .) ∈ K N : supn∈N |λn | bnk < ∞} is equipped with the norm (λ1 , λ2 , . . .) → supn∈N |λn | bnk r Then, the above equality 0 (B) = ∞ (B) is even topological, when we equip these spaces with their projective topologies (indeed, each Ek is isometrically included in Hk ; also, if k and h are as in (9.19) then Hh is contained in Ek with continuous inclusion Hh −→Ek ). Since (Gk )k is compactoid (11.4.2), it is regular (11.3.5(v)). Also, with a simple adaptation of the proof carried out in 2.5.11, we have that each Gk can be identified with Hk . Then applying 11.1.13 we get that ( 0 (B), τi0 )b (= ( ∞ (B), τi∞ )b ) is naturally isomorphic to ( ∞ (B), τp∞ ) (= ( 0 (B), τp0 )), and we obtain the desired descriptions of duals. In 9.7.2 we saw that for matrixes B satisfying (9.19), the equality τi∞ = n( ∞ (B), 0 (B)) holds. But this equality of the inductive and the normal topologies on ∞ (B) does not always happen (as we have announced in 9.5.14); the next counterexample is the non-Archimedean version of the famous classical Grothendieck-K¨othe one (see e.g. [144], 31.7). Example 11.4.6 There exists a matrix B for which τi∞ = n(

∞ (B),

0

(B)).

Proof. To describe the required example it is convenient to replace the index n by a double index (m, n). To do that take any bijection N × N−→N and identify each (m, n) ∈ N × N with its image under this map. Now, define the matrix B by n if m ≤ k k := (11.2) b(m,n) 1 otherwise. To prove that τi∞ = n( ∞ (B), 0 (B)), let for each k, Ck be the closed unit ball of Fk and let U := k Ck . Then U is a zero neighbourhood in ( ∞ (B), τi∞ ) (11.1.2) which contains no element (u(m,n) )(m,n) with the property that (11.3) for every m there is a coordinate u(m,n) with u(m,n) > 1. Suppose U contains a n( ∞ (B), 0 (B))-zero neighbourhood; we derive a contradiction. For this we need (bi)polars; they are understood to be taken with respect to the duality between ( 0 (B), n( 0 (B), ∞ (B))) and 0 (B)× , see 9.4.5. By 9.4.9 there exists a n( 0 (B), ∞ (B))∞ (B) = compactoid subset A of 0 (B) such that U ⊃ A◦ . From 9.4.7(i) we eas ily deduce that A is contained in a set k∈N (λk Ck )◦ for suitably chosen λk ∈ K × . Now, k∈N (λk Ck )◦ = ( k λk Ck )◦ , so that we obtain U ⊃ V ◦◦ with V := k λk Ck . To arrive at the desired contradiction, let ρ ∈ K, 0 < |ρ| < 1,

11.4 Inductive topologies on sequence spaces

433

let e(m,n) , m, n ∈ N, be the unit vectors of K N×N . For each k, the element ρ −(k+1) e(k,nk ) lies in λk Ck for sufficiently large nk . Consequently k −(k+1) ( m e(k,nk ) )m is a sequence in V whose σ ( ∞ (B), 0 (B))k=1 ρ ρ limit is the element u of ∞ (B) with coordinate ρ −1 in place (k, nk ), k ∈ N, and 0 otherwise. By applying 5.2.7 to the locally convex space ( ∞ (B), σ ( ∞ (B), 0 (B))) we conclude that u belongs to V ◦◦ and hence to U ; a contradiction, because u satisfies (11.3). On the other hand, we now prove that the inductive topology τi0 on always coincides with the normal one n( 0 (B), ∞ (B)). Theorem 11.4.7 τi0 = n(

0 (B),

∞

0 (B)

(B)).

Proof. For each k, the canonical inclusion Gk −→( 0 (B), n( 0 (B), ∞ (B))) is easily seen to be continuous, hence n( 0 (B), ∞ (B)) ≤ τi0 . Let us prove that τi0 ≤ n( 0 (B), ∞ (B)). In order to get this we show 0 that every x = (λ1 , λ2 , . . .) ∈ 0 (B) can be written as x = ∞ n=1 λn en in τi , N where e1 , e2 , . . . are the unit vectors of K (then proceed as in the proof of 9.7.2 with the obvious modifications). For that observe that given such an x . Since the there exists a k such that x ∈ Gk and so x = ∞ n=1 λn en in G k inclusion Gk −→( 0 (B), τi0 ) is continuous, we obtain that x = ∞ n=1 λn en in 0 ( 0 (B), τi ) and we are done. In 9.4.1 we announced to present non-perfect sequence spaces. In 11.4.9 we will see that 0 (B), for the matrix B of (11.2), is, indeed, not perfect. At the same time we will prove that, for this B, ( 0 (B), τi0 ) is non-complete and non-regular. However, we know that ∞ (B) is perfect; also, one verifies that, for any matrix B, ( ∞ (B), τi∞ ) is complete and regular, see 11.6.4. For the construction we need a preliminary lemma. Lemma 11.4.8 If a sequence x 1 , x 2 , . . . in (i) x k ∈ Gk for all k, (ii) limk (supn∈N xnk /bn1 ) = 0 (where then limk x k = 0 in ( 0 (B), τi0 ).

0 (B)

satisfies:

x k = (x1k , x2k , . . .)

for

each

k),

Proof. Let U be an absolutely convex zero neighbourhood in ( 0 (B), τi0 ). By 11.1.2 we may assume that there exist λ1 , λ2 , . . . in K × such that U = k λk Bk , where for each k, Bk stands for the closed unit ball of Gk . By (ii), there exists a k ∗ such that sup xnk /bn1 ≤ |λ1 | for all k ≥ k ∗ . (11.4) n∈N

434

Inductive limits

Fix k ≥ k ∗ ;we will prove that x k ∈ U . For each n, let µn ∈ K be defined by µn := 1 if xnk ≤ |λk | bnk and µn := 0 otherwise, and set un := (xnk /λk ) µn , vn := (xnk /λ1 ) (1 − µn ). −1 k From (i) it follows that u := (u1 , u2 , . . .) ∈ Gk (because k k |un | ≤ |λk | xn for all n) and that v := (v1 , v2 , . . .) ∈ G1 (because xn /bn ≤ |λk | for almost all n, i.e., µn = 1 for almost all n). Also, it follows from the definition of µn and (11.4) that u ∈ Bk and v ∈ B1 . Hence x k = λk u + λ1 v ∈ λk Bk + λ1 B1 ⊂ U . Next, we explain the promised example, whose classical counterpart can be found in [171], p. 211. Example 11.4.9 For the matrix B of (11.2) we have the following: (i) 0 (B) is not perfect. (ii) ( 0 (B), τi0 ) is not ( weakly) quasicomplete. (iii) (Gk )k is not regular. There is even a null sequence x 1 , x 2 , . . . in ( such that {x 1 , x 2 , . . .} ⊂ Gk for no k.

0 0 (B), τi )

Proof. It suffices to prove (iii). (Then, by 11.1.12, ( 0 (B), τi0 ) is not quasicomplete, and applying 9.4.6 and 11.4.7 we get (i) and the rest of (ii).) To this end, let us define a sequence x 1 , x 2 , . . . in 0 (B) as follows: −1 if m = k µk k x(m,n) := 0 otherwise, where µ1 , µ2 , . . . is a sequence in K × with limk |µk | = ∞. k k k For each k we have x k ∈ Gk (since lim(m,n) |x(m,n) |/b(m,n) = limn |x(k,n) |/ −1 k+1 k k k+1 −1 ∈ Gk (since |x(k+1,n) |/b(k+1,n) = |µk+1 | b(k,n) = limn |µk |/n = 0) and x for all n). k 1 Also, for each k, supm,n∈N |x(m,n) |/b(m,n) ≤ |µk |−1 , which implies k 1 limk supm,n∈N |x(m,n) |/b(m,n) = 0. Thus, applying 11.4.8 we obtain that x 1 , x 2 , . . . is a sequence tending to 0 in ( 0 (B), τi0 ). But {x 1 , x 2 , . . .} ⊂ Gk for no k, and we are done. Remarks 11.4.10 (a) With this example we have an inductive sequence (Gk )k of Banach spaces that is not α-regular (by (iii)) and for which lim Gk is not quasicomplete −→ (by (ii)) (promised in 11.1.4(b), 11.2.6). Now suppose K is not spherically complete. Then each Gk is a Banach space of countable type, hence reflexive by 7.4.30, and so weakly quasicomplete by 7.4.13. But, by (ii), lim Gk is not weakly quasicomplete, −→ neither reflexive (promised in 11.2.6, 11.2.17(a)).

11.5 Compactoid sets in inductive limits

435

(b) The existence, for non-spherically complete K, of a non-regular inductive sequence of reflexive Banach spaces, proved above, is in contrast with the classical situation. Indeed, in Theorem 4 of [148] it was proved that any real or complex inductive limit of reflexive Banach spaces is regular. However, we have shown in the result given in 11.2.17(a) that, for spherically complete base fields, the non-Archimedean version of this classical result is true.

