Fundamentals of functional analysis

Contents

Preface to the English Translation Preface to the First Russian Edition Chapter 1. An Excursus into Set Theory § 1.1. Correspondences

1 1

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§ 1.3. Filters

Chapter 2. Vector Spaces § 2.1. Spaces and Subspaces § 2.2. Linear Operators

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§ 2.3. Equations in Operators Exercises

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§ 1.2. Ordered Sets

Exercises

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Contents

Chapter 3. Convex Analysis § 3.1. Sets in Vector Spaces § 3.2. Ordered Vector Spaces

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§ 3.3. Extension of Positive Functionals and Operators

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§ 3.5. The Hahn–Banach Theorem

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§ 3.6. The Kre–ın–Milman Theorem

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§ 3.4. Convex Functions and Sublinear Functionals

§ 3.7. The Balanced Hahn–Banach Theorem

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§ 3.8. The Minkowski Functional and Separation Exercises

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Chapter 4. An Excursus into Metric Spaces § 4.1. The Uniformity and Topology of a Metric Space

§ 4.3. Semicontinuity § 4.4. Compactness

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§ 4.5. Completeness Completeness

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§ 4.6. Compactness and Completeness § 4.7. Baire Spaces

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§ 4.2. Continuity and Uniform Continuity

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§ 4.8. The Jordan Curve Theorem and Rough Drafts Exercises

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Chapter 5. Multinormed and Banach Spaces § 5.1. Seminorms and Multinorms

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§ 5.2. The Uniformity and Topology of a Multinormed Space

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§ 5.3. Comparison Between Topologies

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§ 5.4. Metrizable and Normable Spaces

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§ 5.5. Banach Spaces

§ 5.6. The Algebra of Bounded Operators Exercises

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Contents

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Chapter 6. Hilbert Spaces

85

§ 6.1. Hermitian Forms and Inner Products § 6.2. Orthoprojections § 6.3. A Hilbert Basis

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§ 6.4. The Adjoint of an Operator § 6.5. Hermitian Operators

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§ 6.6. Compact Hermitian Operators Exercises

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Chapter 7. Principles of Banach Spaces § 7.1. Banach’s Fundamental Principle § 7.2. Boundedness Principles

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§ 7.3. The Ideal Correspondence Principle

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§ 7.4. Open Mapping and Closed Graph Theorems

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§ 7.5. The Automatic Continuity Principle § 7.6. Prime Principles Exercises

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Chapter 8. Operators in Banach Spaces

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§ 8.1. Holomorphic Functions and Contour Integrals

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§ 8.3. The Approximation Property

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§ 8.4. The Riesz–Schauder Theory

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§ 8.2. The Holomorphic Functional Calculus

§ 8.5. Fredholm Operators Exercises

Chapter 9. An Excursus into General Topology § 9.1. Pretopologies and Topologies § 9.2. Continuity

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§ 9.3. Types of Topological Spaces § 9.4. Compactness

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§ 9.5. Uniform and Multimetric Spaces

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§ 9.6. Covers, and Partitions of Unity

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Exercises

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Contents

Chapter 10. Duality and Its Applications § 10.1. Vector Topologies

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§ 10.2. Locally Convex Topologies

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§ 10.3. Duality Between Vector Spaces

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§ 10.4. Topologies Compatible with Duality § 10.5. Polars

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§ 10.6. Weakly Compact Convex Sets § 10.8. Reflexive Spaces

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§ 10.8. The Space C(Q, R) § 10.9. Radon Measures

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§ 10.10. The Spaces D(Š) and D 0 (Š)

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§ 10.11. The Fourier Transform of a Distribution Exercises

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Chapter 11. Banach Algebras

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§ 11.1. The Canonical Operator Representation

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§ 11.2. The Spectrum of an Element of an Algebra

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§ 11.3. The Holomorphic Functional Calculus in Algebras

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§ 11.4. Ideals of Commutative Algebras § 11.5. Ideals of the Algebra C(Q, C) § 11.6. The Gelfand Transform

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§ 11.7. The Spectrum of an Element of a C ∗ -Algebra

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§ 11.8. The Commutative Gelfand–Na–ımark Theorem

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§ 11.9. Operator ∗-Representations of a C ∗ -Algebra Exercises References

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Notation Index

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Subject Index

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Preface to the English Translation

This is a concise guide to basic sections of modern functional analysis. Included are such topics as the principles of Banach and Hilbert spaces, the theory of multinormed and uniform spaces, the Riesz–Dunford holomorphic functional calculus, the Fredholm index theory, convex analysis and duality theory for locally convex spaces. With standard provisos the presentation is self-sufficient, whereas exposing about a hundred of celebrated “named” theorems furnished with complete proofs and culminating in the Gelfand–Na˘ımark–Segal construction for C ∗ -algebras. The first Russian edition was printed by the Siberian Division of the “Nauka” Publishers in 1983. Since then the monograph has served as the standard textbook on functional analysis at the University of Novosibirsk. This volume is translated from the second Russian edition printed by the Sobolev Institute of Mathematics of the Siberian Division of the Russian Academy of Sciences in 1995. It incorporates new sections on Radon measures, the Schwartz spaces of distributions, and a supplementary list of theoretical exercises and problems for drill. This edition was typeset using AMS-TEX, the American Mathematical Society’s TEX system. To clear my conscience completely, I also confess that := stands for the definor, the assignment operator, / marks the beginning of a (possibly empty) proof, and . signifies the end of the proof. S. Kutateladze

Preface to the First Russian Edition

As the title implies, this book treats of functional analysis. At the turn of the century the term “functional analysis” was coined by J. Hadamard who is famous among mathematicians for the formula of the radius of convergence of a power series. The term “functional analysis” was universally accepted then as related to the calculus of variations, standing for a new direction of analysis which was intensively developed by V. Volterra, C. Arzel` a, S. Pincherle, P. Levy, and other representatives of the French and Italian mathematical schools. J. Hadamard’s contribution to the recent discipline should not be reduced to the invention of the word “functional” (or more precisely to the transformation of the adjective into a proper noun). J. Hadamard was fully aware of the relevance of the rising subject. Working hard, he constantly advertised problems, ideas, and methods just evolved. In particular, to one of his students, M. Fr´echet, he suggested the problem of inventing something that is now generally acclaimed as the theory of metric spaces. In this connection it is worthy to indicate that neighborhoods pertinent to functional analysis in the sense of Hadamard and Volterra served as precursors to Hausdorff’s famous research heralded the birth of general topology. For the sequel it is essential to emphasize that one of the most attractive, difficult, and important sections of classical analysis, the calculus of variations, became the first source of functional analysis. The second source of functional analysis was provided by the study directed to creating some algebraic theory for functional equations or, stated strictly, to simplifying and formalizing the manipulations of “equations in functions” and, in particular, linear integral equations. Ascending to H. Abel and J. Liouville, the theory of the latter was considerably expanded by works of I. Fredholm, K. Neumann, F. Noether, A. Poincar´e, et al. The efforts of these mathematicians fertilized soil for D. Hilbert’s celebrated research into quadratic forms in infinitely many variables. His ideas, developed further by F. Riesz, E. Schmidt, et al., were the immediate predecessors of the axiomatic presentation of Hilbert space theory which was un-

Preface

xi

dertaken and implemented by J. von Neumann and M. Stone. The resulting section of mathematics has vigorously influenced theoretical physics, first of all, quantum mechanics. In this regard it is instructive as well as entertaining to mention that both terms, “quantum” and “functional,” originated in the same year, 1900. The third major source of functional analysis encompassed Minkowski’s geometric ideas. His invention, the apparatus for the finite-dimensional geometry of convex bodies, prepared the bulk of spatial notions ensuring the modern development of analysis. Elaborated by E. Helly, H. Hahn, C. Carath´eodory, I. Radon, et al., the idea of convexity has eventually shaped the fundamentals of the theory of locally convex spaces. In turn, the latter has facilitated the spread of distributions and weak derivatives which were recognized by S. L. Sobolev as drastically changing all tools of mathematical physics. In the postwar years the geometric notion of convexity has conquered a new sphere of application for mathematics, viz., social sciences and especially economics. An exceptional role in this process was performed by linear programming discovered by L. V. Kantorovich. The above synopsis of the strands of functional analysis is schematic, incomplete, and approximate (for instance, it casts aside the line of D. Bernoulli’s superposition principle, the line of set functions and integration theory, the line of operational calculus, the line of finite differences and fractional derivation, the line of general analysis, and many others). These demerits notwithstanding, the three sources listed above reflect the main, and most principal, regularity: functional analysis has synthesized and promoted ideas, concepts, and methods from the classical sections of mathematics: algebra, geometry, and analysis. Therefore, although functional analysis verbatim means analysis of functions and functionals, even a superficial glance at its history gives grounds to claim that functional analysis is algebra, geometry, and analysis of functions and functionals. A more viable and penetrating explanation for the notion of functional analysis is given by the Soviet Encyclopedic Dictionary: “Functional analysis is one of the principal branches of modern mathematics. It resulted from mutual interaction, unification, and generalization of the ideas and methods stemming from all parts of classical mathematical analysis. It is characterized by the use of concepts pertaining to various abstract spaces such as vector spaces, Hilbert spaces, etc. It finds diverse applications in modern physics, especially in quantum mechanics.” The S. Banach treatise The´orie des Operationes Lin´eares, printed half a century ago, inaugurated functional analysis as an essential activity in mathematics. Its influence on the development of mathematics is seminal: Ubiquitous and omnipresent, Banach’s ideas, propounded in the book, captivate the realm of modern mathematics. Outstanding contribution toward progress in functional analysis belongs to the renowned Soviet scientists: I. M. Gelfand, L. V. Kantorovich, M. V. Keldysh,

xii

Preface

A. N. Kolmogorov, M. G. Kre˘ın, L. A. Lyusternik, and S. L. Sobolev. The characteristic feature of the Soviet school is that its research on functional analysis is always conducted in connection with profound applied problems. The research has expanded the scope of functional analysis which becomes the prevailing language of the applications of mathematics. The next fact is demonstrative: In 1948 considered as paradoxical was even the title of Kantorovich’s insightful article Functional Analysis and Applied Mathematics which provided a basis for numerical mathematics of today. Whereas in 1974 S. L. Sobolev stated that “to conceive the theory of calculations without Banach spaces is as impossible as without electronic computers.” The exponential accumulation of knowledge within functional analysis is now observed alongside a sharp rise in demand for the tools and concepts of the discipline. A thus resulting conspicuous gap widens permanently between the state-of the art of analysis as such and as reflected in the literature accessible to the reading community. To alter this ominous trend is the purpose of the present book.

Preface to the Second Russian Edition More than a decade the monograph serves as a reference book for the compulsory and optional courses in functional analysis which are delivered at Novosibirsk State University. The span of time proves that the principles of compiling the book are legitimate. The present edition is enlarged with sections addressing fundamentals of distribution theory. Theoretical exercises are supplemented and the list of references is updated. Also, the inaccuracies are improved that were mostly indicated by my colleagues. I seize the opportunity to express my gratitude to all those who helped me in preparation of the book. My pleasant debt is to acknowledge the financial support of the Sobolev Institute of Mathematics of the Siberian Division of the Russian Academy of Sciences, the Russian Foundation for Basic Research, the International Science Foundation and the American Mathematical Society during the compilation of the second edition. S. Kutateladze March, 1995

Chapter 1 An Excursus into Set Theory

1.1. Correspondences 1.1.1. Definition. Let A and B be sets and let F be a subset of the product A × B := {(a, b) : a ∈ A, b ∈ B}. Then F is a correspondence with the set of departure A and the set of arrival B or just a correspondence from A (in)to B. 1.1.2. Definition. For a correspondence F ⊂ A × B the set dom F := D(F ) := {a ∈ A : (∃ b ∈ B) (a, b) ∈ F } is the domain (of definition) of F and the set im F := R (F ) := {b ∈ B : (∃ a ∈ A) (a, b) ∈ F } is the codomain of F , or the range of F , or the image of F . 1.1.3. Examples. (1) If F is a correspondence from A into B then F −1 := {(b, a) ∈ B × A : (a, b) ∈ F } is a correspondence from B into A which is called inverse to F or the inverse of F . It is obvious that F is the inverse of F −1 . (2) A relation F on A is by definition a subset of A2 , i.e. a correspondence from A to A (in words: “F acts in A”). (3) Let F ⊂ A × B. Then F is a single-valued correspondence if for all a ∈ A the containments (a, b1 ) ∈ F and (a, b2 ) ∈ F imply b1 = b2 . In particular, if U ⊂ A and IU := {(a, a) ∈ A2 : a ∈ U }, then IU is a single-valued correspondence acting in A and called the identity relation (over U on A). A single-valued

2

Chapter 1

correspondence F ⊂ A × B with dom F = A is a mapping of A into B or a mapping from A (in)to B. The terms “function” and “map” are also in current usage. A mapping F ⊂ A × B is denoted by F : A → B. Observe that here dom F always coincides with A whereas im F may differ from B. The identity relation IU on A is a mapping if and only if A = U and in this case IU is called the identity mapping in U or the diagonal of U 2 . The set IU may be treated as a subset of U × A. The resulting mapping is usually denoted by ι : U 0 = 0 O and is called the identical embedding of U into A. It is said that “F is a correspondence from A onto B” if im F = B. Finally, a correspondence F ⊂ X × Y is one-to-one whenever the correspondence F −1 ⊂ B × A is single-valued. (4) Occasionally the term “family” is used instead of “mapping.” Namely, a mapping F : A0 = 0 O is a family in B (indexed in A by F ), also denoted by (ba )a∈A or a 7→ ba (a ∈ A) or even (ba ). Specifically, (a, b) ∈ F if and only if b = ba . In the sequel, a subset U of A is often treated as indexed in itself by the identical embedding of U into A. It is worth recalling that in set theory a is an element or a member of A whenever a ∈ A. In this connection a family in B is also called a family of elements of B or a family of members of B. By way of expressiveness a family or a set of numbers is often addressed as numeric. Also, common abusage practices the identification of a family and its range. This sin is very enticing. (5) Let F ⊂ A × B be a correspondence and U ⊂ A. The restriction of F to U , denoted by F |U , is the set F ∩ (U × B) ⊂ U × B. The set F (U ) := im F |U is the image of U under F . If a and b are elements of A and B then F (a) = b is usually written instead of F ({a}) = {b}. Often the parentheses in the symbol F (a) are omitted or replaced with other symbols. For a subset U of B the image F −1 (U ) of U under F −1 is the inverse image of U or the preimage of U under F . So, inverse images are just images of inverses. (6) Given a correspondence F ⊂ A × B, assume that A is the product of A1 and A2 , i.e. A = A1 × A2 . Fixing a1 in A1 and a2 in A2 , consider the sets F (a1 , · ) := {(a2 , b) ∈ A2 × B : ((a1 , a2 ), b) ∈ F }; F ( · , a2 ) := {(a1 , b) ∈ A1 × B : ((a1 , a2 ), b) ∈ F }. These are the partial correspondences of F . In this regard F itself is often symbolized as F ( · , · ) and referred to as a correspondence in two arguments. This beneficial agreement is effective in similar events. 1.1.4. Definition. The composite correspondence or composition of correspondences F ⊂ A × B and G ⊂ C × D is the set G ◦ F := {(a, d) ∈ A × D : (∃ b) (a, b) ∈ F & (b, d) ∈ G}.

An Excursus into Set Theory

3

The correspondence G ◦ F is considered as acting from A into D. 1.1.5. Remark. The scope of the concept of composition does not diminish if it is assumed in 1.1.4 from the very beginning that B = C. 1.1.6. Let F be a correspondence. Then F ◦ F −1 ⊃ I imF . Moreover, the equality F ◦ F −1 = I imF holds if and only if F |dom F is a mapping. CB 1.1.7. Let F ⊂ A × B and G ⊂ B × C. Further assume U ⊂ A. Then the correspondence G ◦ F ⊂ A × C satisfies the equality G ◦ F (U ) = G(F (U )). CB 1.1.8. Let F ⊂ A×B, G ⊂ B ×C, and H ⊂ C ×D. Then the correspondences H ◦ (G ◦ F ) ⊂ A × D and (H ◦ G) ◦ F ⊂ A × D coincide. CB 1.1.9. Remark. By virtue of 1.1.8, the symbol H ◦ G ◦ F and the like are defined soundly. 1.1.10. Let F, G, and H be correspondences. Then H ◦G◦F =

[

F −1 (b) × H(c).

(b,c)∈G

C (a, d) ∈ H ◦ G ◦ F ⇔ (∃ (b, c) ∈ G) (c, d) ∈ H & (a, b) ∈ F ⇔ (∃ (b, c) ∈ G) a ∈ F −1 (b) & d ∈ H(c) B 1.1.11. Remark. The claim of 1.1.10, together with the calculation intended as its proof, is blatantly illegitimate from a formalistic point of view as based on ambiguous or imprecise information (in particular, on Definition 1.1.1!). Experience justifies treating such criticism as petty. In the sequel, analogous convenient (and, in fact, inevitable) violations of formal purity are mercilessly exercised with no circumlocution. 1.1.12. Let G and F be correspondences. Then G◦F =

[

F −1 (b) × G(b).

b∈ imF

C Insert H := G, G := I imF and F := F into 1.1.10. B 1.2. Ordered Sets 1.2.1. Definition. Let σ be a relation on a set X, i.e. σ ⊂ X 2 . Reflexivity for σ means the inclusion σ ⊃ IX ; transitivity, the inclusion σ◦σ ⊂ σ; antisymmetry, the inclusion σ ∩ σ −1 ⊂ IX ; and, finally, symmetry, the equality σ = σ −1 .