11.5 Compactoid sets in inductive limits A locally convex space is called an (LM)-space (resp. (LF)-space) if it is the inductive limit of an inductive sequence of metrizable (resp. Fr´echet) spaces. In this section we show (11.5.5) that every compactoid in a Hausdorff (LM)space is metrizable. For the classical counterpart see the chapter notes. We need the following concept. Definition 11.5.1 A resolution of a set X is a family (Xα )α∈NN of subsets of X such that: (i) Xα ⊂ Xβ for all α, β ∈ NN with α ≤ β (α ≤ β means that αn ≤ βn for each n ∈ N, where α = (α1 , α2 , . . .), β = (β1 , β2 , . . .)), (ii) {Xα : α ∈ NN } = X. We also need a preliminary lemma. Lemma 11.5.2 Let (Xα )α∈NN be a resolution of an uncountable set X. Then for some γ ∈ NN the set Xγ is infinite. Proof. We can choose inductively a sequence k1 , k2 , . . . of natural numbers such that, for each n, the set {Xα : α ∈ NN , αi = ki for all i ∈ {1, . . . , n}} Yn := is uncountable. Also, inductively, one can select x1 , x2 , . . . ∈ X such that xn ∈ Yn \ {x1 , . . . , xn−1 } for each n. Let n ∈ N. By the definition of Yn there exists a βn = (βn1 , βn2 , . . .) ∈ NN such that xn ∈ Xβn and βni = ki for all i ∈ {1, . . . , n}. For i ∈ N put γi := maxn∈N βni . Then γ := (γ1 , γ2 , . . .) ∈ NN and for each n we have βn ≤ γ , hence xn ∈ Xβn ⊂ Xγ , and so Xγ is infinite. Now we can give the first stepping stone for 11.5.5. We will use also the concept of a t-frame, see 3.10.4.

436

Inductive limits

Theorem 11.5.3 Let E a locally convex space with a resolution (Xα )α∈NN such that each Xα is a compactoid. Then E is of countable type. Proof. Let p be a continuous seminorm on E, let πp : E−→Ep be the natural map. It suffices to show that the normed space (Ep , p) is of countable type. For that, suppose that Ep contains an uncountable t-frame Y for some t ∈ (0, 1); we derive a contradiction. (Then we are done by 3.10.5.) We may assume (multiplying by suitable scalars) that p(y) ≥ 1 for all y ∈ Y . Now, for each α ∈ NN , let Zα := πp (Xα ). Then (Zα )α∈NN is a resolution of Ep , so (Zα ∩ Y )α∈NN is a resolution of Y . By 11.5.2 Zγ ∩ Y is infinite for some γ ∈ NN . But Zγ = πp (Xγ ) is a compactoid in Ep , which conflicts 3.10.7. Our second step is the following. Theorem 11.5.4 Let E be a Hausdorff locally convex space with a base (Uα )α∈NN of zero neighbourhoods such that Uβ ⊂ Uα for all α, β ∈ NN with α ≤ β. Then each compactoid in E is metrizable. Proof. Let A ⊂ E be a compactoid. Then F := [A] is of countable type (4.2.2), hence polar (4.4.3(ii)), and A is a compactoid in F (3.8.9). For α ∈ NN put Vα := Uα ∩ F . Then (Vα◦ )α∈NN (where the polars are taken in F ) is a resolution of F . Each Vα◦ is equicontinuous and is a compactoid with respect to the weak∗ topology on F , so from 3.8.13 and 8.4.11 we have that Vα◦ is a c(F , F )-compactoid (recall, 8.4.10, that c(F , F ) is the topology on F of uniform convergence on the compactoid subsets of F ). By 11.5.3 we obtain that (F , c(F , F )) is of countable type. Therefore, the normed space (F , . A ), where f A := supx∈A |f (x)| (f ∈ F ), is of countable type. Then following a similar reasoning as in the first part of the proof of 7.6.12, replacing E by F and B by A, we derive that A is σ (F, F )-metrizable. Now apply 5.2.12 to conclude that A is metrizable. Next we get the main purpose of this section. Theorem 11.5.5 Every compactoid in a Hausdorff (LM)-space is metrizable. Proof. Let E := lim En be a Hausdorff inductive limit of an inductive sequence −→

of metrizable spaces En . For each n, let (Vkn )k∈N be a decreasing base of absolutely convex zero neighbourhoods in En (3.5.2). For α = (α1 , α2 , . . .) ∈ NN put Uα := n Vαnn . Then (Uα )α∈NN is a base of zero neighbourhoods in E (11.1.2), and clearly Uβ ⊂ Uα for all α, β ∈ NN with α ≤ β. Now the conclusion follows from 11.5.4. Remark 11.5.6 As an application of 3.8.24, 3.8.25 and 11.5.5 we obtain that the compactoids in Hausdorff (LM)-spaces are those sets contained in the

11.6 Notes

437

closed absolutely convex hulls of sequences tending to 0. This gives a partial answer to the question considered in 3.8.18. Another application of 11.5.5 is the next corollary. Corollary 11.5.7 Let E := lim En be an (LF)-space. Then the following are −→ equivalent: (α) For every compactoid A in E there is an n such that A is contained and compactoid in En . (β) For every sequence x1 , x2 , . . . tending to 0 in E there is an n such that xm ∈ En for all m and x1 , x2 , . . . tends to 0 in En . Proof. Let τ be the inductive topology on E and, for each n, let τn be the topology on En . (α) =⇒ (β). From (α) and with the same reasoning as in 11.2.4(ii) we have that E is Hausdorff. Let x1 , x2 , . . . be a null sequence in E. Then the τ -closed absolutely convex hull A of this sequence is a compactoid in E. By (α), there is an n such that A ⊂ En and A is a compactoid in En . Clearly A is metrizable and complete in En . Then we can use 3.8.38 to conclude that τ |A = τn |A. Hence τn xm −→ 0. (β) =⇒ (α). From (β) it is easily seen that E is Hausdorff. Let A be a compactoid in E. We may assume that A is absolutely convex. By 11.5.5 A is metrizable. Thus, applying 3.8.25 there exists a null sequence e1 , e2 , . . . in E such that A ⊂ acoτ {e1 , e2 , . . . }. By (β) there is an n such that em ∈ En for τn all m, and em −→ 0. Then B := acoτn {e1 , e2 , . . . } is a complete, absolutely convex, metrizable and compactoid in En . Using 3.8.38 one gets τ |B = τn |B, so B is a τ -complete set with A ⊂ B. This implies that A is contained and compactoid in En . Problem Does this equivalence also hold for (LM)-spaces? Remark 11.5.8 11.4.9(iii) provides an example of an inductive sequence of Banach spaces for which the two above equivalent properties fail.

11.6 Notes Inductive limits form an important class of locally convex spaces with interesting applications. We point out some of them. The space (Cc (X), τi ) (see 11.1.15(I)) is an inductive limit and the elements of its dual are precisely the integrals defined by Monna and Springer in [162]. They are the non-Archimedean counterpart of the well-known Radon measures of the classical theory.

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Inductive limits

The inductive limit A† (0) (see 11.1.15(III)) is a crucial tool to define a p-adic Laplace Transform, [82], as we mentioned in the notes to Chapter 9. In the same notes we saw that C ∞ (Zp )b , called the space of distributions, and used to define a p-adic Fourier Transform in [83], is isomorphic to ( ∞ (B), n( ∞ (B), 0 (B))), with B := (bnk )k,n , bnk := nk , k ∈ N, n ∈ N ∪ {0}. This matrix B satisfies (9.19), so we have n( ∞ (B), 0 (B)) = τi∞ on ∞ (B) (9.7.2). Therefore, this strong dual is an inductive limit. More examples of (countable and uncountable) inductive limits are in [3], [4], [5], [122], [123], [130], [133], concerning spaces of continuous functions, and in [35], [36], [189], concerning spaces of analytic functions and p-adic differential equations. Inductive sequences (En )n satisfying σn+1 |En = σn for all n (see e.g. 11.1.8, 11.2.5(iii)), are called weakly strict. This and other variants of strictness, as well as the relation between them, were studied in [78] and [92]. In the last paper the relation between these strictness properties and some closedness properties in inductive limits is also investigated. Further, in [93] and [94], regularity enters into this discussion, leading to some extensions of the p-adic Dieudonn´e-Schwartz Theorem 11.1.7, and to an example of an (α)-regular inductive sequence that is not regular (see 11.1.4(a)). In 11.1.13 we proved that the dual of an inductive limit is a projective limit. The converse (see 11.1.14(c)) is considered in the next result. Theorem 11.6.1 ([78], 1.3.7) Let (Fn , πn )n be a projective system with πn ∈ L(Fn+1 , Fn ) (n ∈ N), let F := lim Fn . Suppose that ϕn (F ) is dense in Fn for ←− each n, where ϕn : F → Fn is the n-th coordinate map. Then the strong duals (Fn )b , with the maps πn : Fn → Fn+1 , form an inductive sequence whose inductive limit is algebraically isomorphic to F . If, in addition, for every sequence B1 , B2 , . . ., where Bn is bounded in Fn for each n, there exists a bounded set B in F such that ϕn (B) ⊃ Bn for all n (where the closure is with respect to Fn ), then (lim Fn )b ∼ lim(Fn )b . ←−

−→

The duality between projective and inductive limits was also studied in [129] and [163]. The proof of 11.2.5(i) is taken from [210], 7.9. We think that it is more pleasant for the reader than the possibility considered in [78], 1.4.18 of just following the proof given in the classical case ([101], Theorem 2.12.3). In non-Archimedean analysis it is natural to consider the following polar version of the notion of inductive limit, which was introduced in [91]. A polar inductive sequence is an inductive sequence (En )n of polar spaces. Its polar inductive limit is the space E := n∈N En equipped with the strongest polar locally convex topology for which all the inclusions En → E are continuous.

11.6 Notes

439

It is easy to see that exists. In fact, it is the locally convex topology induced by the collection of all polar seminorms p on E for which p|En is continuous on each En . One verifies that is the polar topology associated to the inductive topology on E (associated polar topologies were defined in the notes to Chapter 4). For a general theory about polar inductive limits we refer to [78], where one can find a discussion about the polar versions of the concepts given in this chapter as well as about the corresponding results. Here we just present some examples of polar inductive limits. Clearly, if we have an inductive sequence of polar spaces and its inductive limit happens to be polar then it coincides with the polar inductive limit. This occurs e.g. when K is spherically complete, or all the En are of countable type, or in case we have a compactoid inductive sequence of polar Banach spaces. These three kinds of sequences cover the most important examples in the theory of inductive limits, which has been our main reason to leave polar inductive limits for the notes. Suppose K is not spherically complete. In 11.2.12 we have met an example of an inductive sequence (En )n of polar Banach spaces (all subspaces of ∞ ) for which the polar inductive limit (E, ) differs from the inductive limit (E, τ ). But there are more interesting topological differences. First of all, (E, ) is normable, which is quite surprising (compare with 11.2.8). Also, (En )n is regular (11.1.7), but the unit ball of (E, ) lies in no En . Further, (E, τ ) is complete (11.2.5,(i)), but (E, ) cannot be because of the Baire Category Theorem. Finally, we know that (E, τ ), being an inductive limit of Banach spaces, is barrelled, and that (E, ) is polarly barrelled but not barrelled (7.1.10). Another example of different behaviour for the polar and the “ordinary” inductive limits is provided by the space ∞ (B), where B is the matrix of (11.2). For details, see [78], 3.2.25. As we said in 11.3.2.b, compactoid inductive sequences can be defined for arbitrary locally convex steps. In order to get this it is convenient to use the concept of semicompact operator considered at the beginning of the notes to Chapter 8, which extends the concept of compactoid operator between Banach spaces. Then, an inductive sequence (En )n of Hausdorff locally convex spaces is called semicompact if for each n there exists an m ≥ n such that the inclusion En −→Em is semicompact. As we announced also in 11.3.2(b), it turns out that these semicompact inductive sequences are equivalent to the compactoid ones with Banach steps treated in Section 11.3, in the sense explained below. Theorem 11.6.2 ([78], 3.1.4) For every semicompact inductive sequence (En )n of Hausdorff locally convex spaces there exists a compactoid inductive sequence

440

Inductive limits

(Fn )n of Banach spaces of countable type having the following property. For each n there are m, r ≥ n, r ≥ m and semicompact inclusions in,m : En −→ Fm , jm,r : Fm −→ Er such that jm,r ◦ in,m is the inclusion En −→Er . The above theorem is the starting point of the theory about semicompact inductive sequences developed in Section 3.1 of [78] which, among other things, contains 11.3.5 as well as some natural converses of this result. The particular case of inductive sequences of locally convex spaces over spherically complete fields such that 11.3.1 holds, after replacing “En −→Em is compactoid” by “En −→Em is compact”, was treated in Section 16 of [210] (again, for the concept of compact operator see the beginning of the notes to Chapter 8). In [182] semicompact inductive sequences were applied to prove the failure, for non-locally compact K, of the non-Archimedean counterpart of the Banach– Dieudonn´e Theorem. (Recall that this classical result states that if E is a metrizable locally convex space over R or C, then the strongest topology on E coinciding with σ (E , E) on equicontinuous sets is the locally convex topology of uniform convergence on the precompact subsets of E, [101], 3.10.1.) For some weaker non-Archimedean versions of the Banach-Dieudonn´e Theorem, see also [182]. More topological properties of the inductive limits ( ∞ (B), τi∞ ) and ( 0 (B), τi0 ) of Section 11.4 were studied in [78]. Among them we point out the following two. First, a result showing that strictness of these inductive limits is a restrictive condition that only holds in the trivial cases. Theorem 11.6.3 ([78], 3.2.16, 3.2.22) The following are equivalent: (α) (β) (γ ) (δ) (ε)

(Fk )k is strict. (Gk )k is strict. Fk = Fk+1 for all k. Gk = Gk+1 for all k. The matrix B has the following property. For each k there exists a ck > 0 such that bnk+1 ≤ ck bnk for all n.

Secondly a result on completeness and regularity of ( with 11.4.9). Theorem 11.6.4 ([78], 3.2.20, 3.2.21) (

∞ ∞ (B), τi )

∞ ∞ (B), τi )

(compare

is complete and regular.

As for the fact that every compactoid in a Hausdorff (LM)-space is metrizable (11.5.5), we have the following historical comments.

11.6 Notes

441

Its classical counterpart, “precompact subsets of Hausdorff (LM)-spaces over R or C are metrizable”, was raised by Floret in [57]. The conclusion was proved first for (LF)-spaces in [32], next for (LM)-spaces in [33]. Latter on, a more simple proof appeared in [55]. Returning to the non-Archimedean case, it was proved in [78] that compactoids in Hausdorff inductive limits of Banach spaces are metrizable. This result was extended to (LM)-spaces in [77], by means of rather complicated techniques. A much more elegant and simple proof was given in [108], which is the one we used in our book. Inductive sequences (En )n satisfying that for every compactoid X in the inductive limit E there is an n such that X is contained and compactoid in En (resp. for every sequence x1 , x2 , . . . tending to 0 in E there is an n such that xm ∈ En for all m and x1 , x2 , . . . tends to 0 in En ) are called compactoid regular (resp. sequentially retractive). Corollary 11.5.7 tells us that when the steps are Fr´echet spaces these two properties are equivalent. Examples of inductive sequences of Banach spaces for which these two equivalent properties hold (resp. do not hold) are provided by 11.3.5(vi) (resp. 11.4.9(iii)). Some connections between ordinary regularity and sequential retractivity can be found in [78], 1.1.14. For information about other interesting subjects related to inductive limits that have not been treated in this chapter, such as Baire-like conditions and Berezanskii-duals, see [78].

appendix a Glossary of terms

A.1 Sets Let X be a set. We denote the collection of all its subsets by P(X). A subcollection C ⊂ P(X) has the finite (or binary) intersection property if C is closed for finite intersections. C is called a nested collection if C is linearly ordered by inclusion. In the same spirit we define nested sequences of subsets of X. A partition of X is a covering of X by mutually disjoint subsets of X. A non-empty subset R of P(X) is called a ring (of subsets of X) if Y, Z ∈ R implies Y ∩ Z ∈ R, Y ∪ Z ∈ R, and Y \ Z := {y ∈ Y : y ∈ Z} ∈ R. A filter (on X) is a non-empty collection F ⊂ P(X) that does not contain the empty set ∅, has the finite intersection property, and such that Y ∈ F, Y ⊂ Z ⊂ X implies Z ∈ F. A filter F is called (a) principal (filter) if there exists an x ∈ X such that F = {Y ∈ P(X) : x ∈ Y }. A non-empty subcollection B of a filter F on X is called a base (of F) if for each Y, Z ∈ B there is a T ∈ B with T ⊂ Y ∩ Z, and if for each Y ∈ F there is a Z ∈ B with Z ⊂ Y . Then F = {Y ∈ P(X) : Y contains a member of B} and F, being the smallest filter containing B, is called the filter generated by B. Conversely, if some non-empty collection B ⊂ P(X) is given with the properties that ∅ ∈ B and for Y, Z ∈ B there is a T ∈ B with T ⊂ Y ∩ Z then B generates a filter F. In fact, F = {Y ∈ P(X) : Y contains a member of B}. Let X1 , X2 be sets, let f : X1 −→X2 , and let F be a filter on X1 . Then {f (Y ) : Y ∈ F} is a base of a filter on X2 (since f (Y ) ∩ f (Z) ⊃ f (Y ∩ Z) for all Y, Z ∈ F). This filter is called f (F). An ultrafilter (on X) is a maximal filter (on X), i.e., F is an ultrafilter if for any filter F ⊃ F on X we have F = F. By Zorn’s Lemma each filter can be extended to an ultrafilter. Theorem A.1.1 ([48], X.7.2 for (α) ⇔ (β); (α) ⇔ (γ ) is trivial) For a filter F on a set X the following are equivalent: (α) F is an ultrafilter. (β) For each Y ∈ P(X), either Y ∈ F or X \ Y ∈ F. (γ ) A set Y ⊂ X is in F if and only if Y ∩ Z = ∅ for all Z ∈ F. A sequence (in X) is a map N−→X, often denoted by x1 , x2 , . . . or (xn )n . A generalization is the concept of a net (in X), which is a map I −→X, where I is a so-called