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Chapter 1

1.2.2. Definition. A preorder is a reflexive and transitive relation. A symmetric preorder is an equivalence. An order (partial order, ordering, etc.) is an antisymmetric preorder. For a set X, the pair (X, σ), with σ an order on X, is an ordered set or rarely a poset. The notation x ≤σ y is used instead of y ∈ σ(x). The terminology and notation are often simplified and even abused in common parlance: The underlying set X itself is called an ordered set, a partially ordered set, or even a poset. It is said that “x is less than y,” or “y is greater than x,” or “x ≤ y,” or “y ≥ x,” or “x is in relation σ to y,” or “x and y belong to σ,” etc. Analogous agreements apply customarily to a preordered set, i.e. to a set furnished with a preorder. The convention is very propitious and extends often to an arbitrary relation. However, an equivalence is usually denoted by the signs like ∼. 1.2.3. Examples. (1) The identity relation; each subset X0 of a set X bearing a relation σ is endowed with the (induced ) relation σ0 := σ ∩ X0 × X0 . (2) If σ is an order (preorder) in X, then σ −1 is also an order (preorder) which is called reverse to σ. (3) Let f : X0 = 0 O be a mapping and let τ be a relation on Y . Consider the relation f −1 ◦ τ ◦ f appearing on X. By 1.1.10, [ f −1 ◦ τ ◦ f = f −1 (y1 ) × f −1 (y2 ). (y1 ,y2 )∈τ

Hence it follows that (x1 , x2 ) ∈ f −1 ◦ τ ◦ f ⇔ (f (x1 ), f (x2 )) ∈ τ. Thus, if τ is a preorder then f −1 ◦ τ ◦ f itself is a preorder called the preimage or inverse image of τ under f . It is clear that the inverse image of an equivalence is also an equivalence. Whereas the preimage of an order is not always antisymmetric. In particular, this relates to the equivalence f −1 ◦ f (= f −1 ◦ IY ◦ f ). (4) Let X be an arbitrary set and let ω be an equivalence on X. Define a mapping ϕ : X0 = 0 O P(X) by ϕ(x) := ω(x). Recall that P(X) stands for the powerset of X comprising all subsets of X and also denoted by 2X . Let X := X/ω := im ϕ be the quotient set or factor set of X by ω or modulo ω. A member of X/ω is usually referred to as a coset or equivalence class. The mapping ϕ is the coset mapping (canonical projection, quotient mapping, etc.). Note that ϕ is treated as acting onto X. Observe that [ ω = ϕ−1 ◦ ϕ = ϕ−1 (x) × ϕ−1 (x). x∈X

Now let f : X0 = 0 O be a mapping. Then f admits factorization through X; i.e., there is a mapping f : X0 = 0 O (called a quotient of f by ω) such that f ◦ ϕ = f if and only if ω ⊂ f −1 ◦ f . CB

An Excursus into Set Theory

5

(5) Let (X, σ) and (Y, τ ) be two preordered sets. A mapping f : X0 = 0 O is increasing or isotone (i.e., x ≤σ y ⇒ f (x) ≤τ f (y)) whenever σ ⊂ f −1 ◦ τ ◦ f . That f decreases or is antitone means σ ⊂ f −1 ◦ τ −1 ◦ f . CB 1.2.4. Definition. Let (X, σ) be an ordered set and let U ⊂ X. An element x of X is an upper bound of U (write x ≥ U ) if U ⊂ σ −1 (x). In particular, x ≥ ∅. An element x of X is a lower bound of U (write x ≤ U ) if x is an upper bound of U in the reverse order σ −1 . In particular, x ≤ ∅. 1.2.5. Remark. Throughout this book liberties are taken with introducing concepts which arise from those stated by reversal, i.e. by transition from a (pre)order to the reverse (pre)order. Note also that the definitions of upper and lower bounds make sense in a preordered set. 1.2.6. Definition. An element x of U is greatest or last if x ≥ U and x ∈ U . When existent, such element is unique and is thus often referred to as the greatest element of U . A least or first element is defined by reversal. 1.2.7. Let U be a subset of an ordered set (X, σ) and let πσ (U ) be the collection of all upper bounds of U . Suppose that a member x of X is the greatest element of U . Then, first, x is the least element of πσ (U ); second, σ(x) ∩ U = {x}. CB 1.2.8. Remark. The claim of 1.2.7 gives rise to two generalizations of the concept of greatest element. 1.2.9. Definition. Let X be a (preordered) set and let U ⊂ X. A supremum of U in X is a least upper bound of U , i.e. a least element of the set of all upper bounds of U . This element is denoted by supX U or in short sup U . Certainly, in a poset an existing supremum of U is unique and so it is in fact the supremum of U . An infimum, a greatest lower bound inf U or inf X U is defined by reversal. 1.2.10. Definition. Let U be a subset of an ordered set (X, σ). A member x of X is a maximal element of U if σ(x) ∩ U = {x}. A minimal element is again defined by reversal. 1.2.11. Remark. It is important to make clear the common properties and distinctions of the concepts of greatest element, maximal element, and supremum. In particular, it is worth demonstrating that a “typical” set has no greatest element while possibly possessing a maximal element. 1.2.12. Definition. A lattice is an ordered set with the following property: each pair (x1 , x2 ) of elements of X has a least upper bound, x1 ∨x2 := sup{x1 , x2 }, the join of x1 and x2 , and a greatest lower bound, x1 ∧ x2 := inf{x1 , x2 }, the meet of x1 and x2 .

6

Chapter 1

1.2.13. Definition. A lattice X is complete if each subset of X has a supremum and an infimum in X. 1.2.14. An ordered set X is a complete lattice if and only if each subset of X has a least upper bound. CB 1.2.15. Definition. An ordered set (X, σ) is filtered upward provided that X = σ −1 ◦ σ. A downward-filtered set is defined by reversal. A nonempty poset is a directed set or simply a direction if it is filtered upward. 2

1.2.16. Definition. Let X be a set. A net or a (generalized) sequence in X is a mapping of a direction into X. A mapping of the set N of natural numbers, N = {1, 2, 3 . . . }, furnished with the conventional order, is a (countable) sequence. 1.2.17. A lattice X is complete if and only if each upward-filtered subset of X has a least upper bound. CB 1.2.18. Remark. The claim of 1.2.17 implies that for calculating a supremum of each subset it suffices to find suprema of pairs and increasing nets. 1.2.19. Definition. Let (X, σ) be an ordered set. It is said that X is ordered linearly whenever X 2 = σ ∪ σ −1 . A nonempty linearly-ordered subset of X is a chain in X. A nonempty ordered set is called inductive whenever its every chain is bounded above (i.e., has an upper bound). 1.2.20. Kuratowski–Zorn Lemma. Each inductive set contains a maximal element. 1.2.21. Remark. The Kuratowski–Zorn Lemma is equivalent to the axiom of choice which is accepted in set theory. 1.3. Filters 1.3.1. Definition. Let X be a set and let B, a nonempty subset of P(X), consist of nonempty elements. Such B is said to be a filterbase (on X) if B is filtered downward. Recall that P(X) is ordered by inclusion. It means that a greater subset includes a smaller subset by definition; this order is always presumed in P(X). 1.3.2. A subset B of P(X) is a filterbase if and only if 1 B 6= ∅ and ∅ 6∈ B; (2) B1 , B2 ∈ B ⇒ (∃ B ∈ B) B ⊂ B1 ∩ B2 . 1.3.3. Definition. A subset F of P(X) is a filter (on X) if there is a filterbase B such that F is the set of all supersets of B; i.e., F = fil B := {C ∈ P(X) : (∃ B ∈ B) B ⊂ C}. In this case B is said to be a base for the filter F (so each filterbase is a filter base).

An Excursus into Set Theory

7

1.3.4. A subset F in P(X) is a filter if and only if (1) F 6= ∅ and ∅ 6∈ F ; (2) (A ∈ F & A ⊂ B ⊂ X) ⇒ B ∈ F ; (3) A1 , A2 ∈ F ⇒ A1 ∩ A2 ∈ F . CB 1.3.5. Examples. (1) Let F ⊂ X × Y be a correspondence and let B be a downwardfiltered subset of P(X). Put F (B) := {F (B) : B ∈ B}. It is easy to see that F (B) is filtered downward. The notation is alleviated by putting F (B) := fil F (B). If F is a filter on X and B ∩ dom F 6= ∅ for all B ∈ F , then F (F ) is a filter (on Y ) called the image of F under F . In particular, if F is a mapping then the image of a filter on X is a filter on Y . (2) Let (X, σ) be a direction. Clearly, B := {σ(x) : x ∈ X} is a filterbase. For a net F : X0 = 0 O , the filter fil F (B) is the tail filter of F . Let (X, σ) and F : X → Y be another direction and another net in Y . If the tail filter of F includes the tail filter of F then F is a subnet (in a broad sense) of the net F . If there is a subnet (in a broad sense) G : X → X of the identity net ((x)x∈X in the direction (X, σ)) such that F = F ◦ G, then F is a subnet of F (sometimes F is addressed as a Moore subnet or a strict subnet of F ). Every subnet is a subnet in a broad sense. It is customary to speak of a net having or lacking a subnet with some property. 1.3.6. Definition. Let F (X) be the collection of all filters on X. Take F1 , F2 ∈ F (X). Say that F1 is finer than F2 or F2 refines F1 (in other words, F1 is coarser than F2 or F1 coarsens F2 ) whenever F1 ⊃ F2 . 1.3.7. The set F (X) with the relation “to be finer” is a poset. CB 1.3.8. Let N be a direction in F (X). Then N has a supremum F0 := sup N . Moreover, F0 = ∪{F : F ∈ N }. C To prove this, it is necessary to show that F0 is a filter. Since N is not empty it is clear that F0 6= ∅ and ∅ ∈ / F0 . If A ∈ F0 and B ⊃ A then, choosing F in N for which A ∈ F , conclude that B ∈ F ⊂ F0 . Given A1 , A2 ∈ F0 , find an element F of N satisfying A1 , A2 ∈ F , which is possible because N is a direction. By 1.3.4, A1 ∩ A2 ∈ F ⊂ F0 . B 1.3.9. Definition. An ultrafilter is a maximal element of the ordered set F (X) of all filters on X. 1.3.10. Each filter is coarser than an ultrafilter. C By 1.3.8, the set of filters finer than a given filter is inductive. Recalling the Kuratowski–Zorn Lemma completes the proof. B

8

Chapter 1

1.3.11. A filter F is an ultrafilter if and only if for all A ⊂ X either A ∈ F or X \ A ∈ F . C ⇒: Suppose that A 6∈ F and B := X \ A 6∈ F . Note that A 6= ∅ and B 6= ∅. Put F1 := {C ∈ P(X) : A ∪ C ∈ F }. Then A 6∈ F ⇒ ∅ 6∈ F1 and B ∈ F1 ⇒ F1 6= ∅. The checking of 1.3.4 (2) and 1.3.4 (3) is similar. Hence, F1 is an ultrafilter. By definition, F1 ⊃ F . In addition, F is an ultrafilter and so F1 = F . Observe that B 6∈ F and B ∈ F , a contradiction. ⇐: Take F1 ∈ F (X) and let F1 ⊃ F . If A ∈ F1 and A 6∈ F then X \ A ∈ F by hypothesis. Hence, X \A ∈ F1 ; i.e., ∅ = A∩(X \A) ∈ F1 , which is impossible. B 1.3.12. If f is a mapping from X into Y and F is an ultrafilter on X then f (F ) is an ultrafilter on Y . CB 1.3.13. Let X := XF0 := {F ∈ F (X) : F ⊂ F0 } for F0 ∈ F (X). Then X is a complete lattice. C It is obvious that F0 is the greatest element of X and {X} is the least element of X . Therefore, the empty set has a supremum and an infimum in X : in fact, sup ∅ = inf X = {X} and inf ∅ = sup X = F0 . By 1.2.17 and 1.3.8, it suffices to show that the join F1 ∨ F2 is available for all F1 , F2 ∈ X . Consider F := {A1 ∩ A2 : A1 ∈ F1 , A2 ∈ F2 }. Clearly, F ⊂ F0 while F ⊃ F1 and F ⊃ F2 . Thus to verify the equality F = F1 ∨ F2 , it is necessary to establish that F is a filter. Plainly, F 6= ∅ and ∅ 6∈ F . It is also immediate that (B1 , B2 ∈ F ⇒ B1 ∩ B2 ∈ F ). Moreover, if C ⊃ A1 ∩ A2 where A1 ∈ F1 and A2 ∈ F2 , then C = {A1 ∩ A2 } ∪ C = (A1 ∪ C) ∩ (A2 ∪ C). Since A1 ∪ C ∈ F1 and A2 ∪ C ∈ F2 , conclude that C ∈ F . Appealing to 1.3.4, complete the proof. B Exercises 1.1. Give examples of sets and nonsets as well as set-theoretic properties and non-settheoretic properties. 1.2. Is it possible for the interval [0, 1] to be a member of the interval [0, 1]? For the interval [0, 2]? 1.3. Find compositions of the simplest correspondences and relations: squares, disks and circles with coincident or distinct centers in RM × RN for all feasible values of M and N . 1.4. Given correspondences R, S, and T , demonstrate that (R ∪ S)−1 = R−1 ∪ S −1 ; (R ∪ S) ◦ T = (R ◦ T ) ∪ (S ◦ T ); (R ∩ S) ◦ T ⊂ (R ◦ T ) ∩ (S ◦ T );

(R ∩ S)−1 = R−1 ∩ S −1 ; R ◦ (S ∪ T ) = (R ◦ S) ∪ (R ◦ T ); R ◦ (S ∩ T ) ⊂ (R ◦ S) ∩ (R ◦ T ).

1.5. Assume X ⊂ X × X. Prove that X = ∅. 1.6. Find conditions for the equations XA = B and AX = B to be solvable for X in correspondences or in functions.

An Excursus into Set Theory

9

1.7. Find the number of equivalences on a finite set. 1.8. Is the intersection of equivalences also an equivalence? And the union of equivalences? 1.9. Find conditions for commutativity of equivalences (with respect to composition). 1.10. How many orders and preorders are there on two-element and three-element sets? List all of them. What can you say about the number of preorders on a finite set? 1.11. Let F be an increasing idempotent mapping of a set X into itself. Assume that F dominates the identity mapping: F ≥ IX . Such an F is an abstract closure operator or, briefly, an (upper) envelope. Study fixed points of a closure operator. (Recall that an element x is a fixed point of F if F (x) = x.) 1.12. Let X and Y be ordered sets and M (X, Y ), the set of increasing mappings from X to Y with the natural order (specify the latter). Prove that (1) (M (X, Y ) is a lattice) ⇔ (Y is a lattice); (2) (M (X, Y ) is a complete lattice) ⇔ (Y is a complete lattice). 1.13. Given ordered sets X, Y , and Z, demonstrate that (1) M (X, Y × Z) is isomorphic with M (X, Y ) × M (Y, Z); (2) M (X × Y, Z) is isomorphic with M (X, M (Y, Z)). 1.14. How many filters are there on a finite set? 1.15. How do the least upper and greatest lower bounds of a set of filters look like? 1.16. Let f be a mapping from X onto Y . Prove that each ultrafilter on Y is the image of some ultrafilter on X under f . 1.17. Prove that an ultrafilter refining the intersection of two filters is finer than either of them. 1.18. Prove that each filter is the intersection of all ultrafilters finer than it. 1.19. Let A be an ultrafilter on N containing cofinite subsets (a cofinite subset is a subset with finite complement). Given x, y ∈ s := RN , put x ∼A y := (∃ A ∈ A ) x|A = y|A . Denote ∗R := RN /∼A . For t ∈ R the notation ∗t symbolizes the coset with the constant sequence t defined as t(n) := t (n ∈ N). Prove that ∗R \ {∗t : t ∈ R} 6= ∅. Furnish ∗R with algebraic and order structures. How are the properties of R and ∗R related to each other?

Chapter 2 Vector Spaces

2.1. Spaces and Subspaces 2.1.1. Remark. In algebra, in particular, modules over rings are studied. A module X over a ring A is defined by an abelian group (X, +) and a representation of the ring A in the endomorphism ring of X which is considered as left multiplication · : A × X0 = 0 O by elements of A. Moreover, a natural agreement is presumed between addition and multiplication. With this in mind, the following phrase is interpreted: “A module X over a ring A is described by the quadruple (X, A, +, · ).” Note also that A is referred to as the ground ring of X. 2.1.2. Definition. A basic field is the field R of real numbers or the field C of complex numbers. The symbol F stands for a basic field. Observe that R is treated as embedded into C in a standard (and well-known) fashion so that the operation Re of taking the real part of a number sends C onto the real axis, R. 2.1.3. Definition. Let F be a field. A module X over F is a vector space (over F). An element of the ground field F is a scalar in X and an element of X is a vector in X or a point in X. So, X is a vector space with scalar field F. The operation + : X × X0 = 0 O is addition in X and · : F × X0 = 0 O is scalar multiplication in X. We refer to X as a real vector space in case F = R and as a complex vector space, in case F = C. A more complete nomenclature consists of (X, F, +, · ), (X, R, +, · ), and (X, C, +, · ). Neglecting these subtleties, allow X to stand for every vector space associated with the underlying set X. 2.1.4. Examples. (1) A field F is a vector space over F. (2) Let (X, F, +, · ) be a vector space. Consider (X, F, +, ·∗ ), where ·∗ : (λ, x) 7→ λ∗ x for λ ∈ F and x ∈ X, the symbol λ∗ standing for the conventional complex conjugate of λ. The so-defined vector space is the twin of X denoted by X∗ . If F := R then the space X and the twin of X, the space X∗ , coincide.

Vector Spaces

11

(3) A vector space (X0 , F, +, · ) is called a subspace of a vector space (X, F, +, · ), if X0 is a subgroup of X and scalar multiplication in X0 is the restriction of that in X to F × X0 . Such a set X0 is a linear set in X, whereas X is referred to as an ambient space (for X0 ). It is convenient although not perfectly puristic to treat X0 itself as a subspace of X. Observe the important particularity of terminology: a linear set is a subset of a vector space (whose obsolete title is a linear space). To call a subset of whatever space X a set in X is a mathematical idiom of long standing. The same applies to calling a member of X a point in X. Furthermore, the neutral element of X, the zero vector of X, or simply zero of X, is considered as a subspace of X and is denoted by 0. Since 0 is not explicitly related to X, all vector spaces including the basic fields may seem to have a point in common, zero. (4) Take (Xξ )ξ∈„ , a family of vector spaces over F, and let X := Q X be the product of the underlying sets, i.e. the collection of mappings ξ∈„ ξ O x : „0 = 0 ξ∈„ Xξ such that xξ := x(ξ) ∈ Xξ as ξ ∈ „ (of course, here „ is not empty). Endow X with the coordinatewise or pointwise operations of addition and scalar multiplication: (x1 + x2 )(ξ) := x1 (ξ) + x2 (ξ) (x1 , x2 ∈ X , ξ ∈ „); (λ · x)(ξ) := λ · x(ξ) (x ∈ X , λ ∈ F, ξ ∈ „) (below, as a rule, we write λx and sometimes xλ rather than λ · x). The soconstructed vector space X over F is the product of (Xξ )ξ∈„ . If „ := {1, 2, . . . , N } then X1 × X2 × . . . × XN := X . In the case Xξ = X for all ξ ∈ „, the designation X „ := X is used. Given „ := {1, 2, . . . , N }, put X N := X . (5) LetP (Xξ )ξ∈„ be a family of vector spaces over F. Consider Q their direct sum X0 := ξ∈„ Xξ . By definition, X0 is the subset of X := ξ∈„ Xξ which comprises all x0 such that x0 („\„0 ) ⊂ 0 for a finite subset „0 ⊂ „ (routinely speaking, „0 is dependent on x0 ). It is easily seen that X0 is a linear set in X . The vector space associated with X0 presents a subspace of the product of (Xξ )ξ∈„ and is the direct sum of (Xξ )ξ∈„ . (6) Given a subspace (X, F, +, · ) of a vector space (X0 , F, +, · ), introduce ∼X0 := {(x1 , x2 ) ∈ X 2 : x1 − x2 ∈ X0 }. Then ∼X0 is an equivalence on X. Denote X := X/∼X0 and let ϕ : X0 = 0 O X be the coset mapping. Define operations on X by letting x1 + x2 := ϕ(ϕ−1 (x1 ) + ϕ−1 (x2 )) (x1 , x2 ∈ X ); λx := ϕ(λϕ−1 (x)) (x ∈ X , λ ∈ F).

12

Chapter 2

Here, as usual, for subsets S1 and S2 of X, a subset ƒ of F, and a scalar λ, a member of F, it is assumed that S1 + S2 := +{S1 × S2 }; ƒS1 := · (ƒ × S1 );

λS1 := {λ}S1 .