442

A.2 Real numbers

443

directed set (i.e., there is a reflexive, transitive relation ≤ on I such that for all i1 , i2 ∈ I there is an i ∈ I with i1 ≤ i, i2 ≤ i). We denote such a net by (xi )i∈I or by (xi )i . The cartesian product of a collection {Xi : i ∈ I } of sets is written i∈I Xi . If Xi = X for all i ∈ I the product is denoted by XI and sometimes called X to the power In. For a finite number of sets X , . . . , X we often write X × · · · × X , 1 n 1 n i=1 Xi , 1≤i≤n Xi , or instead of i∈{1,...,n} Xi . Let X, Y be sets, let f : X−→Y . The set f (X) = {f (x) : x ∈ X} is usually called Im f . The restriction of f to some subset T of X is denoted by f |T . For Z ⊂ Y we write f −1 (Z) := {x ∈ X : f (x) ∈ Z}. The graph of f is the set {(x, f (x)) ∈ X × Y : x ∈ X}. If f is bijective we denote its inverse Y −→X by f −1 . Let Z be a third set. If there are g : X−→Z and h : Z−→Y such that f is the composition of h and g we write f = h ◦ g to indicate this, and we say that this formula is a factorization of f . A family F ⊂ Y X is said to separate the points of X if for each x1 , x2 ∈ X, x1 = x2 there is an f ∈ F such that f (x1 ) = f (x2 ). The identity map X−→X is denoted by IX , or by I if no confusion is expected. Let n ∈ N, n ≥ 2, let f : X n −→Y or f : X n \ {(x, . . . , x) : x ∈ X}−→Y . f is called a symmetric map (of n variables) if for every bijection σ : {1, . . . , n}−→{1, . . . , n} we have f ((x1 , . . . , xn )) = f ((xσ (1) , . . . , xσ (n) )). A set that has precisely one member x is called a singleton (set) and is denoted by {x}. A set X is countable if there exists a surjection ϕ : N−→X (thus, “countable” includes “finite”). If ϕ is a bijection it is called an enumeration (of X). We will often use the well-known fact that the countable union of countable sets is countable. The cardinality of a set X is denoted by X; sometimes we write ℵ0 for N. In this book we call a set X small if every ultrafilter on X that is closed for countable intersections is principal (i.e., in a more common language, if X has non-measurable cardinality). It is straightforward to check that this notion of “smallness” is equivalent to the one in page 32 of [193]. Recall ([193], 2.6) that subsets of small sets are small, and that P(X) is small if X is small. Also, it is easy to see that N is small. Hence, the sets we meet in daily mathematical life are small. In fact, believing that all sets are small will not lead to a disaster, see [193], p. 31–33 for further discussions and references.

A.2 Real numbers We denote by N, Z, Q, R, C the set of the natural numbers, integers, rational numbers, real numbers, complex numbers, respectively. A subset X ⊂ {α ∈ R : α > 0} is called bounded away from 0 if there exists a δ > 0 such that X ⊂ {α ∈ R : α ≥ δ}. In the same spirit we define “bounded away from 0” for nets in {α ∈ R : α > 0}. For a subset X of R that is not bounded above (below) we write sup X := ∞ (inf X := −∞). We proceed in the same spirit for nets in R. The entire part of an x ∈ R is [x] := max{m ∈ Z : m ≤ x}. The natural logarithm of a real number x > 0 is denoted by log x. Let X be a set, let f, g : X−→R. We write f ≤ g if f (x) ≤ g(x) for all x ∈ X. In general max(f, g) is the function x → max(f (x), g(x)); similarly we define min(f, g).

444

Glossary of terms

Apart from the usual matrixes of linear maps in finite-dimensional spaces with respect to certain algebraic bases, we sometimes consider infinite real matrixes (bnk )k,n where k, n ∈ N, without connection with an underlying linear map.

A.3 Groups, rings and fields We assume that the reader is familiar with the notions of group, ring, field and vector space over a field K (K-vector space, K-linear space). From now on in this section all groups are abelian and written additively. A group G is called divisible if for each x ∈ G and n ∈ N there is a y ∈ G such that n y = x. A coset of a subgroup H of G is a set of the form a + H := {a + h : h ∈ H } for some a ∈ G. Throughout this section rings and fields are commutative with identity 1. Let R1 , R2 be rings, let f : R1 −→R2 . If f (1) = 1, f (x + y) = f (x) + f (y), f (x y) = f (x) f (y) for all x, y ∈ R1 , f is called a (ring) homomorphism. If, in addition, f is a bijection then f is called a (ring) isomorphism, if also R1 = R2 , a (ring) automorphism. A subset D of a ring R is called a subring if D is an additive subgroup of R, 1 ∈ D, and x, y ∈ D implies x y ∈ D. Then D, with the operations inherited from R, is itself a ring. A subset I of R is called an ideal if I is an additive subgroup of R, and x ∈ R, y ∈ I implies x y ∈ I . Then, if I = R, the quotient R/I is in a natural way again a ring. If, in addition, there are no ideals J with I J R the ideal I is called maximal. I is maximal if and only if R/I is a field. An inverse of an x ∈ R is an element y ∈ R such that x y = 1. If such an inverse exists it is unique and denoted by x −1 . Ring homomorphisms (isomorphisms, automorphisms) f between fields satisfy f (x −1 ) = f (x)−1 for all nonzero x and are also called field homomorphisms (isomorphisms, automorphisms). Notice that f is automatically injective. Two fields are called isomorphic if there is a field isomorphism between them. Let K be a field. An embedding (of K into a field L) is a field homomorphism K−→L. A field extension of K is a field L containing K as a subfield. The degree of the extension L is the dimension (see A.4) of L, viewed as a K-vector space (possibly infinite). For each a ∈ L the smallest subfield of L containing K and {a} is denoted K(a).The element a is called algebraic (over K) if the degree of the extension K ⊂ K(a) is finite. L is called algebraic (over K) if each element of L is algebraic over K. K is called algebraically closed if every algebraic field extension of K equals K. An algebraic closure of K is an algebraic field extension of K that is algebraically closed. It is well-known that each field K has an algebraic closure and that two algebraic closures are linked by an isomorphism leaving K pointwise fixed. Thus we may speak about “the” algebraic closure of K; we denote it by K a . The map n → n.1 is a ring homomorphism Z−→K. If it is injective we say that the characteristic of K is 0, notation char K = 0. If not then the kernel is easily seen to be the ideal of all multiples of some prime number p, and in this case we say that the characteristic of K is p, notation char K = p. The smallest subfield of K is called the prime field of K. It is isomorphic to the field of rational numbers if char K = 0 and to the field of p elements if char K = p = 0. For m, n ∈ Z, the expression m ≡ n (mod p) means, as usual, that m − n is divisible by p.

A.4 Vector spaces

445

A.4 Vector spaces Let E, F be vector spaces over a field K. For a linear map T : E−→F we write Ker T := {x ∈ E : T (x) = 0}; it is a subspace of E. If T is bijective we say that T is an (algebraic) isomorphism and that E, F are (algebraically) isomorphic. For a subspace D of E the quotient space E/D is in a natural way again a vector space over K. The canonical map E−→E/D is called the quotient map and usually denoted by π . A subspace D1 of E is called an (algebraic) complement of D if E = D + D1 and D ∩ D1 = {0}. Such a D1 exists and is algebraically isomorphic to E/D. Let {Ei : i ∈ I } be a collection of vector spaces over K. Then the product i∈I Ei , under coordinatewise operations, is again a vector space over K. We define alg Ei := {(xi )i∈I ∈ i∈I

Ei : xi = 0 only for finitely many i}.

i∈I

This is a subspace of the product and called the (algebraic) direct sum of the Ei . In alg case Ei = E for all i ∈ I we write E (I ) for i∈I Ei . Let (Di )i∈I be a collection of subalg spaces of E. The map i∈I Di −→E, given by (di )i∈I → i∈I di is linear. Its image is the smallest subspace of E containing all Di , denoted by i∈I Di , called the sum (of the Di ). If the above map is injective we sometimes identify the sum of the Di with the direct sum of the Di . Let E be a K-vector space. For a subset X of E, the smallest subspace containing X is written [X], the subspace generated by (the elements of) X, or the linear hull of X. To avoid complicated notations, for {ei : i ∈ I } ⊂ E we write [ei : i ∈ I ] instead of [{ei : i ∈ I }]. Similarly, for a sequence e1 , e2 , . . . in E we write [e1 , e2 , . . .] instead of [{e1 , e2 , . . .}]. For x ∈ E we sometimes write Kx for [x]. A collection {ei : i ∈ I } is called an algebraic base of E if the ei are linearly independent and [ei : i ∈ I ] = E. It is not hard to see that {ei : i ∈ I } is an algebraic base of E if and only if each element of E can uniquely be written as a (finite) linear combination of the ei . Two algebraic bases have the same cardinality, called the (algebraic) dimension of E, dim E. The codimension of a subspace D of E, is defined as codim D := dim E/D. A subset X of E is finite-dimensional if dim [X] < ∞. A linear manifold is a subset of E of the form a + D, where a ∈ E and D is a subspace. If codim D = 1, the set a + D is called a hyperplane. A projection E−→E is a linear map P : E−→E with P 2 := P ◦ P = P . The algebraic dual E ∗ of E is defined as {f : E−→K : f is linear}. It is a vector space over K in a natural way. The elements of E ∗ are called (linear) functionals. If X1 , X2 , . . . are subsets of E, by n Xn we denote the set of the elements of E that can be written as finite sums, x1 + · · · + xn , with x1 ∈ X1 , . . . , xn ∈ Xn , n ∈ N. Let E, F be K-vector spaces. Let T : E−→F be linear. If E = F , a subspace D of E is called invariant (for T ) if T (D) ⊂ D. T is of finite rank if T (E) is finite-dimensional. A map S : E−→F is called an affine transformation if S(λ x + µ y) = λ S(x) + µ S(y) for all x, y ∈ E, λ, µ ∈ K, λ + µ = 1. An example is the translation (over a):

446

Glossary of terms

x → a + x (x ∈ E), where a ∈ E. For a set X ⊂ E the set a + X is called the translate of X (by a). Let E1 , E2 , F be vector spaces over K. A map B : E1 × E2 −→F is called bilinear if for each x0 ∈ E1 , y0 ∈ E2 the maps x → B(x, y0 ), y → B(x0 , y) are linear. In case F = K, B is often called a bilinear form. A K-algebra is a set A that is a ring and at the same time a K-vector space, such that λ (x y) = (λ x) y = x (λ y) for all λ ∈ K, x, y ∈ A. The (K-valued) Kronecker delta is defined as usual by the formula 0 ∈ K if m, n ∈ Z, m = n δmn := 1 ∈ K if m, n ∈ Z, m = n. The (K-valued) characteristic function (of a subset Y of a set X) is defined by 1 ∈ K if x ∈ Y ξY (x) := 0 ∈ K if x ∈ X \ Y .

A.5 Topological spaces Let X be a set. A topology on X is a subcollection τ of P(X) containing ∅ and X, that is closed for finite intersections and arbitrary unions. X = (X, τ ) is called a topological space, the members of τ are called (τ -)open. A subset of X is called (τ -)closed if its complement is open. If, for two topologies τ1 , τ2 on X, we have τ1 ⊂ τ2 , we say that τ1 is weaker (or smaller) than τ2 , and τ2 is said to be stronger (or larger) than τ1 , and we often write τ1 ≤ τ2 rather than τ1 ⊂ τ2 . Among all topologies on X there is a weakest one, {∅, X}, called the indiscrete topology and also a strongest one, P(X), called the discrete topology, and in the last case we say that X is discrete. Let X = (X, τ ) be a topological space. A subcollection B ⊂ τ is called a base (for the topology τ ) if each open set is a union of members of B. A subcollection S ⊂ τ is called a subbase (for the topology τ ) if τ is the smallest topology containing S. The union of a collection {τi : i ∈ I } of topologies on X is the topology having i∈I τi as a subbase; it is, among all topologies that are stronger than each τi , the weakest one. A (τ -)neighbourhood of an x ∈ X is a set V ⊂ X such that there is an open set U with x ∈ U ⊂ V . X (and τ ) are called Hausdorff if any two different points of X have disjoint neighbourhoods. A neighbourhood base (or base of neighbourhoods) at x is a collection B of neighbourhoods of x such that each neighbourhood of x contains a member of B. τ For a subset Y of X its closure, denoted by Y or Y , is the smallest closed set containing Y ; its interior, denoted by int Y , is the largest open set contained in Y . The set Y \ int Y is called the boundary of Y . Y is called dense (in X) if Y = X. A point x ∈ X is called an accumulation point of Y if each neighbourhood of x contains points of Y other than x. A point x ∈ X is called isolated if it is not an accumulation point of X, i.e., if {x} is open. The collection {U ∩ Y : U ∈ τ } forms a topology on Y ⊂ X, called the restriction of τ (to Y ), or the inherited topology on Y , and is denoted by τ |Y . The members of

A.5 Topological spaces

447

τ |Y are called relatively open (with respect to Y ) or open in Y . It is customary to call Y = (Y, τ |Y ) a (topological) subspace of X. A set in Y which is τ |Y -closed is called relatively closed or closed in Y . A net (xi )i∈I in X is said to converge to x, and x is called a limit of the net if for each neighbourhood U of x there is an i0 ∈ I such that xi ∈ U for all i ≥ i0 . We indicate this τ convergence by the notations x = limi∈I xi = limi xi or by xi → x or xi −→ x if we want to specify the topology τ . If X is Hausdorff each net has at most one limit. Convergent nets determine the topology in the following sense. Theorem A.5.1 ([144], 2.3.(1)) A subset Y of a topological space X is closed if and only if for every net in Y that converges to x ∈ X we have x ∈ Y . We say that Y ⊂ X is sequentially closed if for any sequence in Y converging to x ∈ X we have x ∈ Y . Let X, Y be topological spaces. A map f : X−→Y is continuous at x ∈ X if for each neighbourhood V of f (x) the set f −1 (V ) is a neighbourhood of x. If f is continuous at all x ∈ X we say that f is continuous. It is easily seen that f is continuous if and only if for each open set V ⊂ Y the set f −1 (V ) is open. A surjection f : X−→Y is called (an) open map if for each open U in X the image f (U ) is open. A bijection f : X−→Y that is both continuous and open is called a homeomorphism. Then f −1 is also a homeomorphism and X, Y are called homeomorphic. We have the following characterization of continuity in terms of nets. Theorem A.5.2 ([144], 2.3.(2)) Let X, Y be topological spaces, let f : X−→Y , let x ∈ X. Then f is continuous at x if and only if for each net (xi )i in X converging to x we have f (xi ) → f (x). f : X−→Y is called sequentially continuous if for each x ∈ X and each sequence x1 , x2 , . . . converging to x we have f (xn ) → f (x). Alternatively, we may use filters to express continuity as follows. We say that a filter F on X converges to x ∈ X, notation F → x, if F contains the filter of all neighbourhoods of x. Such an x is unique if X is Hausdorff. Theorem A.5.3 ([144], 2.5.(5)) Let X, Y, f, x be as in A.5.2. Then f is continuous at x if and only if for each filter F on X, F → x implies f (F) → f (x). A map f : X−→Y is called an embedding (of X into Y ), and we say that X is (homeomorphically) embedded into Y (by f ), if f , viewed as a map X−→f (X) is a homeomorphism, where f (X) carries the relative topology. Let X be a topological space. X is called separable if it has a countable dense subset, (ultra)metrizable if there exists an (ultra)metric on X inducing the topology (see 1.1.1). As usual, a subset Y of a metric space is open if for every a ∈ Y there is a ball about a contained in Y (balls in metric spaces are defined at the beginning of Section 1.1). A subset of X is called clopen if it is both closed and open. X is called connected if ∅ and X are the only clopen sets, totally disconnected if the only non-empty connected subspaces are the singleton sets. X is called zero-dimensional if there is a base for the topology consisting of clopen sets. Clearly, a Hausdorff zero-dimensional space is totally disconnected. X is compact if each open covering has a finite subcovering.

448

Glossary of terms

A point x of X is called a cluster point of a filter F on X if x ∈ V for every V ∈ F. It is easily seen that such a cluster point is a limit of F as soon as F is an ultrafilter. We have the following characterization of compactness in terms of filters. Theorem A.5.4 ([144], 3.1.(1), 3.1.(3)) For a topological space X the following are equivalent: (α) X is compact. (β) Each filter on X has a cluster point. (γ ) Each ultrafilter on X converges. X is called locally compact if every point of X has a compact neighourhood, σ -compact if X is the union of countably many compact subsets, N-compact if X can be embedded into some product of countable discrete spaces. A subset Y of X is called relatively compact if Y is compact (the terminology is in some discord with “relatively closed”). Let {Xi : i ∈ I } be a collection of topological spaces, let X := i∈I Xi . Let, for each i ∈ I , πi : X−→X i be the canonical coordinate map. The product topology on X is the one that has i∈I {πi−1 (Ui ) : Ui open in Xi } as a subbase; it is the weakest topology on X making the maps πi continuous. We quote the following fundamental result. Theorem A.5.5 (Tychonov Theorem, [144], 3.3.(1)) Products of compact spaces are compact. Let X be a topological space, let Y be a set, let f : X−→Y be a surjection. Then {V ⊂ Y : f −1 (V ) is open} is a topology on Y , called the quotient topology. It is the strongest topology on Y making f continuous. The space Y , equipped with the quotient topology, is often called a quotient of X. The quotient map f is usually denoted by π . A topological space X is called a Baire space if the intersection of countably many open dense subsets of X is dense in X. Theorem A.5.6 (Baire Category Theorem, [144], 4.6.(4)) A complete metric space (see A.6) is a Baire space.

A.6 Metric spaces Let (X1 , d1 ) and (X2 , d2 ) be metric spaces (see the beginning of Section 1.1). A map f : X1 −→X2 is called an isometry if d2 (f (x), f (y)) = d1 (x, y) for all x, y ∈ X1 , and we say that X1 is isometrically embedded into X2 (by f ). If, in addition, f is surjective we call f an isometrical isomorphism, and we say that X1 is isometrically isomorphic to X2 . f : X1 −→X2 is called Lipschitz if there is an M > 0, called a Lipschitz constant, such that d2 (f (x), f (y)) ≤ M d1 (x, y) for all x, y ∈ X1 . f : X1 −→X2 is called uniformly continuous if for each ε > 0 there exists a δ > 0 such that x, y ∈ X1 , d1 (x, y) < δ implies d2 (f (x), f (y)) < ε. Each isometry is Lipschitz, each Lipschitz map is uniformly continuous. Let X = (X, d) be a metric space. A subset Y of X is called bounded if {d(y, z) : y, z ∈ Y } is a bounded subset of R. In the same spirit we define bounded sequences, and bounded maps Z−→X, where Z is any set. Also, a family F of maps Z−→X is called

A.7 Topological vector spaces

449

pointwise bounded if for each x ∈ Z the set {f (x) : f ∈ F} is bounded in X. In case X = R (with the usual metric), for such F we define sup F to be the function Z−→R, x → sup{f (x) : f ∈ F}. A sequence x1 , x2 , . . . in X is called Cauchy sequence if limm,n→∞ d(xm , xn ) = 0. If each Cauchy sequence in X converges we say that X is complete. Each metric space (X, d) has a completion, i.e., a complete metric space X ∧ = (X ∧ , d ∧ ) containing X such that d ∧ extends d, and such that X is dense in X ∧ . It is well-known that each uniformly continuous map f : X−→Y , where X, Y are metric spaces, can uniquely be extended to a (uniformly) continuous map f ∧ : X ∧ −→Y ∧ . A metric d on an (abelian) group G is called invariant if d(x, y) = d(a + x, a + y) for all a, x, y ∈ G.