Thus X is furnished with the structure of a vector space. This space, denoted by X/X0 , is the quotient (space) of X by X0 or the factor space of X modulo X0 . 2.1.5. Let X be a vector space and let Lat(X) stand for the collection of all subspaces of X. Ordered by inclusion, Lat(X) presents a complete lattice. C It is clear that inf Lat(X) = 0 and sup Lat(X) = X. Further, the intersection of a nonempty set of subspaces is also a subspace. By 1.2.17, the proof is complete. B 2.1.6. Remark. With X1 , X2 ∈ Lat(X), the equality X1 ∨ X2 = X1 + X2 holds. It is evident that inf E = ∩{X0 : X0 ∈ E } for a nonempty subset E of Lat(X). Provided that E is filtered upward, sup E = ∪ {X0 : X0 ∈ E }. CB 2.1.7. Definition. Subspaces X1 and X2 of a vector space X split X into (algebraic) direct sum decomposition (in symbols, X = X1 ⊕ X2 ), if X1 ∧ X2 = 0 and X1 ∨ X2 = X. In this case X2 is an (algebraic) complement of X1 to X, and X1 is an (algebraic) complement of X2 to X. It is also said that X1 and X2 are (algebraically) complementary to one another. 2.1.8. Each subspace of a vector space has an algebraic complement. C Take a subspace X1 of X. Put E := {X0 ∈ Lat(X) : X0 ∧ X1 = 0}. Obviously, 0 ∈ E . Given a chain E i0 in E , from 2.1.6 infer that X1 ∧ sup E i0 = 0, i.e. sup E i0 ∈ E . Thus E is inductive and, by 1.2.20, E has a maximal element, say, X2 . If x ∈ X \ (X1 + X2 ) then (X2 + {λx : λ ∈ F}) ∧ X1 = 0. Indeed, if x2 + λx = x1 with x1 ∈ X1 , x2 ∈ X2 and λ ∈ F, then λx ∈ X1 + X2 and so λ = 0. Hence, x1 = x2 = 0 as X1 ∧ X2 = 0. Therefore, X2 + {λx : λ ∈ F} = X2 because X2 is maximal. It follows that x = 0. At the same time, it is clear that x 6= 0. Finally, X1 ∨ X2 = X1 + X2 = X. B

Vector Spaces

13 2.2. Linear Operators

2.2.1. Definition. Let X and Y be vector spaces over F. A correspondence T ⊂ X × Y is linear, if T is a linear set in X × Y . A linear operator on X (or simply an operator, with linearity apparent from the context) is a mapping of X and a linear correspondence simultaneously. If need be, we distinguish such T from a linear single-valued correspondence S with dom S 6= X and say that T is given on X or T is defined everywhere or even T is a total operator, whereas S is referred to as not-everywhere-defined or partially-defined operator or even a partial operator. In the case X = Y , a linear operator from X to Y is also called an operator in X or an endomorphism of X. 2.2.2. A correspondence T ⊂ X × Y is a linear operator from X to Y if and only if dom T = X and T (λ1 x1 + λ2 x2 ) = λ1 T x1 + λ2 T x2

(λ1 , λ2 ∈ F; x1 , x2 ∈ X). CB

2.2.3. The set L (X, Y ) of all linear operators carrying X into Y constitutes a vector space, a subspace of Y X . CB 2.2.4. Definition. A member of L (X, F) is a linear functional on X, and the space X # := L (X, F) is the (algebraic) dual of X. A linear functional on X∗ is a ∗-linear or conjugate-linear functional on X. If the nature of F needs specifying, then we speak of real linear functionals, complex duals, etc. Evidently, when F = R the term “∗-linear functional” is used rarely, if ever. 2.2.5. Definition. A linear operator T , a member of L (X, Y ), is an (algebraic) isomorphism (of X and Y , or between X and Y ) if the correspondence T −1 is a linear operator, a member of L (Y, X). 2.2.6. Definition. Vector spaces X and Y are (algebraically) isomorphic, in symbols X ' Y , provided that there is an isomorphism between X and Y . 2.2.7. Vector spaces X and Y are isomorphic if and only if there are operators T ∈ L (X, Y ) and S ∈ L (Y, X) such that S ◦ T = IX and T ◦ S = IY (in this event S = T −1 and T = S −1 ). CB 2.2.8. Remark. Given vector spaces X, Y , and Z, take T ∈ L (X, Y ) and S ∈ L (Y, Z). The correspondence S◦T is undoubtedly a member of L (X, Z). For simplicity every composite operator S ◦ T is denoted by juxtaposition ST . Observe also that the taking of composition (S, T ) 7→ ST is usually treated as the mapping ◦ : L (Y, Z) × L (X, Y )0 = 0 O L(X, Z). In particular, if E ⊂ L (Y, Z) and T ∈ L (X, Y ) then we let E ◦ T := ◦(E × {T }). One of the reasons behind the convention is that juxtaposition in the endomorphism space L (X) := L (X, X) of X which comprises all endomorphisms of X transforms L (X) into a ring (and even into an algebra, the endomorphism algebra of X, cf. 5.6.2).

14

Chapter 2 2.2.9. Examples. (1) If T is a linear correspondence then T −1 is also a linear correspon-

dence. (2) If X1 is a subspace of a vector space X and X2 is an algebraic complement of X1 then X2 is isomorphic with X/X1 . Indeed, if ϕ : X0 = 0 O /X1 is the coset mapping then its restriction to X2 , i.e. the operator x2 7→ ϕ(x2 ) with x2 ∈ X2 , implements a desired isomorphism.CB Q (3) Consider X := ξ∈„ Xξ , the product of vector spaces (Xξ )ξ∈„ . Take a coordinate projection, i.e. a mapping Prξ : X 0 = 0 O ξ defined by Prξ x := xξ . Clearly, Prξ is a linear operator, Prξ ∈ L (X , Xξ ). Such an operator is often treated as an endomorphism of X , a memberQof L (X ), on implying a natural isomorphism between Xξ and Xξ , where Xξ := η∈„ X η with X η := 0 if η 6= ξ and X ξ := Xξ . (4) Let X := X1 ⊕ X2 . Since +−1 is an isomorphism between X and X1 × X2 , we may define P1 , P2 ∈ L (X) as P1 := PX1 ||X2 := Pr1 ◦(+−1 ) and P2 := PX2 ||X1 := Pr2 ◦(+−1 ). The operator P1 is the projection of X onto X1 along X2 and P2 is the complementary projection to P1 or the complement of P1 (in symbols, P2 = P1d ). In turn, P1 is complementary to P2 , and P2 projects X onto X2 along X1 . Observe also that P1 + P2 = IX . Moreover, P12 := P1 P1 = P1 , and so a projection is an idempotent operator. Conversely, every idempotent P belonging to L (X) projects X onto P (X) along P −1 (0). For T ∈ L (X) and P a projection, the equality P T P = T P holds if and only if T (X0 ) ⊂ X0 with X0 = im P (read: X0 is invariant under T ). CB The equality T PX1 ||X2 = PX1 ||X2 T holds whenever both X1 and X2 are invariant under T , and in this case the direct sum decomposition X = X1 ⊕ X2 reduces T . The restriction of T to X1 is acknowledged as an element T1 of L (X1 ) which is called the part of T in X1 . If T2 ∈ L (X2 ) is the part of T in X2 , then T is expressible in matrix form   T1 0 T ∼ . 0 T2 Namely, an element x of X1 ⊕ X2 is regarded as a “column vector” with components x1 and x2 , where x1 = PX1 ||X2 x and x2 = PX1 ||X2 x; matrices are multiplied according to the usual rule, “rows by columns.” The product of T and the column vector x, i.e. the vector with components T1 x1 and T2 x2 , is certainly considered as T x (in this case, we also write T x1 and T x2 ). In other words, T is identified with the mapping from X1 × X2 to X1 × X2 acting as 

x1 x2



 7→

T1 0

0 T2



x1 x2

 .

Vector Spaces

15

In a similar way we can introduce matrix presentation for general operators contained in L (X1 ⊕ X2 , Y1 ⊕ Y2 ). CB (5) A finite subset E of X is linearly independent (in X) provided that e∈E λe e = 0, with λe ∈ F (e ∈ E ), implies λe = 0 for all e ∈ E . An arbitrary subset E of X is linearly independent if every finite subset of E is linearly independent. A Hamel basis (or an algebraic basis) for X is a linearly independent set in X maximal by inclusion. Each linearly independent set is contained in a Hamel basis. All Hamel bases have the same cardinality called the dimension of X and denoted by dim X. Every vector space is isomorphic to the direct sum of a family (F )ξ∈„ with „ of cardinality dim X. Suppose that X1 is a subspace of X. The codimension of X1 is the dimension of X/X1 , with codim X1 standing for the former. If X = X1 ⊕ X2 then codim X1 = dim X2 and dim X = dim X1 + codim X1 . CB P

2.3. Equations in Operators 2.3.1. Definition. Given T ∈ L (X, Y ), define the kernel of T as ker T := T −1 (0), the cokernel of T as coker T := Y / im T , and the coimage of T as coim T := X/ ker T . Agree that an operator T is an monomorphism whenever ker T = 0. An operator T is an epimorphism in the case of the equality im T = Y . 2.3.2. An operator is an isomorphism if and only if it is a monomorphism and an epimorphism simultaneously. CB 2.3.3. Remark. Below use is made of the concept of commutative diagram. So the phrase, “The following diagram commutes,” α1 - Y  X @ α2 α4 α3@ R ? @ V α- W 5 encodes the containments α1 ∈ L (X, Y ), α2 ∈ L (Y, W ), α3 ∈ L (X, W ), α4 ∈ L (V, Y ) and α5 ∈ L (V, W ), as well as the equalities α2 α1 = α3 and α5 = α2 α4 . T

S

2.3.4. Definition. A diagram X −→ Y −→ Z is an exact sequence (at the term Y ), if ker S = im T . A sequence . . . 0 = 0 O k−1 0 = 0 O k 0 = 0 O k+1 0 = 0 O is exact at Xk , if for all k the subsequence Xk−1 0 = 0 O k 0 = 0 O k+1 (symbols of operators are omitted), and is exact if it is exact at every term (except the first and the last, if any).

16

Chapter 2

2.3.5. Examples. T S (1) An exact sequence X −→ Y −→ Z is semi-exact, i.e. ST = 0. The converse is not true. T (2) A sequence 00 = 0 O −→ Y is exact if and only if T is a monomorphism. (Throughout the book 00 = 0 O certainly denotes the sole element of L (0, X), zero (cf. 2.1.4 (3)).) T (3) A sequence X −→ Y 0 = 0 O is exact if and only if T is an epimorphism. (Plainly, the symbol Y 0 = 0 O again stands for zero, the single element of L (Y, 0).) (4) An operator T from X to Y , a member of L (X, Y ), is an isomorT phism if and only if 00 = 0 O −→ Y 0 = 0 O is exact. (5) Suppose that X0 is a subspace of X. Let ι : X0 0 = 0 O stand for the identical embedding of X0 into X. Consider the quotient space X/X0 and let ϕ : X0 = 0 O /X0 be the corresponding coset mapping. Then the sequence ϕ ι 00 = 0 O 0 −→ X −→ X/X0 0 = 0 O

is exact. (On similar occasions the letters ι and ϕ are usually omitted.) In a sense, this sequence is unique. Namely, consider a so-called short sequence T S 00 = 0 O −→ Y −→ Z0 = 0 O

and assume that it is exact. Putting Y0 := im T , arrange the following commutative diagram T S - X - Y - Z - 0 0

0

α ? - Y0

γ β ? ? - Y - Y /Y0

- 0

where α, β, and γ are some isomorphisms. In other words, a short exact sequence actually presents a subspace and the corresponding quotient space. CB (6) Every T in L (X, Y ) generates the exact sequence 00 = 0 O T 0 = 0 O −→ Y 0 = 0 O T 0 = 0 O T

which is called the canonical exact sequence for T . 2.3.6. Definition. Given T0 ∈ L (X0 , Y ), with X0 a subspace of X, call an operator T from X to Y an extension of T0 (onto X, in symbols, T ⊃ T0 ), provided that T0 = T ι, where ι : X0 0 = 0 O is the identical embedding of X0 into X.

Vector Spaces

17

2.3.7. Let X and Y be vector spaces and let X0 be a subspace of X. Then each T0 in L (X0 , Y ) has an extension T in L (X, Y ). C Putting T := T0 PX0 , where PX0 is a projection onto X0 , settles the claim. B 2.3.8. Theorem. Let X, Y , and Z be vector spaces. Take A ∈ L (X, Y ) and B ∈ L (X, Z). The diagram X

A - Y @ X B@@ R ? Z

is commutative for some X in L (Y, Z) if and only if ker A ⊂ ker B. C ⇒: It is evident that ker A ⊂ ker B in case B = X A. ⇐: Set X := B◦A−1 . Clearly, X ◦A(x) = B◦(A−1 ◦A)x = B(x+ker A) = Bx. Show that X0 := X |im A is a linear operator. It suffices to check that X is singlevalued. Suppose that y ∈ im A and z1 , z2 ∈ X (y). Then z1 = Bx1 , z2 = Bx2 and Ax1 = Ax2 = y. By hypothesis B(x1 − x2 ) = 0; therefore, z1 = z2 . Applying 2.3.7, find an extension X of X0 to Y . B 2.3.9. Remark. Provided that the operator A is an epimorphism, there is a unique solution X in 2.3.8. It is the right place to emphasize that the phrase, “There is a unique X ,” implies that X is available as well as unique. 2.3.10. Every linear operator T admits a unique factorization through its coimage; i.e., there is a unique quotient T of T by the equivalence ∼codim T . C Immediate from 2.3.8 and 2.3.9. B 2.3.11. Remark. The operator T is sometimes called the monoquotient of T and treated as acting onto im T . In this connection, observe that T is expressible as the composition T = ιT ϕ, with ϕ an epimorphism, T an isomorphism, and ι a monomorphism; i.e., the following diagram commutes: coim T ϕ 6 X

T im T T

ι ? - Y

2.3.12. Let X be a vector space and let f0 , f1 , . . . , fNP belong to X # . The N functional f0 is a linear combination of f1 , . . . , fN (i.e., f0 = j=1 λj fj with λj in F ) if and only if ker f0 ⊃ ∩N j=1 ker fj .

18

Chapter 2

C Define the linear operator (f1 , . . . , fN ) : X0 = 0 O F N by (f1 , . . . , fN )x := (f1 (x), . . . , fN (x)). Obviously, ker(f1 , . . . , fN ) = ∩N j=1 ker fj . Now apply 2.3.8 to the problem X

(f1 ,... ,fN )

- FN

@ @ f0@ @ R ? @ F on recalling what F N # is. B 2.3.13. Theorem. Let X, Y , and Z be vector spaces. Take A ∈ L (Y, X) and B ∈ L (Z, X). The diagram A X 

Y

6 @ I @ X B @ Z is commutative for some X in L (Z, Y ) if and only if im A ⊃ im B. C ⇒: im B = B(Z) = A(X (Z)) ⊂ A(Y ) = im A. ⇐: Let Y0 be an algebraic complement of ker A to Y and A0 := A|Y0 . Then A0 is an isomorphism between Y0 and im A. The operator X := ιA−1 0 B is obviously a sought solution, with ι the identical embedding of Y0 into Y . B 2.3.14. Remark. Provided that the operator A is a monomorphism, there is a unique operator X in 2.3.13. CB 2.3.15. Remark. Theorems 2.3.8 and 2.3.13 are in “formal duality.” One results from the other by “reversing arrows,” “interchanging kernels and images,” and “passing to reverse inclusion.” 2.3.16. Snowflake Lemma. Let S ∈ L (Y, Z) and T ∈ L (X, Y ). There are unique operators α1 , . . . , α6 such that the following diagram commutes:

Vector Spaces

19 0

0

S w /  α2 ker ST ker S  S α3 α1  7 S  S  S w S /  w S T  - X - Y - coker T - 0 0 - ker T S  S  S   S ST SS w  / S  Z  α4 α6 S S   S S   S w  S / / w  S  coker S coker ST α5 w S /  0 0 Moreover, the sequence α1 α2 α3 α4 α5 00 = 0 O T −→ ker ST −→ ker S −→ coker T −→ coker ST −→ coker S0 = 0 O

is exact. CB Exercises 2.1. Give examples of vector spaces and nonvector spaces. Which constructions lead to vector spaces? 2.2. Study vector spaces over the two-element field Z2 . 2.3. Describe vector spaces with a countable Hamel basis. 2.4. Prove that there is a discontinuous solution f : R0 = 0 O R to the function equation f (x + y) = f (x) + f (y)

(x, y ∈ R).

How to visualize such an f graphically? 2.5. Prove that the algebraic dual to the direct sum of (Xξ ) is presentable as the product of the algebraic duals of (Xξ ). 2.6. Let X ⊃ X0 ⊃ X00 . Prove that the spaces X/X00 and (X/X0 )/(X00 /X0 ) are isomorphic. 2.7. Define the “double sharp” mapping by the rule x## : x# 7→ hx | x# i

(x ∈ X, x# ∈ X # ).

Show that this mapping embeds a vector space X into the second dual X ## .

20

Chapter 2 2.8. Prove that finite-dimensional spaces and only such spaces are algebraically reflexive;

i.e., ##

(X) = X ## ⇔ dim X < +∞.

2.9. Are there any analogs for a Hamel basis in general modules? 2.10. When does a sum of projections present a projection itself? 2.11. Let T be an endomorphism of some vector space which satisfies the conditions T n−1 6= 0 and T n = 0 for a natural n. Prove that the operators T 0 , T, . . . , T n−1 are linearly independent. 2.12. Describe the structure of a linear operator defined on the direct sum of spaces and acting into the product of spaces. 2.13. Find conditions for unique solvability of the following equations in operators XA = B and AX = B (here the operator X is unknown). 2.14. What is the structure of the spaces of bilinear operators? 2.15. Characterize the real vector space that results from neglecting multiplication by imaginary scalars in a complex vector space (cf. 3.7.1). 2.16. Given a family of linearly independent vectors (xe )e∈E , find a family of functionals (x# ) e e∈E satisfying the following conditions: hxe | x# e i=1 hxe | x# e i

=0

(e ∈ E );

(e, e0 ∈ E, e 6= e0 ).

2.17. Given a family of linearly independent functionals (x# e )e∈E , find a family of vectors (xe )e∈E satisfying the following conditions: hxe | x# e i=1 hxe | x# e i=0

(e ∈ E );

(e, e0 ∈ E, e 6= e0 ).

2.18. Find compatibility conditions for simultaneous linear equations and linear inequalities in real vector spaces. 2.19. Consider the commutative diagram T

W −→ X −→ Y −→ Z α↓ β↓ γ↓ δ↓ T

W −→ X −→ Y −→ Z with exact rows and such that α is an epimorphism, and δ is a monomorphism. Prove that ker γ = T (ker β) and T −1 (im γ) = im β.

Chapter 3 Convex Analysis

3.1. Sets in Vector Spaces 3.1.1. Definition. Let € be a subset of F 2. A subset U of a vector space is a € -set in this space (in symbols, U ∈ (€ )) if (λ1 , λ2 ) ∈ € ⇒ λ1 U + λ2 U ⊂ U. 3.1.2. Examples. (1) Every set is in (∅). (Hence, (∅) is not a set.) (2) If € := F 2 then a nonempty € -set is precisely a linear set in a vector space. (3) If € := R2 then a nonempty € -set in a vector space X is a real subspace of X. 2 . In other (4) By definition, a cone is a nonempty € -set with € := R+ words, a nonempty set K is a cone if and only if K + K ⊂ K and αK ⊂ K for all 2 α ∈ R+ . A nonempty R+ \ 0-set is sometimes referred to as a nonpointed cone; and a nonempty R+ × 0-set, as a nonconvex cone. (From now on we use the convenient notation R+ := {t ∈ R : t ≥ 0}.)

(5) A nonempty € -set, for € := {(λ1 , λ2 ) ∈ F 2 : λ1 + λ2 = 1}, is an affine variety or a flat. If X0 is a subspace of X and x ∈ X then x+X0 := {x}+X0 is an affine variety in X. Conversely, if L is an affine variety in X and x ∈ L then L − x := L + {−x} is a linear set in X. CB (6) If € := {(λ1 , λ2 ) ∈ F 2 : |λ1 | + |λ2 | ≤ 1} then a nonempty € -set is absolutely convex. (7) If € := {(λ, 0) ∈ F 2 : |λ| ≤ 1} then a nonempty € -set is balanced. (In the case F := R the term “star-shaped” is occasionally employed; the word “symmetric” can also be found.) (8) A set is convex if it is a € -set for € := {(λ1 , λ2 ) ∈ R2 : λ1 ≥ 0, λ2 ≥ 0, λ1 + λ2 = 1}.