A.7 Topological vector spaces Throughout this section, K = (K, | . |) is a valued field, complete with respect to the metric induced by the valuation | . |, see Section 1.2. We assume the valuation to be non-trivial. Let E, F be topological vector spaces over K (i.e., E, F are equipped with a topology making the vector operations continuous). → ni=1 xi A sequence x1 , x2 , . . . in E is said to be summable with sum x ∈ E if n ∞ converges to x. If E is Hausdorff such a sum is unique and we write x = i=1 xi , as usual. A sequence in E that converges to 0 is sometimes called a null sequence. A net (xi )i∈I in E is called Cauchy (net) if for each zero neighbourhood U in E there exists an i0 ∈ I such that xi − xj ∈ U for all i, j ≥ i0 . If E is Hausdorff we say that E is complete if each Cauchy net converges. For such E, a subset X is closed if and only if each Cauchy net in X converges and whose limit is in X. We sometimes call a subset X of a Hausdorff E complete if each Cauchy net in X converges and has its limit in X. By replacing in the above “net” by “sequence” we arrive at the notion of sequentially complete (space, subset). For a subset X of E, the closed linear hull of X is [X]. X is called precompact if for each zero neighbourhood U in E there is a finite set G ⊂ E such that X ⊂ U + G. Each relatively compact set in E is precompact but not conversely. The notions coincide if E is complete. Let X be a set. Let (fi )i∈I be a net of maps X−→E, and let also f : X−→E. We say that (fi )i∈I converges pointwise (to f ) if fi (x) → f (x) for all x ∈ X. We say that (fi )i∈I converges uniformly on Y ⊂ X (to f ) if for each zero neighbourhood U in E there exists an i0 ∈ I such that f (x) − fi (x) ∈ U for all i ≥ i0 and all x ∈ Y . Let X be a topological space. For an f : X−→E we define the support of f , denoted supp f , by {x ∈ X : f (x) = 0}. A family F of maps X−→E is called equicontinuous at x ∈ X if for each zero neighbourhood U in E there exists a neighbourhood V of x such that f (y) − f (x) ∈ U for all y ∈ V and all f ∈ F. We say that F is equicontinuous if it is equicontinuous at all points of X. Let X ⊂ E, Y ⊂ F . A map f : X−→Y is called uniformly continuous if for any zero neighbourhood V in F there is a zero neighbourhood U in E such that for all x1 , x2 ∈ X, x1 − x2 ∈ U implies f (x1 ) − f (x2 ) ∈ V .

450

Glossary of terms

The following extension theorem holds. We give a reference for topological vector spaces over the real or complex field. The same proof works for any ground field K. Theorem A.7.1 ([101], 2.9.5) Let F be complete, let X ⊂ E, let f : X−→F be uniformly continuous. Then f extends uniquely to a (uniformly) continuous map f : X−→F .

appendix b Guide to the examples

The most important examples of locally convex spaces occurring in this book are listed below, together with indications where their properties can be found. References in brackets concern underlying aspects and definitions. Numbers refer to theorems, corollaries, etc. unless indicated.

B.1 Spaces of continuous functions C(X), X compact: (Subsection 2.5.3), (2.5.21), 2.5.22, 2.5.24, 2.5.27, 2.5.30, 3.8.2, 4.4.8, Subsection 7.2.1, 7.5.1, Section 8.7. P C(X): (2.5.20), 2.5.21, 2.5.23, 2.5.37, 2.5.38, 2.5.40, 4.4.8, Subsection 7.2.1, 7.5.1, Section 8.7. C0 (X), X locally compact: (2.5.20), 2.5.21, 2.5.34, 2.5.36, 4.4.8, Subsection 7.2.1, 7.5.1, Section 8.7. c0 , c0 (I) (case X discrete): (2.3.2), 2.3.9, (Subsection 2.5.1), 2.5.4, 2.5.6, 2.5.8, 2.5.9, 2.5.11, 2.5.21, 3.8.31, 4.1.10, 4.1.12, 4.2.8, 5.2.3, 5.2.6, 5.5.5, 7.4.3, 8.1.4, 8.1.11, 8.2.1, 8.2.3, 9.8.1. BC(X): (2.5.20), 2.5.21, 2.5.40, 4.4.8, Subsection 7.2.1, 7.5.2, Section 8.7. ∞ , ∞ (I) (case X discrete): (Subsection 2.5.2), 2.5.11, 2.5.13, 2.5.15, 2.5.21, 4.1.10, 4.1.12, 4.4.9, 5.2.6, 5.5.5, 5.5.7, 5.6.4, 5.7.13, 7.1.10, 7.4.3, 7.4.18, 8.1.11, 8.2.1, 8.2.2, 8.2.3. (C(X), τc ): (Subsection 2.5.3), (3.7.3), 3.7.9, 3.7.10, 4.3.2, 4.3.4, 4.4.8, 5.3.13, 7.2.7, 7.2.8, 7.5.7, 7.5.8, 7.5.9, 8.7.2, 8.7.3. (C(X), τs ): (Subsection 2.5.3), (3.7.1), 3.7.2, 3.7.10, 4.3.4, 4.4.8, 7.2.10, 7.2.11, 7.2.12, 7.5.6, 8.7.1. (BW (X), τW ), (BC W (X), τW ): (3.7.11), (3.7.12), 3.7.13, 4.4.8. (BC(X), β0 ) (case W = W0 ): (3.7.14), 3.7.16, 3.7.17, 3.7.18, 4.3.2, 4.3.7, 4.3.8, 4.3.11, 5.3.14, 7.2.14, 7.5.10, 8.7.2, 8.7.3. (BC Wc (X), τWc ), (C c (X), τWc ), (C c (X), τi ) (case W = Wc ): (3.7.19), 3.7.20, (3.7.21), 3.7.22, 3.7.23, 4.3.12, 4.3.13, 5.3.14, 7.2.15, 7.2.16, 7.5.11, 8.7.4, 8.7.5, 11.1.15(I), 11.2.9. C(X−→E): (10.5.1), 10.5.3, 10.5.4, 10.5.7, 10.5.10, 10.5.11, 10.5.12, 10.5.13.

451

452

Guide to the examples

B.2 Spaces of differentiable functions (C n (X), τcn ), (C ∞ (X), τc∞ ): (2.5.79), (2.5.91), (2.5.92), (3.7.36), 3.7.37, 3.7.38, 3.7.39, 3.7.41, 3.7.49, 3.7.54, 4.3.16, 4.3.17, 4.4.8, 5.3.17, 7.2.18, 7.5.14, 7.5.16, 7.5.17, 7.5.20, 7.5.21, 8.7.8, 8.7.9, 8.7.10. (C n (X), τsn ), (C ∞ (X), τs∞ ): (2.5.79), (2.5.91), (2.5.92), (3.7.42), 3.7.43, 3.7.45, 4.3.17, 4.4.8, 5.3.17, 7.2.17, 7.5.13, 8.7.7. BC n (X): (2.5.79), (2.5.82), 2.5.84, 2.5.85, 4.4.8, Subsection 7.2.1, Section 8.7. BC ∞ (X): (2.5.79), (2.5.82), (3.7.46), 3.7.47, 4.3.19, 4.4.8, 5.3.17, Subsection 7.2.1, 7.5.22, 7.5.23, 8.7.11. C0n (X), C0∞ (X): (2.5.79), (3.7.57), 3.7.61, 3.7.62, 3.7.63, 4.3.21, 4.4.8, 5.3.17, Subsection 7.2.1, 7.5.5, 7.5.24, Section 8.7, 8.7.12. n ∞ (X), τWn ), (BCW (X), τW∞ ): (2.5.79), (3.7.48), 3.7.49, 3.7.51, 4.4.8, 5.3.17. (BC W n ∞ C c (X), C c (X) (case W = C ∞ (X)): 3.7.53, 3.7.54, 4.3.20, 7.2.19, 7.2.21, 7.5.26, 7.5.27, 8.7.14, 8.7.16, 11.1.15(II), 11.2.9. n ∞ BC W (X), BC W (X) = S(X) (case W = {polynomial functions}): 3.7.55, 3.7.56, 3.7.62, 3.7.63, (3.7.64), 4.3.21, Subsection 7.2.1, 7.5.29, 7.5.32, 8.7.17.

B.3 Spaces of analytic functions P S(B): (2.5.51), 2.5.53, 2.5.55, 2.5.68, (2.5.91), 4.4.8, Subsection 7.2.1, 7.5.4. B P S(B): (2.5.57), 2.5.59, 2.5.60, 2.5.68, (2.5.70), (2.5.92), 4.4.8, Subsection 7.2.1, 7.5.4. H( A): (2.5.62), 2.5.64, 2.5.68, 2.5.69, (2.5.70), 2.5.72, 3.7.29, 4.4.8, Subsection 7.2.1, 7.5.4. A(r): (3.7.25), 3.7.26, 4.3.14, 4.4.8, 5.3.15, Subsection 7.2.1, 9.7.1, 9.7.4, 9.7.5, 9.7.6. A† (r): (3.7.27), 4.3.14, 4.4.8, 5.3.15, Subsection 7.2.1, 9.7.3, 9.7.4, 9.7.5, 9.7.6, 9.7.7, 11.1.15(III), 11.2.9, 11.4.3. O( A): (3.7.29), 3.7.31, 3.7.33, 4.3.15, 4.4.8, 5.3.16, Subsection 7.2.1, 7.5.12.

B.4 Valued field extensions K (a): (Subsection 2.5.4), 2.5.42, 2.5.44, 4.4.8, Subsection 7.2.1, 7.5.3, Section 8.7.

B.5 Sequence spaces

0 (B): (9.5.2), 9.5.4, 9.5.7, 9.5.9, 9.5.11, 9.5.12, 9.5.15, 9.6.1, 9.6.2, 9.6.3, 11.4.5.