22

Chapter 3

(9) A conical segment or conical slice is by definition a nonempty € -set 2 with € := {(λ1 , λ2 ) ∈ R+ : λ1 + λ2 ≤ 1}. A set is a conical segment if and only if it is a convex set containing zero. CB (10) Given € ⊂ F 2 , observe that X ∈ (€ ) for every vector space X over F. Note also that in 3.1.2 (1)–3.1.2 (9) the set € is itself a € -set. 3.1.3. Let X be a vector space and let E be a family of € -sets in X. Then ∩{U : U ∈ im E } ∈ (€ ). Provided that im E is filtered upward (by inclusion), ∪{U : U ∈ im E } ∈ (€ ). CB 3.1.4. Remark. The claim of 3.1.3 means in particular that the collection of € -sets of a vector space, ordered by inclusion, presents a complete lattice. 3.1.5. Let X and Y be vector spaces and let U and V be € -sets, with U ⊂ X and V ⊂ Y . Then U × V ∈ (€ ). C If U or V is nonempty then U × V = ∅, and there is nothing to prove. Now take u1 , u2 ∈ U , v1 , v2 ∈ V , and (λ1 , λ2 ) ∈ € . Find λ1 u1 + λ2 u2 ∈ U and λ1 v2 + λ2 v1 ∈ V . Hence, (λ1 u1 + λ2 u2 , λ1 v1 + λ2 v2 ) ∈ U × V . B 3.1.6. Definition. Let X and Y be vector spaces and € ⊂ F 2. A correspondence T ⊂ X × Y is a € -correspondence provided that T ∈ (€ ). 3.1.7. Remark. When a € -set bears a specific attribute, the latter is preserved for naming a € -correspondence. With this in mind, we speak about linear and convex correspondences, affine mappings, etc. The next particularity of the nomenclature is worth memorizing: a convex function of one variable is not a convex correspondence, save trivial cases (cf. 3.4.2). 3.1.8. Let T ⊂ X × Y be a €1 -correspondence and let U ⊂ X be a €2 -set. If €2 ⊂ €1 then T (U ) ∈ (€2 ). C Take y1 , y2 ∈ T (U ). Then (x1 , y1 ) ∈ T and (x2 , y2 ) ∈ T with some x1 , x2 ∈ U . Given (λ1 , λ2 ) ∈ €2 , observe that λ1 (x1 , y1 ) + λ2 (x2 , y2 ) ∈ T , because by hypothesis (λ1 , λ2 ) ∈ €1 . Finally, infer that λ1 y1 + λ2 y2 ∈ T (U ). B 3.1.9. The composition of € -correspondences is also a € -correspondence. C Let F ⊂ X × V and G ⊂ W × Y , with F, G ∈ (€ ). Note that (x1 , y1 ) ∈ G ◦ F ⇔ (∃ v1 )

(x1 , v1 ) ∈ F & (v1 , y1 ) ∈ G;

(x2 , y2 ) ∈ G ◦ F ⇔ (∃ v2 )

(x2 , v2 ) ∈ F & (v2 , y2 ) ∈ G.

To complete the proof, “multiply the first row by λ1 ; the second, by λ2 , where (λ1 , λ2 ) ∈ € ; and sum the results.” B

Convex Analysis

23

3.1.10. If subsets U and V of a vector space are € -sets for some € ⊂ F 2 then αU + βV ∈ (€ ) for all α, β ∈ F. C The claim is immediate from 3.1.5, 3.1.8, and 3.1.9. B 3.1.11. Definition. Let X be a vector space and let U be a subset of X. For € ⊂ F 2 the € -hull of U is the set H€ (U ) := ∩{V ⊂ X : V ∈ (€ ), V ⊃ U }. 3.1.12. The following statements are valid: (1) H€ (U ) ∈ (€ ); (2) H€ (U ) is the least € -set including U ; (3) U1 ⊂ U2 ⇒ H€ (U1 ) ⊂ H€ (U2 ); (4) U ∈ (€ ) ⇔ U = H€ (U ); (5) H€ (H€ (U )) = H€ (U ). CB 3.1.13. The Motzkin formula holds: H€ (U ) = ∪{H€ (U0 ) : U0 is a finite subset of U }. C Denote the right side of the Motzkin formula by V . Since U0 ⊂ U ; applying 3.1.12 (3), deduce that H€ (U0 ) ⊂ H€ (U ); and, hence, H€ (U ) ⊃ V . By 3.1.12 (2), it is necessary (and, surely, sufficient) to verify that V ∈ (€ ). The last follows from 3.1.3 and the inclusion H€ (U0 ) ∪ H€ (U1 ) ⊂ H€ (U0 ∪ U1 ). B 3.1.14. Remark. The Motzkin formula reduces the problem of describing arbitrary € -hulls to calculating € -hulls of finite sets. Observe also that € -hulls for concrete € s have special (but natural) designations. For instance, if € := {(λ1 , λ2 ) ∈ 2 R+ : λ1 + λ2 = 1}, then the term “convex hull” is used and co(U ) stands for H€ (U ). If U 6= ∅, the notation for HF 2 (U ) is lin(U ); moreover, it is natural and convenient to put lin(∅) := 0. The subspace lin(U ) is called the linear span of U . The concepts of affine hull, conical hull, etc. are introduced in a similar fashion. Note also that the convex hull of a finite set of points comprises their convex combinations; i.e., (N ) X co({x1 , . . . , xN }) = λk xk : λk ≥ 0, λ1 + . . . + λN = 1 . /. k=1

3.2. Ordered Vector Spaces 3.2.1. Definition. Let (X, R, +, · ) be a vector space. A preorder σ on X is compatible with vector structure if σ is a cone in X 2 ; in this case X is an ordered vector space. (It is more precise to call (X, R, +, · , σ) a preordered vector space, reserving the term “ordered vector space” for the case in which σ is an order.)

24

Chapter 3

3.2.2. If X is an ordered vector space and σ is the corresponding preorder then σ(0) is a cone and σ(x) = x + σ(0) for all x ∈ X. C By 3.1.3, σ(0) is a cone. The equality (x, y) = (x, x) + (0, y − x) yields the equivalence (x, y) ∈ σ ⇔ y − x ∈ σ(0). B 3.2.3. Let K be a cone in a vector space X. Denote σ := {(x, y) ∈ X 2 : y − x ∈ K}. Then σ is a preorder compatible with vector structure and K coincides with the cone σ(0) of positive elements of X. The relation σ is an order if and only if K ∩ (−K) = 0. C It is clear that 0 ∈ K ⇒ IX ⊂ σ and K + K ⊂ K ⇒ σ ◦ σ ⊂ σ. Furthermore, σ −1 = {(x, y) ∈ X 2 : x − y ∈ K}. Therefore, σ ∩ σ −1 ⊂ IX ⇔ K ∩ (−K) = 0. To show that σ is a cone, take (x1 , y1 ), (x2 , y2 ) ∈ σ and α1 , α2 ∈ R+ . Find α1 y1 + α2 y2 − (α1 x1 + α2 x2 ) = α1 (y1 − x1 ) + α2 (y2 − x2 ) ∈ α1 K + α2 K ⊂ K. B 3.2.4. Definition. A cone K is an ordering cone or a salient cone provided that K ∩ (−K) = 0. 3.2.5. Remark. By virtue of 3.2.2 and 3.2.3, assigning a preorder to X is equivalent to distinguishing some cone of positive elements in it which is the positive cone of X. The structure of an ordered vector space arises from selecting an ordering cone. Keeping this in mind, we customarily call a pair (X, X+ ) with positive cone X+ , as well as X itself, a (pre)ordered vector space. 3.2.6. Examples. (1) The space of real-valued functions R„ with the positive cone R„ + := „ (R+ ) constituted by the functions assuming only positive values has the “natural order.” (2) Let X be an ordered vector space with positive cone X+ . If X0 is a subspace of X then the order induced in X0 is defined by the cone X0 ∩ X+ . In this way X0 is considered as an ordered vector space, a subspace of X. (3) If X and Y are preordered vector spaces then T ∈ L (X, Y ) is a positive operator (in symbols, T ≥ 0) whenever T (X+ ) ⊂ Y+ . The set L+ (X, Y ) of all positive operators is a cone. The linear span of L+ (X, Y ) is denoted by Lr (X, Y ), and a member of Lr (X, Y ) is called a regular operator. 3.2.7. Definition. An ordered vector space is a vector lattice or a Riesz space if the ordered set of its vectors presents a lattice. 3.2.8. Definition. A vector lattice X is called a Kantorovich space or briefly a K-space if X is boundedly order complete or Dedekind complete, which means that each nonempty bounded above subset of X has a least upper bound.

Convex Analysis

25

3.2.9. Each nonempty bounded below subset of a Kantorovich space has a greatest lower bound. C Let U be bounded below: x ≤ U for some x. So, −x ≥ −U . Applying 3.2.8, find sup(−U ). Obviously, − sup(−U ) = inf U . B 3.2.10. If U and V are nonempty bounded above subsets of a K-space then sup(U + V ) = sup U + sup V. C If U or V is a singleton then the equality is plain. The general case follows from the associativity of least upper bounds. Namely, sup(U + V ) = sup{sup(u + V ) : u ∈ U } = sup{u + sup V : u ∈ U } = sup V + sup{u : u ∈ U } = sup V + sup U. . 3.2.11. Remark. The derivation of 3.2.10 is valid for an arbitrary ordered vector space provided that the given sets have least upper bounds. The equality sup λU = λ sup U for λ ∈ R+ is comprehended by analogy. 3.2.12. Definition. For an element x of a vector lattice, the positive part of x is x+ := x ∨ 0, the negative part of x is x− := (−x)+ , and the modulus of x is |x| := x ∨ (−x). 3.2.13. If x and y are elements of a vector lattice then x + y = x ∨ y + x ∧ y. C x + y − x ∧ y = x + y + (−x) ∨ (−y) = y ∨ x B 3.2.14. x = x+ − x− ; |x| = x+ + x− . C The first equality follows from 3.2.13 on letting y := 0. Furthermore, |x| = x ∨ (−x) = −x + (2x) ∨ 0 = −x + 2x+ = (x+ − x− ) + 2x+ = x+ + x− . B 3.2.15. Interval Addition Lemma. If x and y are positive elements of a vector lattice X then [0, x + y] = [0, x] + [0, y]. (As usual, [u, v] := σ(u) ∩ σ −1 (v) is the (order) interval with endpoints u and v.) C The inclusion [0, x]+[0, y] ⊂ [0, x+y] is obvious. Assume that 0 ≤ z ≤ x+y and put z1 := z ∧ x. It is easy that z1 ∈ [0, x]. Now if z2 := z − z1 then z2 ≥ 0 and z2 = z − z ∧ x = z + (−z) ∨ (−x) = 0 ∨ (z − x) ≤ 0 ∨ (x + y − x) = 0 ∨ y = y. B 3.2.16. Remark. The conclusion of the Interval Addition Lemma is often referred to as the Riesz Decomposition Property. 3.2.17. Riesz–Kantorovich Theorem. Let X be a vector space and let Y be a K-space. The space of regular operators Lr (X, Y ), ordered by the cone of positive operators L+ (X, Y ), is a K-space. CB

26

Chapter 3 3.3. Extension of Positive Functionals and Operators

3.3.1. Counterexamples. (1) Let X be B([0, 1], R), the space of the bounded real-valued functions given on [0, 1]; and let X0 be C([0, 1], R), the subspace of X comprising all continuous functions. Put Y := X0 and equip X0 , X, and Y with the natural order (cf. 3.2.6 (1) and 3.2.6 (2)). Consider the problem of extending the identical embedding T0 : X0 0 = 0 O to a positive operator T in L+ (X, Y ). If the problem had a solution T , then each nonempty bounded set E in X0 would have a least upper bound supX0 E calculated in X0 . Namely, supX0 E = T supX E , where supX E is the least upper bound of E in X; whereas, undoubtedly, Y fails to be a K-space. (2) Denote by s := RN the sequence space and furnish s with the natural order. Let c be the subspace of s comprising all convergent sequences, the convergent sequence space. Demonstrate that the positive functional f0 : c0 = 0 O R defined by f0 (x) := lim x(n) has no positive extension to s. Indeed, assume that f ∈ s# , f ≥ 0 and f ⊃ f0 . Put x0 (n) := n and xk (n) := k ∧ n for k, n ∈ N. Plainly, f0 (xk ) = k; moreover, f (x0 ) ≥ f (xk ) ≥ 0, since x0 ≥ xk ≥ 0, a contradiction. 3.3.2. Definition. A subspace X0 of an ordered vector space X with positive cone X+ is massive (in X) if X0 + X+ = X. The terms “coinitial” or “minorizing” are also in common parlance. 3.3.3. A subspace X0 is massive in X if and only if for all x ∈ X there are elements x0 and x0 in X0 such that x0 ≤ x ≤ x0 . CB 3.3.4. Kantorovich Theorem. If X is an ordered vector space, X0 is massive in X, and Y is a K-space; then each positive operator T0 , a member of L+ (X0 , Y ), has a positive extension T , a member of L+ (X, Y ). C Step I. First, let X := X0 ⊕ X1 , where X1 is a one-dimensional subspace, X1 := {αx : α ∈ R}. Since X0 is massive and T0 is positive, the set U := {T0 x0 : x0 ∈ X0 , x0 ≥ x} is nonempty and bounded below. Consequently, there is some y such that y := inf U . Assign T x := {T0 x0 +αy : x = x0 +αx, x0 ∈ X0 , α ∈ R}. It is clear that T is a single-valued correspondence. Further, T ⊃ T0 and dom T = X. So, only the positivity of T needs checking. If x = x0 + αx and x ≥ 0, then the case of α equal to 0 is trivial. If α > 0 then x ≥ −x0 /α. This implies that −T0 x0 /α ≤ y, i.e., T x ∈ Y+ . In a similar way for α < 0 observe that x ≤ −x0 /α. Thus, y ≤ −T0 x0 /α and, finally, T x = T0 x0 + αy ∈ Y+ . Step II. Now let E be the collection of single-valued correspondences S ⊂ X × Y such that S ⊃ T0 and S(X+ ) ⊂ Y+ . By 3.1.3, E is inductive in order by inclusion and so, by the Kuratowski–Zorn Lemma, E has a maximal element T . If x ∈ X \ dom T , apply the result of Step I with X := dom T ⊕ X1 , X0 := dom T, T0 := T and X1 := {αx : α ∈ R} to obtain an extension of T . But T is maximal, a contradiction. Thus, T is a sought operator. B

Convex Analysis

27

3.3.5. Remark. When Y := R, Theorem 3.3.4 is sometimes referred to as the Kre˘ın–Rutman Theorem. 3.3.6. Definition. A positive element x is discrete whenever [0, x] = [0, 1]x. 3.3.7. If there is a discrete functional on (X, X+ ) then X = X+ − X+ . C Let T be such functional and := X+ − X+ . Take f ∈ X # . It suffices to check that ker f ⊃⇒ = . Evidently, T + f ∈ [0, T ]; i.e., T + f = αT for some α ∈ [0, 1]. If T | = 0 then 2T ∈ [0, T ]. Hence, T = 0 and f = 0. Now if T (x0 ) 6= 0 for some x0 ∈, then α = 1 and f = 0. B 3.3.8. Discrete Kre˘ın–Rutman Let X be a massive subspace of an ordered vector space X and let T0 be a discrete functional on X0 . Then there is a discrete functional T on X extending T0 . C Adjust the proof of 3.3.4. Step I. Observe that the exhibited functional T is discrete. For, if T 0 ∈ [0, T ] then there is some α ∈ [0, 1] such that T 0 (x0 ) = αT (x0 ) for all x0 ∈ X0 and so (T − T 0 )(x0 ) = (1 − α)T (x0 ). Estimating, find T 0 (x) ≤ inf{T 0 (x0 ) : x0 ≥ x, x0 ∈ X0 } = αT (x); (T − T 0 )(x) ≤ inf{(T − T 0 )(x0 ) : x0 ≥ x, x0 ∈ X0 } = (1 − α)T (x). Therefore, T 0 = αT and [0, T ] ⊂ [0, 1]T . The reverse inclusion is always true. Thus, T is discrete. Step II. Let E be the same as in the proof of 3.3.4. Consider Ed , the set comprising all S in E such that the restriction S|dom S is a discrete functional on dom S. Show that Ed is inductive. To this end, take a chain E0 in Ed and put S := ∪{S0 : S0 ∈ E0 }; obviously, S ∈ E . Verifying the discreteness of S will complete the proof. Suppose that 0 ≤ S 0 (x0 ) ≤ S(x0 ) for all x0 ∈ (dom S)+ and S 0 ∈ (dom S)# . If S(x0 ) = 0 for all x0 then S 0 = 0S, as was needed. In the case S(x0 ) 6= 0 for some x0 ∈ (dom S)+ choose S0 ∈ E0 such that S0 (x0 ) = S(x0 ). Since S0 is discrete, deduce that S 0 (x0 ) = αS(x0 ) for all x0 ∈ dom S0 . Furthermore, α = S 0 (x0 )/S(x0 ); i.e., α does not depend on the choice of S0 . Since E0 is a chain, infer that S 0 = αS. B 3.4. Convex Functions and Sublinear Functionals 3.4.1. Definition. The semi-extended real line R· is the set R· with some greatest element +∞ adjoined formally. The following agreements are effective: α(+∞) := +∞ (α ∈ R+ ) and +∞ + x := x + (+∞) := +∞ (x ∈ R· ).

28

Chapter 3

3.4.2. Definition. Let f : X0 = 0 O R· be a mapping (also called an extended function). The epigraph of f is the set 0 = 0 OO := {(x, t) ∈ X × R : t ≥ f (x)}. The effective domain of definition or simply the domain of f is the set dom f := {x ∈ X : f (x) < +∞}. 3.4.3. Remark. Inconsistency in overusing the symbol dom f is illusory. Namely, the effective domain of definition of f : X0 = 0 O R· coincides with the domain of definition of the single-valued correspondence f ∩ X × R. We thus continue to write f : X0 = 0 O R, omitting the dot in the symbol R· whenever dom f = X. 3.4.4. Definition. If X is a real vector space then a mapping f : X0 = 0 O R· is a convex function provided that the epigraph 0 = 0 OO is convex. 3.4.5. A function f : X0 = 0 O R· is convex if and only if the Jensen inequality holds: f (α1 x1 + α2 x2 ) ≤ α1 f (x1 ) + α2 f (x2 ) for all α1 , α2 ≥ 0, α1 + α2 = 1 and x1 , x2 ∈ X. C ⇒: Take α1 , α2 ≥ 0, α1 + α2 = 1. If either of x1 and x2 fails to belong to dom f then the Jensen inequality is evident. Let x1 , x2 ∈ dom f . Then (x1 , f (x1 )) ∈ 0 = 0 OO and (x2 , f (x2 )) ∈ 0 = 0 OO . Appealing to 3.1.2 (8), find α1 (x1 , f (x1 )) + α2 (x2 , f (x2 )) ∈ 0 = 0 OO . ⇐: Take (x1 , t1 ) ∈ 0 = 0 OO and (x2 , t2 ) ∈ 0 = 0 OO , i.e. t1 ≥ f (x1 ) and t2 ≥ f (x2 ) (if dom f = ∅ then f (x) = +∞ (x ∈ X) and 0 = 0 OO = ∅). Applying the Jensen inequality, observe the containment (α1 x1 +α2 x2 , α1 t1 +α2 t2 ) ∈ 0 = 0 OO for all α1 , α2 ≥ 0, α1 + α2 = 1. B 3.4.6. Definition. A mapping p : X0 = 0 O R· is a sublinear functional provided that 0 = 0 OO is a cone. 3.4.7. If dom p 6= 0 then the following statements are equivalent: (1) p is a sublinear functional; (2) p is a convex function that is positively homogeneous: p(αx) = αp(x) for all α ≥ 0 and x ∈ dom p; (3) if α1 , α2 ∈ R+ and x1 , x2 ∈ X, then p(α1 x1 + α2 x2 ) ≤ α1 p(x1 ) + α2 p(x2 ); (4) p is a positively homogeneous functional that is subadditive: p(x1 + x2 ) ≤ p(x1 ) + p(x2 ) (x1 , x2 ∈ X). CB

Convex Analysis

29

3.4.8. Examples. (1) A linear functional is sublinear; an affine functional is a convex function. (2) Let U be a convex set in X. Define the indicator function δ(U ) : X0 = 0 O R· of U as the mapping  δ(U )(x) :=

0,

if x ∈ U

+∞, if x 6∈ U .

It is clear that δ(U ) is a convex function. If U is a cone then δ(U ) is a sublinear function. If U is an affine set then δ(U ) is an affine function. (3) The sum of finitely many convex functions and the supremum (or upper envelope) of a family of convex functions (calculated pointwise, i.e. in (R· )X ) are convex functionals. Sublinear functionals have analogous properties. (4) The composition of a convex function with an affine operator, i.e. an everywhere-defined single-valued affine correspondence, is a convex function. The composition of a sublinear functional with a linear operator is sublinear. 3.4.9. Definition. If U and V are subsets of a vector space X then U absorbs V if V ⊂ nU for some n ∈ N. A set U is absorbing (in X) if it absorbs every point of X; i.e., X = ∪n∈N nU . 3.4.10. Let T ⊂ X × Y be a linear correspondence with im T = Y . If U is absorbing (in X) then T (U ) is absorbing (in Y ). / Y = T (X) = T (∪n∈N nU ) = ∪n∈N T (nU ) = ∪n∈N nT (U ) . 3.4.11. Definition. Let U be a subset of a vector space X. An element x belongs to the core of U (or x is an algebraically interior point of U ) if U − x is absorbing in X. 3.4.12. If f : X0 = 0 O R· is a convex function and x ∈ core dom f then for all h ∈ X there is a limit f 0 (x)(h) := lim α↓0

f (x + αh) − f (x) f (x + αh) − f (x) = inf . α>0 α α

Moreover, the mapping f 0 (x) : h 7→ f 0 (x)h is a sublinear functional f 0 (x) : X0 = 0 O R, the directional derivative of f at x. C Set ϕ(α) := f (x + αh). By 3.4.8 (4), ϕ : R0 = 0 O R· is a convex function and 0 ∈ core dom ϕ. The mapping α 7→ (ϕ(α) − ϕ(0))/α (α > 0) is increasing and bounded above; i.e., ϕ0 (0)(1) is available. By definition, f 0 (x)(h) = ϕ0 (0)(1).

30

Chapter 3 Given β > 0 and h ∈ H, successively find f 0 (x)(βh) = inf

f (x + αβh) − f (x) f (x + αβh) − f (x) = β inf = βf 0 (x)(h). α αβ

Moreover, using the above result, for h1 , h2 ∈ X infer that
1 f x + / α(h + h ) − f (x) 2 1 2 f 0 (x)(h1 + h2 ) = 2 lim α↓0 α
1 1 f /2 (x + αh1 ) + /2 (x + αh2 ) − f (x) = 2 lim α↓0 α f (x + αh1 ) − f (x) f (x + αh2 ) − f (x) ≤ lim + lim α↓0 α↓0 α α 0 0 = f (x)(h1 ) + f (x)(h2 ). Appealing to 3.4.7 ends the proof. B 3.5. The Hahn–Banach Theorem 3.5.1. Definition. Let X be a real vector space and let f : X0 = 0 O R· be a convex function. The subdifferential of f at a point x in dom f is the set ∂x (f ) := {l ∈ X # : (∀ y ∈ X) l(y) − l(x) ≤ f (y) − f (x)}. 3.5.2. Examples. (1) Let p : X0 = 0 O R· be a sublinear functional. Put ∂ (p) := ∂0 (p). Then ∂ (p) = {l ∈ X # : (∀ x ∈ X) l(x) ≤ p(x)}; ∂x (p) = {l ∈ ∂ (p) : l(x) = p(x)}. (2) If l ∈ X # then ∂ (l) = ∂x (l) = {l}. (3) Let X0 be a subspace of X. Then ∂ (δ(X0 )) = {l ∈ X # : ker l ⊃ X0 }. (4) If f : X0 = 0 O R· is a convex function then ∂x (f ) = ∂ (f 0 (x)) whenever x ∈ core dom f . CB

Convex Analysis

31

3.5.3. Hahn–Banach Theorem. Let T ∈ L (X, Y ) be a linear operator and let f : Y 0 = 0 O R· be a convex function. If x ∈ X and T x ∈ core dom f then ∂x (f ◦ T ) = ∂T x (f ) ◦ T. C By 3.4.10 it follows that x ∈ core dom f . From 3.5.2 (4) obtain ∂x (f ◦ T ) = ∂ ((f ◦ T )0 (x)). Moreover, (f ◦ T )(x + αh) − (f ◦ T )(x) α↓0 α f (T x + αT h) − f (T x) = lim = f 0 (T x)(T h) α↓0 α

(f ◦ T )0 (x)(h) = lim

for h ∈ X. Put p := f 0 (T x). By 3.4.12, p is a sublinear functional. Whence, on appealing again to 3.5.2 (4), infer that ∂ (p) = ∂ (f 0 (T x)) = ∂ T x (f ); ∂ (p ◦ T ) = ∂ ((f ◦ T )0 (x)) = ∂x (f ◦ T ). Thereby, the only claim left unproven is the equality ∂ (p ◦ T ) = ∂ (p) ◦ T. If l ∈ ∂ (p) ◦ T (i.e., l = l1 ◦ T , where l1 ∈ ∂ (p)) then l1 (y) ≤ p(y) for all y ∈ Y . In particular, l(x) ∈ l1 (T x) ≤ p(T x) = p ◦ T (x) for all x ∈ X and so l ∈ ∂ (p ◦ T ). This argument yields the inclusion ∂ (p) ◦ T ⊂ ∂ (p ◦ T ). Now take l ∈ ∂ (p ◦ T ). If T x = 0 then l(x) ≤ p(T x) = p(0) = 0; i.e., l(x) ≤ 0. The same holds for −x. Therefore, l(x) = 0; in other words, ker l ⊃ ker T . Hence, by 2.3.8, l = l1 ◦ T for some l1 ∈ Y # . Putting Y0 := T (X), observe that the functional l1 ◦ ι belongs to ∂ (p ◦ ι), with ι the identical embedding of Y0 into Y . With ∂ (p ◦ ι) ⊂ ∂ (p) ◦ ι proven, observe that l1 ◦ ι = l2 ◦ ι for some l2 ∈ ∂ (p), which consequently yields l = l1 ◦ T = l1 ◦ ι ◦ T = l2 ◦ ι ◦ T = l2 ◦ T ; i.e., l ∈ ∂ (p) ◦ T . Thus, to complete the proof of the Hahn–Banach Theorem, showing that ∂ (p ◦ ι) ⊂ ∂ (p) ◦ ι is in order. Take an element l0 in ∂ (p◦ι) and consider the functional T0 : (y0 , t) 7→ t−l0 (y0 ) given on the subspace Y0 := Y0 × R of Y := Y × R. Equip Y with the cone Y+ := 0 = 0 OO . Note that, first, Y0 is massive since (y, t) = (0, t − p(y)) + (y, p(y))

(y ∈ Y, t ∈ R).

Second, by 3.4.2, t ≥ p(y0 ) for (y0 , t) ∈ Y0 ∩ Y+ , and so T0 (y0 , t) = t − l0 (y0 ) ≥ 0; i.e., T0 is a positive functional on Y0 . By 3.3.4, there is a positive functional T defined on Y which is an extension of T0 . Put l(y) := T (−y, 0) for y ∈ Y . It is clear that l ◦ ι = l0 . Furthermore, T (0, t) = T0 (0, t) = t. Hence, 0 ≤ T (y, p(y)) = p(y) − l(y), i.e., l ∈ ∂ (p). B

32

Chapter 3

3.5.4. Remark. The claim of Theorem 3.5.3 is also referred to as the formula for a linear change-of-variable under the subdifferential sign or the Hahn–Banach Theorem in subdifferential form. Observe that the inclusion ∂ (p ◦ ι) ⊂ ∂ (p) ◦ ι is often referred to as the Hahn–Banach Theorem in analytical form or the Dominated Extension Theorem and verbalized as follows: “A linear functional given on a subspace of a vector space and dominated by a sublinear functional on this subspace has an extension also dominated by the same sublinear functional.” 3.5.5. Corollary. If X0 is a subspace of a vector space X and p : X0 = 0 O R is a sublinear functional then the (asymmetric) Hahn–Banach formula holds: ∂ (p + δ(X0 )) = ∂ (p) + ∂ (δ(X0 )). C It is obvious that the left side includes the right side. To prove the reverse inclusion, take l ∈ ∂ (p + δ(X0 )). Then l ◦ ι ∈ ∂ (p ◦ ι), where ι is the identical embedding of X0 into X. By 3.5.3, l ◦ ι ∈ ∂ (p) ◦ ι, i.e., l ◦ ι = l1 ◦ ι for some l1 ∈ ∂ (p). Put l2 := l − l1 . By definition, find l2 ◦ ι = (l − l1 ) ◦ ι = l ◦ ι − l1 ◦ ι = 0, i.e., ker l2 ⊃ X0 . As was mentioned in 3.5.2 (3), this means l2 ∈ ∂ (δ(X0 )). B 3.5.6. Corollary. If f : X0 = 0 O R· is a linear functional and x ∈ core dom f then ∂x (f ) 6= ∅. C Let p := f 0 (x) and let ι : 00 = 0 O be the identical embedding of 0 to X. Plainly, 0 ∈ ∂ (p ◦ ι), i.e., ∂ (p ◦ ι) 6= ∅. By the Hahn–Banach Theorem, ∂ (p) 6= ∅ (otherwise, ∅ = ∂ (p) ◦ ι = ∂ (p ◦ ι)). To complete the proof, apply 3.5.2 (4). B 3.5.7. Corollary. Let f1 , f2 : X0 = 0 O R· be convex functions. Assume further that x ∈ core dom f1 ∩ core dom f2 . Then ∂x (f1 + f2 ) = ∂x (f1 ) + ∂x (f2 ). C Let p1 := f10 (x) and p2 := f20 (x). Given x1 , x2 ∈ X, define p(x1 , x2 ) := p1 (x1 ) + p2 (x2 ) and ι(x1 ) := (x1 , x1 ). Using 3.5.2 (4) and 3.5.3, infer that ∂x (f1 + f2 ) = ∂ (p1 + p2 ) = ∂ (p ◦ ι) = ∂ (p) ◦ ι = ∂ (p1 ) + ∂ (p2 ) = ∂x (f1 ) + ∂x (f2 ). . 3.5.8. Remark. Corollary 3.5.6 is sometimes called the Nonempty Subdifferential Theorem. On the one hand, it is straightforward from the Kuratowski– Zorn Lemma. On the other hand, with Corollary 3.5.6 available, demonstrate ∂ (p ◦ T ) = ∂ (p) ◦ T as follows: Define p T (y) := inf{p(y + T x) − l(x) : x ∈ X}, where l ∈ ∂ (p) and the notation of 3.5.3 is employed. Check that pT is sublinear and every l1 in ∂ (p T ) satisfies the equality l = l1 ◦ T . Thus, the Nonempty Subdifferential Theorem and the Hahn–Banach Theorem in subdifferential form constitute a precious (rather than vicious) circle.

Convex Analysis

33 3.6. The Kre˘ın–Milman Theorem

3.6.1. Definition. Let X be a real vector space. Putting seg(x1 , x2 ) := {α1 x1 + α2 x2 : α1 , α2 > 0, α1 + α2 = 1}, define the correspondence seg ⊂ X 2 × X that assigns to each pair of points the open segment joining them. If U is a convex set in X and segU is the restriction of seg 2 to U 2 , then a convex subset V of U is extreme in U if seg−1 U (V ) ⊂ V . An extreme subset of U is sometimes called a face in U . A member x of U is an extreme point in U if {x} is an extreme subset of U . The set of extreme points of U is usually denoted by ext U . 3.6.2. A convex subset V is extreme in U if and only if for all u1 , u2 ∈ U and α1 , α2 > 0, α1 + α2 = 1, the containment α1 u1 + α2 u2 ∈ V implies that u1 ∈ V and u2 ∈ V . CB 3.6.3. Examples. (1) Let p : X0 = 0 O R· be a sublinear functional and let a point x of X belong to dom p. Then ∂x (p) is an extreme subset of ∂ (p). C For, if α1 l1 + α2 l2 ∈ ∂x (p) and l1 , l2 ∈ ∂ (p), where α1 , α2 > 0, α1 + α2 = 1; then 0 = p(x) − (α1 l1 (x) + α2 l2 (x)) = α1 (p(x) − l1 (x)) + α2 (p(x) − l2 (x)) ≥ 0. Moreover, p(x)−l1 (x) ≥ 0 and p(x)−l2 (x) ≥ 0. Hence, l1 ∈ ∂x (p) and l2 ∈ ∂x (p). B (2) Let W be an extreme set in V and let V be in turn an extreme set in U . Then W is an extreme set in U . CB (3) If X is an ordered vector space then a positive element x of X is discrete if and only if the cone {αx : α ∈ R+ } is an extreme subset of X+ . C ⇐: If 0 ≤ y ≤ x then x = 1 /2 (2y) + 1 /2 (2(x − y)). Therefore, by 3.6.2, 2y = αx and 2(x − y) = βx for some α, β ∈ R+ . Thus, 2x = (α + β)x. The case x = 0 is trivial. Now if x 6= 0 then α /2 ∈ [0, 1] and, consequently, [0, x] ⊂ [0, 1]x. The reverse inclusion is evident. ⇒: Let [0, x] = [0, 1]x, and suppose that αx = α1 y1 +α2 y2 for α ≥ 0; α1 , α2 > 0, α1 + α2 = 1 and y1 , y2 ∈ X+ . If α = 0 then α1 y1 ∈ [0, x] and α2 y2 ∈ [0, x]; hence, y1 is a positive multiple of x; the same is valid for y2 . In the case α > 0 observe that (α1 /α )y1 = tx for some t ∈ [0, 1]. Finally, (α2 /α )y2 = (1 − t)x. B (4) Let U be a convex set. A convex subset V of U is a cap of U , if U \ V is convex. A point x in U is extreme if and only if {x} is a cap of U . CB 3.6.4. Extreme and Discrete Lemma. Let p : X0 = 0 O R be a sublinear functional and l ∈ ∂ (p). Assign := X × R, + := = OO , and Tl : (x, t) 7→ t − l(x) (x ∈ X, t ∈ R). Then the functional l is an extreme point of ∂ (p) if and only if Tl is a discrete functional.

34

Chapter 3 C ⇒: Consider T 0 ∈# such that T 0 ∈ [0, Tl ]. Put t1 := T 0 (0, 1), t2 := (Tl − T 0 )(0, 1),

l1 (x) := T 0 (−x, 0); l2 (x) := (Tl − T 0 )(−x, 0).

It is clear that t1 ≥ 0, t2 ≥ 0, t1 + t2 = 1; l1 ∈ ∂ (t1 p), l2 ∈ ∂ (t2 p), and l1 + l2 = l. If t1 = 0 then l1 = 0; i.e., T 0 = 0 and T 0 ∈ [0, 1]Tl . Now if t2 = 0 then t1 = 1; i.e., 1 T 0 = Tl , and so T 0 ∈ [0, 1]Tl . Assume that
t1 , t2 > 0. In
this case /t1 l1 ∈ ∂ (p) 1 1 1 and /t2 l2 ∈ ∂ (p); moreover, l = t1 /t1 l1 + t2 /t2 l2 . Since, by hypothesis, l ∈ ext ∂ (p), from 3.6.2 it follows that l1 = t1 l; i.e., T 0 = t1 Tl . B ⇐: Let l = α1 l1 + α2 l2 , where l1 , l2 ∈ ∂ (p) and α1 , α2 > 0, α1 + α2 = 1. The functionals T 0 := α1 Tl1 and T 00 := α2 Tl2 are positive; moreover, T 0 ∈ [0, Tl ], since T 0 + T 00 = Tl . Therefore, there is some β ∈ [0, 1] such that T 0 = βTl . At the point (0, 1), find α1 = β. Hence, l1 = l. By analogy, l2 = l. CB 3.6.5. Kre˘ın–Milman Let T ∈ L (X, Y ) be a linear operator and let f : Y 0 = 0 O R· be a convex function. If x ∈ X and T x ∈ core dom f then ext ∂x (f ◦ T ) ⊂ (ext ∂ T x (f )) ◦ T. C Start arguing as in the proof of the Hahn–Banach Theorem: Put p := f 0 (T x) and observe that the only claim left to checking is the inclusion ext ∂ (p ◦ T ) ⊂ (ext ∂ (p)) ◦ T. Take l ∈ ext ∂ (p ◦ T ). Since ext ∂ (p ◦ T ) ⊂ ∂ (p ◦ T ) ⊂ ∂ (p) ◦ T , there is some f in ∂ (p) such that l = f ◦ T . Denote by f0 the restriction of f to Y0 := im T and notice that f0 ∈ ext ∂ (p ◦ ι), with ι the identical imbedding of Y0 into Y . Now in Y := Y ×R consider Y+ := 0 = 0 OO and introduce the space Y0 := Y0 ×R. Note that Y+ ∩ Y0 = 0 = 0 OO p ◦ ι). Applying 3.6.4 with X := Y0 , l := f0 , and p := p ◦ ι, observe that Tf0 is a discrete functional on Y0 ; moreover, Y0 is massive in Y (cf. the proof of 3.5.3). By 3.3.8, find a discrete extension S ∈ Y# of Tf0 . Evidently, S = Tg , where g(y) := S(−y, 0) for y ∈ Y . Appealing again to 3.6.4, infer that g ∈ ext ∂ (p). By construction, l(x) = f (T x) = f0 (T x) = g(T x) for all x ∈ X. Finally, l ∈ (ext ∂ (p)) ◦ T . B 3.6.6. Corollary. If p : X0 = 0 O R is a sublinear functional then for every x in X there is an extreme functional l, a member of ext ∂ (p), such that l(x) = p(x). C From 3.6.5 it is easy that ext ∂ (p) 6= ∅ for every p (cf. 3.5.6). Using this and 3.4.12, choose l in ext ∂x (p0 (x)). Applying 3.5.2 (2) and 3.5.2 (4), obtain l ∈ ext ∂x (p). By 3.6.3 (1), ∂x (p) is extreme in ∂ (p). Finally, 3.6.3 (2) implies that l is an extreme point of ∂ (p). B

Convex Analysis

35 3.7. The Balanced Hahn–Banach Theorem

3.7.1. Definition. Let (X, F, +, · ) be a vector space over a basic field F. The vector space (X, R, +, · |R ×X ) is called the real carrier or realification of (X, F, +, · ) and is denoted by XR . 3.7.2. Definition. Given a linear functional f on a vector space X, define the mapping Re f : x 7→ Re f (x) (x ∈ X). The real part map or realifier is the mapping Re : (X # )R 0 = 0 O XR )# . 3.7.3. The real part map Re is a linear operator and, moreover, an isomorphism of (X # )R onto (XR )# . C Only the case F := C needs verifying, because Re is the identity mapping when F := R. Undoubtedly, Re is a linear operator. Check that Re is a monomorphism and an epimorphism simultaneously (cf. 2.3.2). If Re f = 0 then 0 = Re f (ix) = Re(if (x)) = Re(i(Re f (x) + i Im f (x))) = − Im f (x). Hence, f = 0 and so Re is a monomorphism. Now take g ∈ (XR )# and put f (x) := g(x)−ig(ix). Evidently, f ∈ L (XR , C R ) and Re f (x) = g(x) for x ∈ X. It is sufficient to show that f (ix) = if (x), because this equality implies f ∈ X # . Straightforward calculation shows f (ix) = g(ix) + ig(x) = i(g(x) − ig(ix)) = if (x), which enables us to conclude that Re is also an epimorphism. B 3.7.4. Definition. The lear trap map or complexifier is the inverse Re−1 : (XR )# 0 = 0 O X # )R of the real part map. 3.7.5. Remark. By 3.7.3, for the complex scalar field Re−1 g : x 7→ g(x) − ig(ix) (g ∈ (XR )# , x ∈ X). In the case of the reals, F := R, the complexifier Re−1 is the identity operator. 3.7.6. Definition. Let (X, F, +, · ) be a vector space over F. A seminorm on X is a function p : X0 = 0 O R· such that dom p 6= ∅ and p(λ1 x1 + λ2 x2 ) ≤ |λ1 |p(x1 ) + |λ2 |p(x2 ) for x1 , x2 ∈ X and λ1 , λ2 ∈ F. 3.7.7. Remark. A seminorm is a sublinear functional (on the real carrier of the space in question).