∞ (B): (9.5.8), 9.5.9, 9.5.14, 9.6.4, 9.6.5, 9.6.6, 9.7.2, 11.4.2, 11.4.5, 11.4.6, 11.6.3, 11.6.4.

0 (B): (11.4.1), 11.4.2, 11.4.4, 11.4.5, 11.4.7, 11.4.9, 11.6.3.

∞ (B): (11.4.1), 11.4.5.

Notation

1, 444 . , 15 . n m , 292 .∞ , 42, 44, 48 ℵ0 , 443 alg i∈I Ei , 101, 445 i∈I Ei , 101 n X, 71

i∈I Ei , 21 i Ei , 21 ⊗, b374 Ei , 21 i∈I b E , 21 i i 1≤i≤n Xi , 443 i∈I Ei , 99, 445 Xi , 443 i∈I n i=1 Xi , 443 ∅, 442 | . |, 5 | . |p , 9, 10 | . |∞ , 6 Ae , 84 A∧ n , 420 aco X, 84 aco X, 84 A(r), 124 A† (r), 125 B, 356 B(a, r), 1 B(a, r − ), 1 BC(X−→K), 48 BC(X), 48 BC ∞ (X), 133

BC n (X−→K), 73 BC n (X), 73 BCW (X), 118 ∞ (X), 133, 138 BCW n BCW (X), 133, 138 BP S(B), 64 BK , 6 − BK ,6 BX (a, r), 1 BX (a, r − ), 1 BE , 15 BE− , 15 BW (X), 118 (bnk )k,n , 356 β0 , 119 b(E , E), 272 bw, 336 C(X−→E), 396 C(X−→K), 46 C(X), 46 C 1 (X−→K), 70 C 1 (X), 70 C e , 88 C ∞ (X−→K), 72 C ∞ (X), 72 C n (X−→K), 72 C n (X), 72 C0 (X−→K), 48 C0 (X), 48 C0∞ (X), 139 C0n (X), 139 Cc∞ (X), 136 Ccn (X), 136 C, 443 Cp , 10

453

454

c(E , E), 315 c0 , 28 c0 (I ), 42 c0 (I, s), 46 c0 (N), 42 c00 , 42 c00 (I ), 42 co X, 88 co X, 88 codim D, 445 CF (E, F ), 309 C(E, F ), 302 Dn f , 72 δmn , 446 diam Y , 2 dim E, 445 dist (a, Y ), 2 dist (Y, Z), 2 (En )n , 410 (En , τn )n , 410 ˆ π F , 395 E⊗ E ⊗ F , 374 E ⊗ε F , 387 E ⊗π F , 384 E ∼ F , 97 E F , 17 E , 17, 97 E , 275 Eb , 272 Eσ , 272 E ⊗π L, 404 E (I ) , 101, 445 E ∗ , 445 E I , 99 E ∧ , 16, 100 E1 × · · · × En , 21, 99 Ep , 92 E, 15 (ei )i∈I , 25 [e1 , e2 , . . .], 445 [ei : i ∈ I ], 445 ξY (x), 446 ei , 42 f (a), 70 f Y , 63 f n , 73, 418 f r , 63 f − r , 63

Notation

f s , 124 f nφ , 133 f B ,126 f → X f dµ, 52 f |T , 443 f , 70 f (F), 442 f (0) , 70 f (n) , 70 f −1 (Z), 443 f −1 , 443 n f , 74 0 f , 71 1 f , 70 n f , 71 i f Y , 127 1 f , 70 n f , 72 φj , 101 ˜ n f , 74 ϕµ , 52 F → x, 447 H (A), 65 HB , 293 h ◦ g, 443 I , 17, 443 IX , 443 Im f , 443 inf X, 443 int Y , 446 jE , 274 Kx, 445 K, 5 K(X), 57 K(a), 57, 444 K[X], 59 K[a], 58 K H , 38 K×, 7 K ∨ , 11 K∧, 5 K n , 20 Kν2 , 36 |K|, 7 |K × |, 7 k, 7 (K, | . |), 5

Notation

Ker T , 445 K0 , 9 L(E), 17, 97 L(E, F ), 16, 97 L(X), 11 Lb (E, F ), 390 2 , 40, 96 1 , 228 ∞ , 44 ∞ (I ), 44 ∞ (I, s), 46 ∞ (N), 44 , 350 0 (B), 356 ∞ (B), 429 × , 350 0 (B), 429 ∞ (B), 357 λ → λ∗ , 38 λ → λ, 7 lim En , 106 ←−

lim(En , πn ), 106 ←−

lim(En , in ), 107

p ⊗ q, 379 p ∨ q, 83 p ∧ q, 83 pA , 85 py , 351 pY , 114 pYn (f ), 127 pφ , 118 pπ , 98 p∅ , 114 π, 384, 445 πi , 448 πj , 101 πp , 92 P(X), 442 Q, 443 Qp , 9 Qap , 10 q˜Z , 396 R(A), 65 R, 443 R((X)), 40

−→

lim En , 107 −→

limi xi , 447 limi∈I xi , 447 log x, 443 M(X), 52 µ, 51 m ≡ n (mod p), 444 max(f, g), 443 min(f, g), 443 Nϕ , 263 N, 443 n( , × ), 351 (X), 47 ω, 375 O(A), 126 P C(X−→K), 48 P C(X), 48 P S(B), 62 p, 92 ρ, 7 p ⊗ε q, 378 p ⊗π q, 378

(S 0 ), 179 n Xn , 445 Di , 445 i∈I ∞ i=1 xi , 16 i∈I xi , 41 σ , 410 σ (E , E), 272 σ (E, E ), 182, 211 σ (E ∗ , E), 242 σ (c0 , ∞ ), 211 σn , 410 sup X, 443 supp ϕ, 263 supp f , 449 S(X), 143 T , 272 T ∧ , 101 T1 ⊗ T2 , 376 Tz , 391 , 397 T , 16 τ |Y , 446 τc , 115, 396 τc∞ , 128 τcn , 128

455

456

τi , 121 τi0 , 429 τi∞ , 359 τp0 , 356, 431 τp∞ , 432 τs , 114 τsn , 131 τW , 118 ∞ , 133 τW n τW , 133 τπ , 98 τc.t , 207 τnu , 336 τpol , 208 τsm , 336 τs∞ , 131 Vol(x1 , . . . , xn ), 165 W0 , 119 Wc , 121 (X, τ ), 446 (X, d), 1 (X ∧ , d ∧ ), 449 X Y , 377 X ⊗ Y , 377 X, 1, 46 X I , 443 X ◦• , 275 X ◦◦ , 196

Notation

X◦ , 196 X⊥ , 38 X∧ , 4, 449 Xζ , 55 X1 × · · · × Xn , 443 X1 ⊥ X2 , 23 [X], 445 X, 63 X, 443 (xi )i , 443 (xi )i∈I , 443 (xn )n , 442 < x, y >, 38 {x}, 443 {x}, 443, 445 x ⊗ y, 375 x ⊥ X2 , 23 x ⊥ y, 23 x −1 , 444 x1 , x2 , . . ., 442 τ xi −→ x, 447 x i → x, 447 x n , 79 Y , 446 τ Y , 446 Z, 443 Zp , 9 ζx , 55 (ζX , X ζ ), 55

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Index

absolutely convex, 83 absolutely convex hull, 84 absorbing, 85 accumulation point, 446 adjoint, 272 admissible topology, 235 affine transformation, 445 algebraic, 444 algebraic base, 445 algebraic closure, 444 algebraic dual, 445 algebraically closed, 444 α-regular inductive sequence, 411 almost Hahn–Banach property, 199 analytic element, 65 A-normed space, 203 Approximation Theorem for linear forms, 243 Ascoli Theorem, 143 associated norm, 38 associated topologies, 336 automorphism, 444 Baire Category Theorem, 104, 448 Baire space, 448 Banach disk, 414 Banach space (over K), 15 Banach’s Open Mapping Theorem, 21 Banach–Dieudonn´e Theorem, 440 Banach–Steinhaus Theorem, 22, 111, 256 Banach–Stone Theorem, 79 Banaschewski compactification, 55 barrel, 255 barrelled, 255 barycenter, 240 base, 31, 338, 442, 446 base of continuous seminorms, 94

Bessaga–Pelczynski Selection Principle, 234 best approximation, 5 bilinear form, 446 bilinear map, 374, 446 Bipolar Theorem, 215 boundary, 446 bounded away from 0, 443 bounded map, 448 bounded sequence, 112, 448 bounded set, 112, 448 bounded weak topology, 336 C 1 -function, 70 C n -function, 72 C ∞ -function, 72 c-countably generated, 90 c-compact, 154, 206, 244 canonical norm on K n , 20 Cauchy net, 100, 449 Cauchy sequence, 449 Cauchy–Schwarz inequality, 38 centre of a ball, 1 characteristic, 444 characteristic function, 446 clopen, 447 closed, 446 closed ball, 1 Closed Graph Theorem, 22, 111, 413 closed linear hull, 449 closed unit ball, 6, 15 closed unit disk, 6 closely related, 101 closure, 446 cluster point, 448 codimension, 445 coefficient functional, 338

468

Index

commutation relation, 80 compact, 447 compact open topology, 115 compact operator, 334 compactifying operator, 309 compactoid inductive limit, 424 compactoid inductive sequence, 424 compactoid operator, 302 compactoid regular inductive sequence, 441 compactoid set, 143 complement, 24, 105, 445 complemented, 24, 105 complete, 100, 449 completed tensor product, 395 completion, 100, 449 complexification, 404 connected, 447 Continuity Lemma, 384 continuous, 447 continuously differentiable, 70 convergence of filters, 447 convergence of nets, 447 convex, 87 convex filter, 246 convex hull, 88 coordinate map, 338 countable type, 28 Decomposition Lemma, 236, 242 degree, 444 dense, 446 dense valuation, 7 derivative, 70 diameter, 2 Dieudonn´e-Schwartz Theorem, 412 differentiable function, 70 dimension, 445 direct sum, 445 directed set, 443 discrete topological space, 446 discrete topology, 446 discrete valuation, 7 disk, 62 distance between sets, 2 distributions, 371, 438 divisible group, 444 dual, 97 dual of an inductive sequence, 415 dual space, 17, 97 dual-separating, 202, 210, 211

469

˘ Eberlein-Smulian Theorem, 241 edged, 84, 88 edged hull, 84 embedded, 447 embedding, 17, 97, 444, 447 entire function, 62 equicontinuous, 449 equivalent norms, 18 equivalent seminorms, 92 equivalent valuations, 6 ε-dense, 91 ε-topology, 335 extreme point, 206 extreme set, 206 fenced disk, 63 field extension, 444 field homomorphisms, 444 filter, 442 finite rank, 445 finite type, 182, 218 Fr´echet space, 108 free simplex, 207 Full Perturbation Lemma, 33 functionals, 445 fundamental sequence fundamental sequence of bounded sets, 295 fundamental sequence of compact sets, 116 fundamental sequence of complete bounded subsets, 126 fundamental sequence of equicontinuous sets, 295 gauge, 85 germs of analytic functions, 125 Goldstine Theorem, 276 graph, 443 Grothendieck, 413 Grothendieck–K¨othe counterexample, 432 Hahn–Banach property, 199 Hahn–Banach Theorem, 171, 173, 176, 178, 193, 205, 217 Hasse derivatives, 131 Hausdorff, 446 hemicompact, 116 Hilbert-like space, 40 homeomorphic, 447 homeomorphism, 447 homomorphism, 444 hyperplane, 445

470

ideal, 444 immediate valued field extension, 11 indiscrete topology, 446 inductive limit, 107 inductive sequence, 107 inductive system, 107 inductive topology, 121 inherited topology, 446 inner product, 38 interior, 446 invariant metric, 108, 449 invariant subspace, 445 inverse, 444 isolated, 446 isometrical isomorphism, 448 isometrically embedded, 448 isometrically isomorphic, 17, 448 isometry, 448 isomorphic, 97, 444, 445 isomorphism, 444, 445 isosceles triangle principle, 3 K-algebra, 446 k0 -space, 117 K¨othe dual, 350 K¨othe sequence space associated to a matrix, 356 K¨othe space, 349 Krein–Milman compactoids, 206 Krein–Milman Theorem, 206 Kronecker delta, 446 Krull valuation, 13 Lagrange polynomials, 187 (LF)-space, 435 limit, 447 linear hull, 445 linear manifold, 445 linearly homeomorphic, 17 Liouville Theorem, 64 Lipschitz, 448 Lipschitz constant, 448 Lipschitz norm, 16 (LM)-space, 435 local convergence, 305 locally compact, 448 locally constant, 47 locally convex direct sum, 101 locally convex direct sum topology, 101 locally convex space, 93 locally convex topology, 93

Index

Mackey Convergence Condition, 426 Mackey space, 235 Mackey topology, 235 Mahler base, 79 maximal convex filter, 246 maximal ideal, 444 maximal valued field, 11 measure, 51 measures, 263 mercury drop behaviour, 3 metric, 1 metric space, 1 metrizable, 447 metrizable locally convex space, 108 Minkowski function, 85 Montel, 312 multiplicative group of a field, 7 N-compact, 448 n-th derivative, 70 n-th difference quotient, 71 n-th Hasse derivative, 72 natural map, 8 neighbourhood, 446 neighbourhood base, 446 nested collection, 442 net, 442 non-Archimedean valuation, 6 non-Archimedean valued field, 6 non-measurable cardinality, 443 norm, 15 normable, 96 normal topology, 351 normed direct sum, 21 normed product, 21 normed space (over K), 15 nuclear, 312, 320 null sequence, 449 of countable type, 175 of finite type, 218 (O.P.)-space, 240 open, 446 open ball, 1 open map, 447 Open Mapping Theorem, 21, 111, 308 open unit ball, 6, 15 open unit disk, 6 operator, 302 Orlicz–Pettis space, 240 orthocomplemented, 24 orthogonal, 23

Index

orthogonal base, 29 “orthogonal” base, 338 orthogonal complement, 24 orthogonal indexed system, 25 orthogonal projection, 24 orthogonal sequence, 25 “orthogonal” sequence, 162 orthogonal set, 24 orthogonal system, 24 “orthogonal” system, 26 orthogonal with respect to p, 162 orthonormal base, 29 orthonormal system, 24 orthoprojection, 24 overconvergent power series, 125 p-adic complex number, 10 p-adic Fourier Transform, 438 p-adic integer, 9 p-adic Laplace Transform, 438 p-adic metric, 2 p-adic number, 9 p-adic valuation, 8 p-open, 92 p-topology, 92 partition, 442 perfect sequence space, 350 Perturbation Lemma, 25, 33, 42 ϕ-null set, 263 π -topology, 335 pointwise bounded, 256, 449 pointwise convergence, 114, 449 polar inductive limit, 259, 422, 438 polar inductive sequence, 438 polar seminorm, 193 polar set, 196 polar space, 193 polar topology, 193 polar topology associated to τ , 208 polarly barrelled, 256 power series, 62 precompact, 143, 449 prime field, 444 principal, 442 Principle of van Rooij, 23 Product Lemma, 305 product of locally convex spaces, 99 product topology, 448 projection, 24, 105, 445 projective limit, 106 projective sequence, 106 projective system, 106

471

projective tensor product, 384 property (∗), 240 pseudocompact, 267 Pull Back Principle, 155, 156, 323 quasicomplete, 113 quotient, 448 quotient map, 445, 448 quotient norm, 17 quotient seminorm, 98 quotient space, 17, 445 quotient topology, 17, 98, 448 radius, 1 Radon measures, 437 rapidly decreasing at infinity, 143 rapidly decreasing sequences, 371 reflexive, 275 regular inductive limit, 411 regular inductive sequence, 411 relatively closed, 447 relatively compact, 448 relatively open, 447 residue class field, 7 resolution, 435 restriction topology, 446 Riesz Representation Theorem, 52 Riesz Theorem, 145 rigid analytic function, 126 ring, 442 saturated, 93 Schauder base, 31, 163 Schwartz space, 334 semi-metric, 2 semi-Montel, 311 semicompact inductive sequence, 439 semicompact operator, 334 seminorm, 83 seminorm of countable type, 175 seminorm of finite type, 218 semireflexive, 275 separable, 28, 447 separate the points, 443 separation of convex sets, 204, 240 sequence space, 125, 350 sequentially closed, 447 sequentially complete, 114, 449 sequentially continuous, 447 sequentially retractive inductive sequence, 441 sides of a hyperplane, 205 σ -compact, 448

472

Index

simple convergence, 114 small, 443 solid norm, 19 solid seminorm, 83 spherical completion, 11 spherically complete, 4 splitting lemma, 389, 392 step function, 47 steps, 107 ˘ Stone–Cech compactification, 55 strict inductive limit, 411 strict inductive sequence, 411 strict topology, 119 strictly differentiable, 70 strictly of countable type, 176 strong topology, 272 strong triangle inequality, 2, 6 stronger topology, 446 strongest locally convex topology, 103, 113, 195, 216, 220, 319 strongly polar, 201 subbase, 446 subring, 444 subspace, 447 sum, 16, 41 sum of subspaces, 445 summable, 16, 41, 449 support, 449 support of a measure, 263 t-frame, 166 t-frame sequence, 166 t-orthogonal, 27 t-orthogonal base, 29 t-orthogonal indexed system, 30 t-orthogonal sequence, 30 t-orthogonal set, 27 t-orthogonal with respect to p, 162 t-orthogonal system, 27 Taylor formula, 72 tensor product, 374 tensor product of seminorms, 378 Tietze–Urysohn Extension Lemma, 49, 222 topological base, 31 topological field, 5 topological space, 446 topological vector space, 15, 91 topology, 446 topology of countable type, 175 topology of countable type associated to τ , 207 topology of finite type, 218 topology of finite type associated to τ , 209

totally disconnected, 447 translate, 446 translation, 445 transseparable, 175 trivial metric, 3 trivial valuation, 7 twist map, 375 Tychonov Theorem, 448 ultrafilter, 442 ultrametric, 2 ultrametric space, 2 ultrametrizable, 447 unconditional summation, 16 Uniform Boundedness Principle, 22, 111, 256 uniform convergence, 449 uniform convergence on bounded sets, 272, 390 uniform convergence on compact sets, 115, 396 uniform convergence on compact sets of difference quotients, 128 uniform convergence on compactoids, 315 uniformizing element, 7 uniformly continuous, 448, 449 union, 446 unit disk, 6 unit sphere, 6 unit vectors, 29 upper half plane, 127 valuation, 5 value group, 7, 13 valued field, 5 valued field extension, 7, 56 vanish at infinity, 48, 119 vanishing at infinity, 139 Volume Function, 165 W -topology, 118, 133 weak Hahn–Banach property, 199, 212 weak star topology, 272 weak topology, 182, 211 weaker topology, 446 weakly compact, 232 weakly convergent sequence, 228 weakly precompact, 232 weakly strict, 438 Weierstrass Theorem, 11, 190 weight functions, 118 zero-dimensional, 447