36

Chapter 3

3.7.8. Definition. If p : X0 = 0 O R· is a seminorm then the balanced subdifferential of p is the set |∂ |(p) := {l ∈ X # : |l(x)| ≤ p(x) for all x ∈ X}. 3.7.9. Balanced Subdifferential Lemma. Let p : X0 = 0 O R· be a seminorm. Then |∂ |(p) = Re−1 (∂ (p)); Re (|∂ |(p)) = ∂ (p) for the subdifferentials |∂ |(p) and ∂ (p) of p. C If F := R then the equality |∂ |(p) = ∂ (p) is easy. Furthermore, in this case Re is the identity operator. Let F := C. If l ∈ |∂ |(p) then (Re l)(x) = Re l(x) ≤ |l(x)| ≤ p(x) for all x ∈ X; i.e., Re (|∂ |(p)) ⊂ ∂ (p). Take g ∈ ∂ (p) and put f := Re−1 g. If f (x) = 0 then |f (x)| ≤ p(x); for f (x) 6= 0 set θ := |f (x)|/f (x). Thereby |f (x)| = θf (x) = f (θx) = Re f (θx) = g(θx) ≤ p(θx) = |θ|p(x) = p(x), since |θ| = 1. Finally, f ∈ |∂ |(p). B 3.7.10. Let X be a vector space, let p : X0 = 0 O R be a seminorm, and let X0 be a subspace of X. The asymmetric balanced Hahn–Banach formula holds: |∂ |(p + δ(X0 )) = |∂ |(p) + |∂ |(δ(X0 )). C From 3.7.9, 3.5.5, and the results of Section 3.1 obtain |∂ |(p + δ(X0 )) = Re−1 (∂ (p + δ(X0 ))) = Re−1 (∂ (p) + ∂ (δ(X0 ))) = Re−1 (∂ (p)) + Re−1 (∂ (δ(X0 ))) = |∂ |(p) + |∂ |(δ(X0 )). . 3.7.11. If X and Y are vector spaces, T ∈ L (X, Y ) is a linear operator, and p : Y 0 = 0 O R is a seminorm; then p ◦ T is also a seminorm and, moreover, |∂ |(p ◦ T ) = |∂ |(p) ◦ T. C Applying 2.3.8 and 3.7.10, successively infer that |∂ |(p ◦ T ) = |∂ |(p + δ(im T )) ◦ T = (|∂ |(p) + |∂ |(δ(im T ))) ◦ T = |∂ |(p) ◦ T + |∂ |(δ(im T )) ◦ T = |∂ |(p) ◦ T. . 3.7.12. Remark. If T is the identical embedding of a subspace and the ground field is C then 3.7.11 is referred to as the Sukhomlinov–Bohnenblust–Sobczyk Theorem. 3.7.13. Balanced Hahn–Banach Let X be a vector space. Assume further that p : X0 = 0 O R is a seminorm and X0 is a subspace of X. If l0 is a linear functional given on X0 such that |l0 (x0 )| ≤ p(x0 ) for x0 ∈ X0 , then there is a linear functional l on X such that l(x0 ) = l0 (x0 ) whenever x0 ∈ X0 and |l(x)| ≤ p(x) for all x ∈ X. CB

Convex Analysis

37

3.8. The Minkowski Functional and Separation 3.8.1. Definition. Let R stand for the extended real axis or extended reals (i.e., R denotes R· with the least element −∞ adjoined formally). If X is an arbitrary set and f : X0 = 0 O R is a mapping; then, given t ∈ R, put {f ≤ t} := {x ∈ X : f (x) ≤ t}; {f = t} := f −1 (t); {f < t} := {f ≤ t} \ {f = t}. Every set of the form {f ≤ t}, {f = t}, and {f < t} is a level set or Lebesgue set of f . 3.8.2. Function Recovery Lemma. Let T ⊂ R and let t 7→ Ut (t ∈ T ) be a family of subsets of X. There is a function f : X0 = 0 O R such that {f < t} ⊂ Ut ⊂ {f ≤ t} (t ∈ T ) if and only if the mapping t 7→ Ut increases. C ⇒: Suppose that T contains at least two elements s and t. (Otherwise there is nothing left to proof.) If s < t then Us ⊂ {f ≤ s} ⊂ {f < t} ⊂ Ut . ⇐: Define a mapping f : X0 = 0 O R by setting f (x) := inf{t ∈ T : x ∈ Ut }. If {f < t} is empty for t ∈ T then all is clear. If x ∈ {f < t} then f (x) < +∞, and so there is some s ∈ T such that x ∈ Us and s < t. Thus, {f < t} ⊂ Us ⊂ Ut . Continuing, for x ∈ Ut find f (x) ≤ t; i.e., Ut ⊂ {f ≤ t}. B 3.8.3. Function Comparison Lemma. Let f, g : X0 = 0 O R be functions defined by (Ut )t∈T and (Vt )t∈T as follows: {f < t} ⊂ Ut ⊂ {f ≤ t}; {g < t} ⊂ Vt ⊂ {g ≤ t} (t ∈ T ). If T is dense in R (i.e., (∀ r, t ∈ R, r < t) (∃ s ∈ T ) (r < s < t)) then the inequality X f ≤ g (in R ; i.e., f (x) ≤ g(x) for x ∈ X) holds if and only if (t1 , t2 ∈ T & t1 < t2 ) ⇒ Vt1 ⊂ Ut2 . C ⇒: This is immediate from the inclusions Vt1 ⊂ {g ≤ t1 } ⊂ {f ≤ t1 } ⊂ {f < t2 } ⊂ Ut2 . ⇐: Assume that g(x) 6= +∞ (if not, f (x) ≤ g(x) for obvious reasons). Given t ∈ R such that g(x) < t < +∞, choose t1 , t2 ∈ T so as to satisfy the conditions g(x) < t1 < t2 < t. Now x ∈ {g < t1 } ⊂ Vt1 ⊂ Ut2 ⊂ {f ≤ t2 } ⊂ {f < t}. Thus, f (x) < t. Since t is arbitrary, obtain f (x) ≤ g(x). B

38

Chapter 3

3.8.4. Corollary. If T is dense in R and the mapping t 7→ Ut (t ∈ T ) increases then there is a unique function f : X0 = 0 O R such that {f < t} ⊂ Ut ⊂ {f ≤ t} (t ∈ T ). Moreover, the level sets of f may be presented as follows: {f < t} = ∪ {Us : s < t, s ∈ T }; {f ≤ t} = ∩{Ur : t < r, r ∈ T } (t ∈ R). C It is immediate from 3.8.2 and 3.8.3 that f exists and is unique. If s < t and s ∈ T then Us ⊂ {f ≤ s} ⊂ {f < t}. If now f (x) < t then, since T is dense, there is an element s in T such that f (x) < s < t. Therefore, f ∈ {f < s} ⊂ Us , which proves the formula for {f < t}. Suppose that r > t, r ∈ T . Then {f ≤ t} ⊂ {f < r} ⊂ Ur . In turn, if x ∈ Ur for r ∈ T, r > t; then f (x) ≤ r for all r > t. Hence, f (x) ≤ t. B 3.8.5. Let X be a vector space and let S be a conical segment in X. Given t ∈ R, put Ut := ∅ if t < 0 and Ut := tS if t ≥ 0. The mapping t 7→ Ut (t ∈ R) increases. C If 0 ≤ t1 < t2 and x ∈ t1 S then x ∈ (t1 /t2 ) t2 S. Hence, x ∈ t2 S. B 3.8.6. Definition. The Minkowski functional or the gauge function or simply the gauge of a conical segment S is a functional pS : X0 = 0 O R such that {pS < t} ⊂ tS ⊂ {pS ≤ t} (t ∈ R+ ) and {p < 0} = ∅. (Such a functional exists and is unique by 3.8.2, 3.8.4, and 3.8.5.) In other words, pS (x) = inf{t > 0 : x ∈ tS} (x ∈ X). 3.8.7. Gauge Theorem. The Minkowski functional of a conical segment is a sublinear functional assuming positive values. Conversely, if p is a sublinear functional assuming positive values then the sets {p < 1} and {p ≤ 1} are conical segments. Moreover, p is the Minkowski functional of each conical segment S such that {p < 1} ⊂ S ⊂ {p ≤ 1}. C Consider a conical segment S and its Minkowski functional pS . Let x ∈ X. The inequality pS (x) ≥ 0 is evident. Take α > 0. Then  pS (αx) = inf{t > 0 : αx ∈ tS} = inf t > 0 : x ∈ t /α S = inf{αβ > 0 : x ∈ βS, β > 0} = α inf{β > 0 : x ∈ βS} = αpS (x).

Convex Analysis

39

To start checking that pS is subadditive, take x1 , x2 ∈ X. Noting the inclusion t1 S + t2 S ⊂ (t1 + t2 )S for t1 , t2 > 0, in view of the identity  t1 x1 + t2 x2 = (t1 + t2 )

 t1 t2 x1 + x2 , t1 + t2 t1 + t2

successively infer that pS (x1 + x2 ) = inf{t > 0 : x1 + x2 ∈ tS} ≤ inf{t : t = t1 + t2 ; t1 , t2 > 0, x1 ∈ t1 S, x2 ∈ t2 S} = inf{t1 > 0 : x1 ∈ t1 S} + inf{t2 > 0 : x2 ∈ t2 S} = pS (x1 ) + pS (x2 ). Let p : X0 = 0 O R· be an arbitrary sublinear functional with positive values and let {p < 1} ⊂ S ⊂ {p ≤ 1}. Given t ∈ R+ , put Vt := {p < t} and Ut := tS; given t < 0, put Vt := Ut := ∅. Plainly, {pS < t} ⊂ Ut ⊂ {pS ≤ t};

{p < t} ⊂ Vt ⊂ {p ≤ t}

for t ∈ R. If 0 ≤ t1 < t2 then Vt1 = {p < t1 } = t1 {p < 1} ⊂ t1 S = Ut1 ⊂ Ut2 . Moreover, Ut1 ⊂ t1 {p ≤ 1} ⊂ {p ≤ t1 } ⊂ {p < t2 } ⊂ Vt2 . Therefore, by 3.8.3 and 3.8.4, p = pS . B 3.8.8. Remark. A conical segment S in X is absorbing in X if and only if dom pS = X. Also, pS is a seminorm whenever S is absolutely convex. Conversely, for every seminorm p the sets {p < 1} and {p ≤ 1} are absolutely convex. CB 3.8.9. Definition. A subspace H of a vector space X is a hypersubspace in X if X/H is isomorphic to the ground field of X. An element of X/H is called a hyperplane in X parallel to H. By a hyperplane in X we mean an affine variety parallel to some hypersubspace of X. An affine variety H is a hyperplane if H −h is a hypersubspace for some (and, hence, for every) h in H. If necessary, a hyperplane in the real carrier of X is referred to as a real hyperplane in X. 3.8.10. Hyperplanes in X are exactly level sets of nonzero elements of X # . CB 3.8.11. Separation Theorem. Let X be a vector space. Assume further that U is a nonempty convex set in X and L is an affine variety in X. If L ∩ U = ∅ then there is a hyperplane H in X such that H ⊃ L and H ∩ core U = ∅. C Without loss of generality, it may be supposed that core U 6= ∅ (otherwise there is nothing left unproven) and, moreover, 0 ∈ core U . Take x ∈ L and put X0 := L − x. Consider the quotient space X/X0 and the corresponding coset mapping ϕ : X0 = 0 O /X0 . Applying 3.1.8 and 3.4.10, observe that ϕ(U ) is an absorbing

40

Chapter 3

conical segment. Hence, by 3.8.7 and 3.8.8, the domain of the Minkowski functional p := pϕ(U ) is the quotient X/X0 ; moreover, ϕ(core U ) ⊂ core ϕ(U ) ⊂ {p < 1} ⊂ ϕ(U ). In particular, this entails the inequality p(ϕ(x)) ≥ 1, since ϕ(x) 6∈ ϕ(U ). Using 3.5.6, find a functional f in ∂x (p ◦ ϕ); now the Hahn–Banach Theorem implies f ∈ ∂x (p ◦ ϕ) = ∂ ϕ(x) (p) ◦ ϕ. Put H := {f = p ◦ ϕ(x)}. It is clear that H is a real hyperplane and, undoubtedly, H ⊃ L. Appealing to 3.5.2 (1), conclude that H ∩ core U = ∅. Now let f := Re−1 f and H := {f = f (x)}. There is no denying that L ⊂ H ⊂ H. Thus, the hyperplane H provides us with what was required. B 3.8.12. Remark. Under the hypotheses of the Separation Theorem, it may be assumed that core U ∩ L = ∅. Note also that Theorem 3.8.11 is often referred to as the Hahn–Banach Theorem in geometric form or as the Minkowski–Ascoli–Mazur Theorem. 3.8.13. Definition. Let U and V be sets in X. A real hyperplane H in X separates U and V if these sets lie in the different halfspaces defined by H; i.e., if there is a presentation H = {f = t}, where f ∈ (XR )# and t ∈ R, such that V ⊂ {f ≤ t} and U ⊂ {f ≥ t} := {−f ≤ −t}. 3.8.14. Eidelheit Separation Theorem. If U and V are convex sets such that core V 6= ∅ and U ∩ core V = ∅, then there is a hyperplane separating U and V and disjoint from core V . CB Exercises 3.1. Show that a hyperplane is precisely an affine set maximal by inclusion and other than the whole space. 3.2. Prove that every affine set is an intersection of hyperplanes. 3.3. Prove that the complement of a hyperplane to a real vector space consists of two convex sets each of which coincides with its own core. The sets are named open halfspaces. The union of an open halfspace with the corresponding hyperplane is called a closed halfspace. Find out how a halfspace can be prescribed. 3.4. Find possible presentations of the elements of the convex hull of a finite set. What use can be made of finite dimensions? 3.5. Given sets S1 and S2 , let S := so are S1 and S2 .

S 0≤λ≤1

λS1 ∩(1−λ)S2 . Prove that S is convex whenever

3.6. Calculate the Minkowski functional of a halfspace or a cone and of the convex hull of the union or intersection of conical segments.

Convex Analysis

41

3.7. Let S := {p + q ≤ 1}, where p and q are the Minkowski functionals of the conical segments Sp and Sq . Express S via Sp and Sq . 3.8. Describe sublinear functionals with domain RN . 3.9. Calculate the subdifferential of the upper envelope of a finite set of linear or sublinear functionals. 3.10. Let p and q be sublinear functionals in general position, i.e. such that dom p−dom q = dom q − dom p. Prove the symmetric Hahn–Banach formula (cf. 3.5.7): ∂ (p + q) = ∂ (p) + ∂ (q). 3.11. Let p, q : X0 = 0 O R be total (= everywhere-defined) sublinear functionals on X. Then the equality holds: ∂ (p ∨ q) = co(∂ (p) ∪ ∂ (q)). 3.12. Find the Minkowski functional of a ball in a Hilbert space whose center of symmetry is not necessarily the zero of the space. 3.13. A symmetric square 2 × 2-matrix is called positive, provided that its eigenvalues are positive. Does the resulting order in the space of such matrices agree with vector structure? Does it define the structure of a Kantorovich space? 3.14. Each ordered vector space admits a nonzero positive linear functional, doesn’t it? 3.15. What are the means for transforming RN into a Kantorovich space? 3.16. Under what conditions does the claim of the Hahn–Banach Theorem in analytical form hold for a partial (= not-everywhere-defined) sublinear functional? 3.17. Find the extreme points of the subdifferential of the conventional norm on l∞ . 3.18. Find possible generalizations of the Hahn–Banach Theorem for a mapping acting into a Kantorovich space. 3.19. Given a set C in a space X, define the H¨ ormander transform H(C) of C as H(C) = {(x, t) ∈ X × R : x ∈ tC}. Study the properties of the H¨ ormander transform on the collection of all convex sets.

Chapter 4 An Excursus into Metric Spaces

4.1. The Uniformity and Topology of a Metric Space 4.1.1. Definition. A mapping d : X 2 0 = 0 O R+ is a metric on X if (1) d(x, y) = 0 ⇔ x = y; (2) d(x, y) = d(y, x) (x, y ∈ X); (3) d(x, y) ≤ d(x, z) + d(z, y) (x, y, z ∈ X). The real d(x, y) is usually referred to as the distance between x and y. The pair (X, d) is a metric space. In this situation, it is convenient to take the liberty of calling the underlying set X a metric space. An element of a metric space X is also called a point of X. 4.1.2. A mapping d : X 2 0 = 0 O R+ is a metric if and only if (1) {d ≤ 0} = IX ; (2) {d ≤ t} = {d ≤ t}−1 (t ∈ R+ ); (3) {d ≤ t1 } ◦ {d ≤ t2 } ⊂ {d ≤ t1 + t2 } (t1 , t2 ∈ R+ ). C Items 4.1.2 (1)–4.1.2 (3) reformulate 4.1.1 (1)–4.1.1 (3). B 4.1.3. Definition. Let (X, d) be a metric space and take ε ∈ R+ \0, a strictly positive real. The relation Bε := Bd,ε := {d ≤ ε} is the closed cylinder of size ε. The ◦

◦

set B ε := B d,ε := {d < ε} is the open cylinder of size ε. The image Bε (x) of a point x under the relation Bε is called the closed ball with center x and radius ε. By analogy, ◦

the set B ε (x) is the open ball with center x and radius ε. 4.1.4. In a nonempty metric space open cylinders as well as closed cylinders form bases for the same filter. CB

An Excursus into Metric Spaces

43

4.1.5. Definition. The filter on X 2 with the filterbase of all cylinders of a nonempty metric space (X, d) is the metric uniformity on X. It is denoted by UX , Ud , or even U , if the space under consideration is implied. Given X := ∅, put UX := {∅}. An element of UX is an entourage on X. 4.1.6. If U is a metric uniformity then (1) U ⊂ fil {IX }; (2) U ∈ U ⇒ U −1 ∈ U ; (3) (∀ U ∈ U ) (∃ V ∈ U ) V ◦ V ⊂ U ; (4) ∩{U : U ∈ U } = IX . CB 4.1.7. Remark. Property 4.1.6 (4) reflecting 4.1.1 (1) is often expressed as follows: “U is a Hausdorff or separated uniformity.” 4.1.8. Given a space X with uniformity UX , put τ (x) := {U (x) : U ∈ U }. Then τ (x) is a filter for every x in X. Moreover, (1) τ (x) ⊂ fil {x}; (2) (∀ U ∈ τ (x)) (∃ V ∈ τ (x) & V ⊂ U ) (∀ y ∈ V ) V ∈ τ (y). CB 4.1.9. Definition. The mapping τ : x 7→ τ (x) is the metric topology on X. An element of τ (x) is a neighborhood of x or about x. More complete designations for the topology are also in use: τX , τ (U ), etc. 4.1.10. Remark. All closed balls centered at x form a base for the neighborhood filter of x. The same is true of open balls. Note also that there are disjoint (= nonintersecting) neighborhoods of different points in X. This property, ciphered in 4.1.6 (4), reads: “τX is a Hausdorff or separated topology.” 4.1.11. Definition. A subset G of X is an open set in X whenever G is a neighborhood of its every point (i.e., G ∈ Op(τ ) ⇔ ((∀ x ∈ G) G ∈ τ (x))). A subset F of X is a closed set in X whenever its complement to X is open (in symbols, F ∈ Cl(τ ) ⇔ (X \ F ∈ Op(τ ))). 4.1.12. The union of a family of open sets and the intersection of a finite family of open sets are open. The intersection of a family of closed sets and the union of a finite family of closed sets are closed. CB 4.1.13. Definition. Given a subset U of X, put ◦

int U := U := ∪{G ∈ Op(τX ) : G ⊂ U }; cl U := U := ∩{F ∈ Cl(τX ) : F ⊃ U }.

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Chapter 4

The set int U is the interior of U and its elements are interior points of U . The set cl U is the closure of U and its elements are adherent to U . The exterior of U is the interior of X \ U ; the elements of the former are exterior to U . A boundary point of U is by agreement a point of X neither interior nor exterior to U . The collection of all boundary points of U is called the boundary of U or the frontier of U and denoted by fr U or ∂U . 4.1.14. A set U is a neighborhood about x if and only if x is an interior point of U . CB

An Excursus into Metric Spaces

45

4.1.15. Remark. In connection with 4.1.14, the set Op(τX ) is also referred to as the topology of the space X under study, since τX is uniquely determined from Op(τX ). The same relates to Cl(τX ), the collection of all closed sets in X. 4.1.16. Definition. A filterbase B on X converges to x in X or x is a limit of B (in symbols, B0 = 0 O ) if fil B is finer than the neighborhood filter of x; i.e., fil B ⊃ τ (x). 4.1.17. Definition. A net or (generalized) sequence (xξ )ξ∈„ converges to x (in symbols, xξ 0 = 0 O ) if the tail filter of (xξ ) converges to x. Other familiar terms and designations are freely employed. For instance, x = limξ xξ and x is a limit of (xξ ) as ξ ranges over „. 4.1.18. Remark. A limit of a filter, as well as a limit of a net, is unique in a metric space X. This is another way of expressing the fact that the topology of X is separated. CB 4.1.19. For a nonempty set U and a point x the following statements are equivalent: (1) x is an adherent point of U ; (2) there is a filter F such that F 0 = 0 O and U ∈ F ; (3) there is a sequence (xξ )ξ∈„ whose entries are in U and which converges to x. C (1) ⇒ (2): Since x is not exterior to U , the join F := τ (x)∨fil {U } is available of the pair of the filters τ (x) and fil {U }. (2) ⇒ (3): Let F 0 = 0 O and U ∈ F . Direct F by reverse inclusion. Take xV ∈ V ∩ U for V ∈ F . It is clear that xV 0 = 0 O . (3) ⇒ (1): Let V be a closed set. Take a sequence (xξ )ξ∈„ in V such that xξ 0 = 0 O . In this case it suffices to show that x ∈ V , which is happily evident. Indeed, were x in X \ V we would find at least one ξ ∈ „ such that xξ ∈ X \ V . B 4.1.20. Remark. It may be assumed that F has a countable base in 4.1.19 (2), and „ := N in 4.1.19 (3). This property is sometimes formulated as follows: “In metric spaces the first axiom of countability is fulfilled.” 4.2. Continuity and Uniform Continuity 4.2.1. If f : X0 = 0 O and τX and τY are topologies on X and Y then the following conditions are equivalent:

46

Chapter 4 (1) G ∈ Op(τY ) ⇒ f −1 (G) ∈ Op(τX ); (2) F ∈ Cl(τY ) ⇒ f −1 (F ) ∈ Cl(τX ); (3) f (τX (x)) ⊃ τY (f (x)) for all x ∈ X; (4) (x ∈ X & F 0 = 0 O ) ⇒ (f (F )0 = 0 O (x)) for a filter F ; (5) f (xξ )0 = 0 O (x) for every point x and every sequence (xξ ) conver-

gent to x. C The equivalence (1) ⇔ (2) follows from 4.1.11. It remains to demonstrate that (1) ⇒ (3) ⇒ (4) ⇒ (5) ⇒ (2). (1) ⇒ (3): If V ∈ τY (f (x)) then W := int V ∈ Op(τY ) and f (x) ∈ W . Hence, −1 f (W ) ∈ Op(τX ) and x ∈ f −1 (W ). In other words, f −1 (W ) ∈ τX (x) (see 4.1.14). Moreover, f −1 (V ) ⊃ f −1 (W ) and, consequently, f −1 (V ) ∈ τX (x). Finally, V ⊃ f (f −1 (V )). (3) ⇒ (4): Given F 0 = 0 O , by Definition 4.1.16 fil F ⊃ τX (x). From the hypothesis infer that f (F ) ⊃ f (τX (x)) ⊃ τY (f (x)). Appealing to 4.1.16, again reveal the sought instance of convergence, f (F )0 = 0 O (x). (4) ⇒ (5): The image of the tail filter of (xξ )ξ∈„ under f is coarser than the tail filter of (f (xξ ))ξ∈„ . (5) ⇒ (2): Take a closed set F in Y . If F = ∅ then f −1 (F ) is also empty and, hence, closed. Assume F nonempty and let x be an adherent point of f −1 (F ). Consider a sequence (xξ )ξ∈„ converging to x and consisting of points in f −1 (F ) (the claim of 4.1.19 yields existence). Then f (xξ ) ∈ F and f (xξ )0 = 0 O (x). Another citation of 4.1.19 guarantees that f (x) ∈ F and, consequently, x ∈ f −1 (F ). B 4.2.2. Definition. A mapping satisfying one (and hence all) of the equivalent conditions 4.2.1 (1)–4.2.1 (5) is continuous. If 4.2.1 (5) holds at a fixed point x then f is said to be continuous at x. Thus, f is continuous on X whenever f is continuous at every point of X. 4.2.3. Every composition of continuous mappings is continuous. C Apply 4.2.1 (5) thrice. B 4.2.4. Let f : X0 = 0 O and let UX and UY be uniformities on X and Y . The following statements are equivalent: (1) (∀ V ∈ UY ) (∃ U ∈ UX ) ((∀ x, y)(x, y) ∈ U ⇒ (f (x), f (y)) ∈ V ); (2) (∀ V ∈ UY ) f −1 ◦ V ◦ f ∈ UX ; (3) f × (UX ) ⊃ UY , with f × : X 2 0 = 0 O defined as f × : (x, y) 7→ (f (x), f (y)); (4) (∀ V ∈ UY ) f ×−1 (V ) ∈ UX ; i.e., f ×−1 (UY ) ⊂ UX .

An Excursus into Metric Spaces

47

C By 1.1.10, given U ⊂ X 2 and V ⊂ Y 2 , observe that [ f −1 ◦ V ◦ f = f −1 (v1 ) × f −1 (v2 ) (v1 ,v2 )∈V

= {(x, y) ∈ X 2 : (f (x), f (y)) ∈ V } = f ×−1 (V ); [ f ◦ U ◦ f −1 = f (u1 ) × f (u2 ) (u1 ,u2 )∈U

= {(f (u1 ), f (u2 )) : (u1 , u2 ) ∈ U } = f × (U ). . 4.2.5. Definition. A mapping f : X0 = 0 O satisfying one (and hence all) of the equivalent conditions 4.2.4 (1)–4.2.4 (4) is called uniformly continuous (the term “uniform continuous” is also in common parlance). 4.2.6. Every composition of uniformly continuous mappings is uniformly continuous. C Consider f : X0 = 0 O , g : Y 0 = 0 O and h := g ◦ f : X0 = 0 O . Plainly, h× (x, y) = (h(x), h(y)) = (g(f (x)), g(f (y))) = g × (f (x), f (y)) = g × ◦ f × (x, y) for all x, y ∈ X. Hence, by 4.2.4 (3) h× (UX ) = g × (f × (UX )) ⊃ g × (UY ) ⊃ UZ , meaning that h is uniformly continuous. B 4.2.7. Every uniformly continuous mapping is continuous. CB 4.2.8. Definition. Let E be a set of mappings from X into Y and let UX and UY be the uniformities in X and Y . The set E is equicontinuous if \ (∀ V ∈ UY ) f −1 ◦ V ◦ f ∈ UX . f ∈E

4.2.9. Every equicontinuous set consists of uniformly continuous mappings. Every finite set of uniformly continuous mappings is equicontinuous. CB 4.3. Semicontinuity 4.3.1. Let (X1 , d1 ) and (X2 , d2 ) be metric spaces, and := X × X. Given x := (x1 , x2 ) and y := (y1 , y2 ), put d(x, y) := d1 (x1 , y1 ) + d2 (x2 , y2 ). Then d is a metric on . Moreover, for every x := (x1 , x2 ) in the presentation holds: τ(x) = fil{U1 × U2 : U1 ∈ τX1 (x1 ), U2 ∈ τX2 (x2 )}. /. 4.3.2. Definition. The topology τ is called the product of τX1 and τX2 or the product topology of X1 × X2 . This topology on X1 × X2 is denoted by τX1 × τX2 .

48

Chapter 4

4.3.3. Definition. A function f : X0 = 0 O R· is lower semicontinuous if its epigraph 0 = 0 OO is closed in the product topology of X × R. 4.3.4. Examples. (1) A continuous real-valued function f : X0 = 0 O R is lower semicontinuous. (2) If fξ : X0 = 0 O R· is lower semicontinuous for all ξ ∈ „, then the upper envelope f (x) := sup{fξ (x) : ξ ∈ „} (x ∈ X) is also lower semicontinuous. A simple reason behind this is the equality 0 = 0 OO = ∩ξ∈„ 0 = 0 OO ξ . 4.3.5. A function f : X0 = 0 O R· is lower semicontinuous if and only if x ∈ X ⇒ f (x) = lim O inf f (y). y0=0 Here, as usual, lim O inf f (y) := lim f (y) := sup inf f (U ) U ∈τ (x) y0=0 O

y0=0

is the lower limit of f at x (with respect to τ (x)). C ⇒: If x 6∈ dom f then (x, t) 6∈ 0 = 0 OO for all t ∈ R. Hence, there is a neighborhood Ut of x such that inf f (Ut ) > t. This implies that limy0=0 O inf f (y) = +∞ = f (x). Suppose that x ∈ dom f . Then inf f (V ) > −∞ for some neighborhood V of x. Choose ε > 0 and for an arbitrary U in τ (x) included in V find xU in U so that inf f (U ) ≥ f (xU ) − ε. By construction, xU ∈ dom f and, moreover, xU 0 = 0 O (by implication, the set of neighborhoods of x is endowed with the conventional order by inclusion, cf. 1.3.1). Put tU := inf f (U ) + ε. It is clear that tU 0 = 0 O := limy0=0 O inf f (y) + ε. Since (xU , tU ) ∈ 0 = 0 OO , from the closure property of 0 = 0 OO obtain (x, t) ∈ 0 = 0 OO . Thus, lim O inf f (y) + ε ≥ f (x) ≥ lim O inf f (y). y0=0

y0=0

⇐: If (x, t) 6∈ 0 = 0 OO then t < lim O inf f (y) = sup inf f (U ). U ∈τ (x) y0=0 Therefore, inf f (U ) > t for some neighborhood U of x. It follows that the complement of 0 = 0 OO to X × R is open. B 4.3.6. Remark. The property, stated in 4.3.5, may be accepted as an initial definition of lower semicontinuity at a point.

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49

4.3.7. A function f : X0 = 0 O R is continuous if and only if both f and −f are lower semicontinuous. CB 4.3.8. A function f : X0 = 0 O R· is lower semicontinuous if and only if for every t ∈ R the level set {f ≤ t} is closed. C ⇒: If x 6∈ {f ≤ t} then t < f (x). By 4.3.5, t < inf f (U ) in a suitable neighborhood U about x. In other words, the complement of {f ≤ t} to X is open. ⇐: Given limy0=0 O inf f (y) ≤ t < f (x) for some x ∈ X and t ∈ R, choose ε > 0 such that t + ε < f (x). Repeating the argument of 4.3.5, given U ∈ τ (x) take a point xU in U ∩ {f ≤ inf f (U ) + ε}. Undoubtedly xU ∈ {f ≤ t + ε} and xU 0 = 0 O , a contradiction. B 4.4. Compactness 4.4.1. Definition. A subset C of X is called a compact set whenever for every subset E of Op(τX ) with the property C ⊂ ∪{G : G ∈ E } there is a finite subset E0 in E such that still C ⊂ ∪{G : G ∈ E0 }. 4.4.2. Remark. Definition 4.4.1 is verbalized as follows: “A set is compact if its every open cover has a finite subcover.” The terms “covering” and “subcovering” are also in current usage. 4.4.3. Every closed subset of a compact set is also compact. Every compact set is closed. CB 4.4.4. Remark. With regard to 4.4.3, it stands to reason to use the term “relatively compact set” for a set whose closure is compact. 4.4.5. Weierstrass Theorem. The image of a compact set under a continuous mapping is compact. C The inverse images of sets in an open cover of the image compose an open cover of the original set. B 4.4.6. Each lower semicontinuous real-valued function, defined on a nonempty compact set, assumes its least value; i.e., the image of a compact domain has a least element. C Suppose that f : X0 = 0 O R· and X is compact. Let t0 := inf f (X). In the case t0 = +∞ there is nothing left to proof. If t0 < +∞ then put T := {t ∈ R : t > t0 }. The set Ut := {f ≤ t} with t ∈ T is nonempty and closed. Check that ∩{Ut : t ∈ T } is not empty (then every element x of the intersection meets the claim: f (x) = inf f (X)). Suppose the contrary. Then the sets {X \ Ut : t ∈ T } compose an open cover of X. Refining a finite subcover, deduce ∩{Ut : t ∈ T0 } = ∅. However, this equality is false, since Ut1 ∩ Ut2 = Ut1 ∧t2 for t1 , t2 ∈ T . B

50

Chapter 4

4.4.7. Bourbaki Criterion. A space is compact if and only if every ultrafilter on it converges (cf. 9.4.4). 4.4.8. The product of compact sets is compact. C It suffices to apply the Bourbaki Criterion twice. B 4.4.9. Cantor Theorem. Every continuous mapping on a compact set is uniformly continuous. CB 4.5. Completeness Completeness 4.5.1. If B is a filterbase on X then {B 2 : B ∈ B} is a filterbase (and a base for the filter B × ) on X 2 . C (B1 × B1 ) ∩ (B2 × B2 ) ⊃ (B1 ∩ B2 ) × (B1 ∩ B2 ) B 4.5.2. Definition. Let UX be a uniformity on X. A filter F is called a Cauchy filter or even Cauchy (the latter might seem preposterous) if F × ⊃ UX . A net in X is a Cauchy net or a fundamental net if its tail filter is a Cauchy filter. The term “fundamental sequence” is treated in a similar fashion. 4.5.3. Remark. If V is an entourage on X and U is a subset of X then U is V -small whenever U 2 ⊂ V . In particular, U is Bε -small (or simply ε-small) if and only if the diameter of U , diam U := sup(U 2 ), is less than or equal to ε. In this connection, the definition of a Cauchy filter is expressed as follows: “A filter is a Cauchy filter if and only if it contains arbitrarily small sets.” 4.5.4. In a metric space the following conditions are equivalent: (1) every Cauchy filter converges; (2) each Cauchy net has a limit; (3) every fundamental sequence converges. C The implications (1) ⇒ (2) ⇒ (3) are obvious; therefore, we are left with establishing (3) ⇒ (1). Given a Cauchy filter F , let Un in F be a B1/n -small set. Put Vn := U1 ∩. . .∩Un and take xn ∈ Vn . Observe that V1 ⊃ V2 ⊃ . . . and diam Vn ≤ 1 /n . Hence, (xn ) is a fundamental sequence. By hypothesis it has a limit, x := lim xn . Check that F 0 = 0 O . To this end, choose n0 ∈ N such that d(xm , x) ≤ 1 /2n as m ≥ n0 . Then for all n ∈ N deduce that d(xp , y) ≤ diam Vp ≤ 1 /2n and d(xp , x) ≤ 1 /2n whenever p := n0 ∨ 2n and y ∈ Vp . It follows that y ∈ Vp ⇒ d(x, y) ≤ 1 /n ; i.e., Vp ⊂ B1/n (x). In conclusion, F ⊃ τ (x). B 4.5.5. Definition. A metric space satisfying one (and hence all) of the equivalent conditions 4.5.4 (1)–4.5.4 (3) is called complete.

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4.5.6. Cantor Criterion. A metric space X is complete if and only if every nonempty downward-filtered family of nonempty closed subsets of X whose diameters tend to zero has a point of intersection. C ⇒: If B is such family then by Definition 1.3.1 B is a filterbase. By hypothesis, B is a base for a Cauchy filter. Therefore, there is a limit: B0 = 0 O . The point x meets the claim. ⇐: Let F be a Cauchy filter. Put B := {cl V : V ∈ F }. The diameters of the sets in B tend to zero. Hence, there is a point x such that x ∈ cl V for all V ∈ F . Plainly, F 0 = 0 O . Indeed, let V be an ε /2 -small member of F and y ∈ V . Some y 0 in V is such that d(x, y 0 ) ≤ ε /2 . Therefore, d(x, y) ≤ d(x, y 0 ) + d(y 0 , y) ≤ ε. Consequently, V ⊂ Bε (x) and so Bε (x) ∈ F . B 4.5.7. Nested Ball Theorem. A metric space is complete if and only if every nested (= decreasing by inclusion) sequence of balls whose radii tend to zero has a unique point of intersection. CB 4.5.8. The image of a Cauchy filter under a uniformly continuous mapping is a Cauchy filter. C Let a mapping f act from a space X with uniformity UX into a space Y with uniformity UY and let F be a Cauchy filter on X. If V ∈ UY then, by Definition 4.2.5, f −1 ◦ V ◦ f ∈ UX (cf. 4.2.4 (2)). Since F is a Cauchy filter, U 2 ⊂ f −1 ◦ V ◦ f for some U ∈ F . It turns out that f (U ) is V -small. Indeed, f (U )2 =

[ (u1 ,u2

2

=f ◦U ◦f

−1

⊂ f ◦ (f

−1

f (u1 ) × f (u2 )

)∈U 2

◦ V ◦ f ) ◦ f −1 = (f ◦ f −1 ) ◦ V ◦ (f ◦ f −1 ) ⊂ V,

because, by 1.1.6, f ◦ f −1 = Iim f ⊂ IY . B 4.5.9. The product of complete spaces is complete. C The claim is immediate from 4.5.8 and 4.5.4. B 4.5.10. Let X0 be dense in X (i.e., cl X0 = X). Assume further that f0 : X0 0 = 0 O is a uniformly continuous mapping from X0 into some complete space Y . Then there is a unique uniformly continuous mapping f : X0 = 0 O extending f0 ; i.e., f |X0 = f0 . C For x ∈ X, the filter Fx := {U ∩ X0 : U ∈ τX (x)} is a Cauchy filter on X0 . Therefore, 4.5.8 implies that f0 (FX ) is a Cauchy filter on Y . By the completeness of Y , there is a limit y ∈ Y ; that is, f0 (Fx )0 = 0 O . Moreover, this limit is unique (cf. 4.1.18). Define f (x) := y. Checking uniform continuity for f readily completes the proof. B

52

Chapter 4

4.5.11. Definition. An isometry or isometric embedding or isometric mape is a mapping f : (X, d)0 = 0 O X, e de) such that d = de ◦ f × . ping of X into X e if f is an isometry of X into X e and, moreover, A mapping f is an isometry onto X e e im f = X. The expressions, “an isometry of X and X” or “an isometry between X e and the like, are also in common parlance. and X” 4.5.12. Hausdorff Completion Theorem. If (X, d) is a metric space then e de) and an isometry ι : (X, d)0 = 0 O X, e de) there are a complete metric space (X, e de). The space (X, e de) is unique to within isometry onto a dense subspace of (X, in the following sense: The diagram ι - (X, e de) (X, d) @ ι1 ‰ @ R @ ? e1 , de1 ) (X e1 , de1 ), where ι1 : (X, d)0 = e de)0 = 0 O X commutes for some isometry ‰ : (X, e1 , de1 ) is an isometry of X onto a dense subspace of a complete space (X e1 , de1 ). 0OX C Uniqueness up to isometry follows from 4.5.10. For, if ‰0 := ι1 ◦ ι−1 then e onto the dense subspace ι1 (X) ‰0 is an isometry of the dense subspace ι(X) of X e1 . Let ‰ be the unique extension of ‰0 to the whole of X. e It is sufficient of X e e to show that ‰ acts onto X1 . Take x e1 in X1 . This element is the limit of some sequence (ι1 (xn )), with xn ∈ X. Clearly, the sequence (xn ) is fundamental. Thus, e Let x e Proceed as (ι(xn )) is a fundamental sequence in X. e := lim ι(xn ), x e ∈ X. −1 follows: ‰(e x) = lim ‰0 (ι(xn )) = lim ι1 ◦ ι (ι(xn )) = lim ι1 (xn ) = x e1 . e We now sketch out the proof that X exists. Consider the set of all fundamental sequences in X. Define some equivalence relation in X by putting x1 ∼ x2 ⇔ e e := /∼ and d(ϕ(x d(x1 (n), x2 (n))0 = 0 O . Assign X 1 ), ϕ(x2 )) := lim d(x1 (n), x2 (n)), e de) is where ϕ : = O X is the coset mapping. An isometry ι : (X, d)0 = 0 O X, immediate: ι(x) := ϕ(n 7→ x (n ∈ N)). B e de) introduced in Theorem 4.5.12, as well 4.5.13. Definition. The space (X, as each space isomorphic to it, is called a completion of (X, d). 4.5.14. Definition. A set X0 in a metric space (X, d) is said to be complete if the metric space (X0 , d|X02 ), a subspace of (X, d), is complete. 4.5.15. Every closed subset of a complete space is complete. Every complete set is closed. CB 4.5.16. If X0 is a subspace of a complete metric space X then a completion of X0 is isometric to the closure of X0 in X.

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53

e := cl X0 and let ι : X0 0 = 0 O X be the identical embedding. It is C Let X e is evident that ι is an isometry onto a dense subspace. Moreover, by 4.5.15 X complete. Appealing to 4.5.12 ends the proof. B 4.6. Compactness and Completeness 4.6.1. A compact space is complete. CB 4.6.2. Definition. Let U be a subset of X and V ∈ UX . A set E in X is a V -net for U if U ⊂ V (E). 4.6.3. Definition. A subset U of X is a totally bounded set in X if for every V in UX there is a finite V -net for U . 4.6.4. If for every V in U a set U in X has a totally bounded V -net then U is totally bounded. C Let V ∈ UX and W ∈ UX be such that W ◦ W ⊂ V . Take a totally bounded W -net F for U ; i.e., U ⊂ W (F ). Since F is totally bounded, there is a finite W -net E for F ; that is, F ⊂ W (E). Finally, U ⊂ W (F ) ⊂ W (W (E)) = W ◦ W (E) ⊂ V (E); i.e., E is a finite V -net for U . B 4.6.5. A subset U of X is totally bounded if and only if for every V in UX there is a family U1 , . . . , Un of subsets of U such that U = U1 ∪ . . . ∪ Un and each of the sets U1 , . . . , Un is V -small. CB 4.6.6. Remark. The claim of 4.6.5 is verbalized as follows: “A set is totally bounded if and only if it has finite covers consisting of arbitrarily small sets.” 4.6.7. Hausdorff Criterion. A set is compact if and only if it is complete and totally bounded. CB 4.6.8. Let C(Q, F ) be the space of continuous functions with domain a compact set Q and range a subset of F. Furnish this space with the Chebysh¨ev metric d(f, g) := sup dF (f (x), g(x)) := sup |f (x) − g(x)| (f, g ∈ C(Q, F)); x∈Q

x∈Q

and, given θ ∈ UF , put  Uθ := (f, g) ∈ C(Q, F)2 : g ◦ f −1 ⊂ θ . Then Ud = fil {Uθ : θ ∈ UF }. CB

54

Chapter 4 4.6.9. The space C(Q, F) is complete. CB

4.6.10. Ascoli–Arzel` a Theorem. A subset E of C(Q, F ) is relatively compact if and only if E is equicontinuous and the set ∪{g(Q) : g ∈ E } is totally bounded in F. C ⇒: It is beyond a doubt that ∪{g(Q) : g ∈ E } is totally bounded. To show equicontinuity for E take θ ∈ UF and choose a symmetric entourage θ0 such that θ0 ◦ θ0 ◦ θ0 ⊂ θ. By the Hausdorff Criterion, there is a finite Uθ0 -net E 0 for E . Consider the entourage U ∈ UQ defined as \ U := f −1 ◦ θ0 ◦ f f ∈E 0

(cf. 4.2.9). Given g ∈ E and f ∈ E 0 such that g ◦ f −1 ⊂ θ0 , observe that θ0 = θ0−1 ⊃ (g ◦ f −1 )−1 = (f −1 )−1 ◦ g −1 = f ◦ g −1 . Moreover, the composition rules for correspondences and 4.6.8 imply g × (U ) = g ◦ U ◦ g −1 ⊂ g ◦ (f −1 ◦ θ0 ◦ f ) ◦ g −1 ⊂ (g ◦ f −1 ) ◦ θ0 ◦ (f ◦ g −1 ) ⊂ θ0 ◦ θ0 ◦ θ0 ⊂ θ. Since g is arbitrary, the resulting inclusion guarantees that E is equicontinuous. ⇐: By 4.5.15, 4.6.7, 4.6.8, and 4.6.9, it is sufficient to construct a finite Uθ -net for E given θ ∈ UF . Choose θ0 ∈ UF such that θ0 ◦ θ0 ◦ θ0 ⊂ θ and find an open symmetric entourage U ∈ UQ from the condition \ U⊂ g −1 ◦ θ0 ◦ g g∈E

(by the equicontinuity property of E , such an U is available). The family {U (x) : x ∈ Q} clearly forms an open cover of Q. By the compactness of Q, refine a finite subcover {U (x0 ) : x0 ∈ Q0 }. In particular, from 1.1.10 derive IQ ⊂

[

U (x0 ) × U (x0 )

x0 ∈Q0

=

[

U −1 (x0 ) × U (x0 ) = U ◦ IQ0 ◦ U.

(x0 ,x0 )∈IQ0

The set {g|Q0 : g ∈ E } is totally bounded in F Q0 . Consequently, there is a finite θ0 -net for this set. Speaking more precisely, there is a finite subset E 0 of E with the following property: g ◦ IQ0 ◦ f −1 ⊂ θ0

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55

for every g in E and some f in E 0 . Using the estimates, successively infer that g ◦ f −1 = g ◦ IQ ◦ f −1 ⊂ g ◦ (U ◦ IQ0 ◦ U ) ◦ f −1 ⊂ g ◦ (g −1 ◦ θ0 ◦ g) ◦ IQ0 ◦ (f −1 ◦ θ0 ◦ f ) ◦ f −1 = (g ◦ g −1 ) ◦ θ0 ◦ (g ◦ IQ0 ◦ f −1 ) ◦ θ0 ◦ (f ◦ f −1 ) = Iimg ◦ θ0 ◦ (g ◦ IQ0 ◦ f −1 ) ◦ θ0 ◦ Iimf ⊂ θ0 ◦ θ0 ◦ θ0 ⊂ θ. Thus, by 4.6.8, E 0 is a finite Uθ -net for E . B 4.6.11. Remark. It is an enlightening exercise to translate the proof of the Ascoli–Arzel`a Theorem into the “ε-δ” language. The necessary vocabulary is in hand: “θ and Uθ stand for ε,” “θ0 is ε /3 ,” and “δ is U .” It is also rewarding and instructive to find generalizations of the Ascoli–Arzel`a Theorem for mappings acting into more abstract spaces. 4.7. Baire Spaces 4.7.1. Definition. A set U is said to be nowhere dense or rare whenever its closure lacks interior points; i.e., int cl U = ∅. A set U is meager or a set of first category if U is included into a countable union of rare sets; i.e., U ⊂ ∪n∈N Un with int cl Un = ∅. A nonmeager set (which is not of first category by common parlance) is also referred to as a set of second category. 4.7.2. Definition. A space is a Baire space if its every nonempty open set is nonmeager. 4.7.3. The following statements are equivalent: (1) X is a Baire space; (2) every countable union of closed rare sets in X lacks interior points; (3) the intersection of a countable family of open everywhere dense sets (i.e., dense in X) is everywhere dense; (4) the complement of each meager set to X is everywhere dense. C (1) ⇒ (2): If U := ∪n∈N Un , Un = cl Un , and int Un = ∅ then U is meager. Observe that int U ⊂ U and int U is open; hence, int U as a meager set is necessarily empty, for X is a Baire space. (2) ⇒ (3): Let U := ∩n∈N Gn , where Gn ’s are open and cl Gn = X. Then X \U = X \∩n∈N Gn = ∪n∈N (X \Gn ). Moreover, X \Gn is closed and int(X \Gn ) = ∅ (since cl Gn = X). Therefore, int(X \ U ) = ∅, which implies that the exterior of U is empty; i.e., U is everywhere dense.

56

Chapter 4

(3) ⇒ (4): Let U be a meager set in X; i.e., U ⊂ ∪n∈N Un and int cl Un = ∅. It may be assumed that Un = cl Un . Then Gn := X \ Un is open and everywhere dense. By hypothesis, ∩n∈N Gn = X \ ∪n∈N Un is everywhere dense. Moreover, the set is included into X \ U , and so X \ U is everywhere dense. (4) ⇒ (1): If U is nonempty open set in X then X \ U is not everywhere dense. Consequently, U is nonmeager. B 4.7.4. Remark. In connection with 4.7.3 (4), observe that the complement of a meager set is (sometimes) termed a residual or comeager set. A residual set in a Baire space is nonmeager. 4.7.5. Osgood Theorem. Let X be a Baire space and let (fξ : X0 = 0 O R)ξ∈„ be a family of lower semicontinuous functions such that sup{fξ (x) : ξ ∈ „} < +∞ for all x ∈ X. Then each nonempty open set G in X includes a nonempty open subset G0 on which (fξ )ξ∈„ is uniformly bounded above; i.e., supx∈G0 sup{fξ (x) : ξ ∈ „} ≤ +∞. CB 4.7.6. Baire Category Theorem. A complete metric space is a Baire space. C Let G be a nonempty open set and x0 ∈ G. Suppose by way of contradiction that G is meager; i.e., G ⊂ ∪n∈N Un , where int Un = ∅ and Un = cl Un . Take ε0 > 0 satisfying Bε0 (x0 ) ⊂ G. It is obvious that U1 is not included into Bε0 /2 (x0 ). Consequently, there is an element x1 in Bε0 /2 (x0 ) \ U1 . Since U1 is closed, find ε1 satisfying 0 < ε1 ≤ ε0 /2 and Bε1 (x1 ) ∩ U1 = ∅. Check that Bε1 (x1 ) ⊂ Bε0 (x0 ). For, if d(x1 , y1 ) ≤ ε1 , then d(y1 , x0 ) ≤ d(y1 , x1 ) + d(x1 , x0 ) ≤ ε1 + ε0 /2 , because d(x1 , x0 ) ≤ ε0 /2 . The ball Bε1 /2 (x1 ) does not lie in U2 entirely. It is thus possible to find x2 ∈ Bε1 /2 (x1 ) \ U2 and 0 < ε2 ≤ ε1 /2 satisfying Bε2 (x2 ) ∩ U2 = ∅. It is easy that again Bε2 (x2 ) ⊂ Bε1 (x1 ). Proceeding by induction, obtain the sequence of nested balls Bε0 (x0 ) ⊃ Bε1 (x1 ) ⊃ Bε2 (x2 ) ⊃ . . . ; moreover, εn+1 ≤ εn /2 and Bεn (xn ) ∩ Un = ∅. By the Nested Ball Theorem, the balls have a common point, x := lim xn . Further, x 6= ∪n∈N Un ; and, hence, x 6∈ G. On the other hand, x ∈ Bε0 (x0 ) ⊂ G, a contradiction. B 4.7.7. Remark. The Baire Category Theorem is often used as a “pure existence theorem.” As a classical example, consider the existence problem for continuous nowhere differentiable functions. Given f : [0, 1]0 = 0 O R and x ∈ [0, 1), put f (x + h) − f (x) ; h↓0 h f (x + h) − f (x) D+ f (x) := lim sup . h↓0 h D+ f (x) := lim inf

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57

The elements D+ f (x) and D+ f (x) of the extended axis R are the lower right and upper right Dini derivatives of f at x. Further, let D stand for the set of functions f in C([0, 1], R) such that for some x ∈ [0, 1) the elements D+ f (x) and D+ f (x) belong to R; i.e., they are finite. Then D is a meager set. Hence, the functions lacking derivatives at every point of (0, 1) are everywhere dense in C([0, 1], R). However, the explicit examples of such functions are not at all easy to find and grasp. Below a few of the most popular are listed: ∞ X hh4n xii van der Waerden’s function: , 4n n=0

with hhxii := (x − [x]) ∧ (1 + [x] − x), the distance from x to the whole number nearest to x; +∞ X 1 Riemann’s function: sin (n2 πx); 2 n n=0 and, finally, the historically first Weierstrass’s function:

∞ X

bn cos (an πx),

n=0

with a an odd positive integer, 0 < b < 1 and ab > 1 + 3π /2 . 4.8. The Jordan Curve Theorem and Rough Drafts 4.8.1. Remark. In topology, in particular, many significant and curious facts of the metric space R2 are scrutinized. Here we recall those of them which are of use in the sequel and whose role is known from complex analysis. 4.8.2. Definition. A (Jordan) arc is a homeomorphic image of a (nondegenerate) interval of the real axis. Recall that a homeomorphism or a topological mapping is by definition a one-to-one continuous mapping whose inverse is also continuous. A (simple Jordan) loop is a homeomorphic image of a circle. Concepts like “smooth loop” are understood naturally. 4.8.3. Jordan Curve Theorem. If γ is a simple loop in R2 then there are open sets G1 and G2 such that G1 ∪ G2 = R2 \ γ; γ = ∂G1 = ∂G2 . /. 4.8.4. Remark. Either G1 or G2 in 4.8.3 is bounded. Moreover, each of the two sets is connected; i.e., it cannot be presented as the union of two nonempty disjoint open sets. In this regard the Jordan Curve Theorem is often read as follows: “A simple loop divides the plane into two domains and serves as their mutual boundary.”

58

Chapter 4

4.8.5. Definition. Let D1 , . . . , Dn and D be closed disks (= closed balls) in the plane which satisfy D1 , . . . , Dn ⊂ int D and Dm ∩ Dk = ∅ as m 6= k. The set n [ D\ int Dk k=1

is a holey disk or, more formally, a perforated disk. A subset of the plane which is diffeomorphic (= “smoothly homeomorphic”) to a holey disk is called a connected elementary compactum. The union of a finite family of pairwise disjoint connected elementary compacta is an elementary compactum. 4.8.6. Remark. The boundary ∂F of an elementary compactum F comprises finitely many disjoint smooth loops. Furthermore, the embedding of F into the (oriented) plane R2 induces on F the structure of an (oriented) manifold with (oriented) boundary ∂F . Observe also that, by 4.8.3, it makes sense to specify the positive orientation of a smooth loop. This is done by indicating the orientation induced on the boundary of the compact set surrounded by the loop. 4.8.7. If K is a compact subset of the plane and G is a nonempty open set that includes K then there is a nonempty elementary compactum F such that K ⊂ int F ⊂ F ⊂ G. /. 4.8.8. Definition. Every set F appearing in 4.8.7 is referred to as a rough draft for the pair (K, G) or an oriented envelope of K in G. Exercises 4.1. Give examples of metric spaces. Find methods for producing new metric spaces. 4.2. Which filter on X 2 coincides with some metric uniformity on X? 4.3. Let S be the space of measurable functions on [0, 1] endowed with the metric

Z1 d(f, g) :=

|f (t) − g(t)| dt 1 + |f (t) − g(t)|

(f, g ∈ S)

0

with some natural identification implied (specify it!). Find the meaning of convergence in the space. 4.4. Given α, β ∈ NN , put d(α, β) := 1/ min {k ∈ N : αk 6= βk }. Check that d is a metric and the space NN is homeomorphic with the set of irrational numbers.

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59

4.5. Is it possible to metrize pointwise convergence in the sequence space? In the function space F[0,1] ? 4.6. How to introduce a reasonable metric into the countable product of metric spaces? Into an arbitrary product of metric spaces? 4.7. Describe the function classes distinguishable by erroneous definitions of continuity and uniform continuity. 4.8. Given nonempty compact subsets A and B of the spaces RN, define

 d(A, B) :=

  sup inf |x − y| x∈A y∈B

∨

 sup inf |x − y|

.

y∈B x∈A

Show that d is a metric. The metric is called the Hausdorff metric. What is the meaning of convergence in this metric? 4.9. Prove that nonempty compact convex subsets of a compact convex set in RN constitute a compact set with respect to the Hausdorff metric. How does this claim relate to the Ascoli–Arzel` a Theorem? 4.10. Prove that each lower semicontinuous function on RN is the upper envelope of some family of continuous functions. 4.11. Explicate the interplay between continuous and closed mappings (in the product topology) of metric spaces. 4.12. Find out when a continuous mapping of a metric space into a complete metric spaces is extendible onto a completion of the initial space. 4.13. Describe compact sets in the product of metric spaces. 4.14. Let (Y, d) be a complete metric spaces. A mapping F : Y 0 = 0 O is called expanding whenever d(F (x), F (y)) ≥ βd(x, y) for some β > 1 and all x, y ∈ Y . Assume that an expanding mapping F : Y 0 = 0 O acts onto Y . Prove that F is one-to-one and possesses a sole fixed point. 4.15. Prove that no compact set can be mapped isometrically onto a proper part of it. 4.16. Show normality of an arbitrary metric space (see 9.3.11). 4.17. Under what conditions a countable subset of a complete metric spaces is nonmeager? 4.18. Is it possible to characterize uniform continuity in terms of convergent sequences? 4.19. In which metric spaces does each continuous real-valued function attain the supremum and infimum of its range? When is it bounded?

Chapter 5 Multinormed and Banach Spaces

5.1. Seminorms and Multinorms 5.1.1. Let X be a vector space over a basic field F and let p : X0 = 0 O R· be a seminorm. Then (1) dom p is a subspace of X; (2) p(x) ≥ 0 for all x ∈ X; (3) the kernel ker p := {p = 0} is a subspace in X; ◦

(4) the sets B p := {p < 1} and Bp := {p ≤ 1} are absolutely convex; ◦

moreover, p is the Minkowski functional of every set B such that B p ⊂ B ⊂ Bp ; ◦

(5) X = dom p if and only if B p is absorbing. C If x1 , x2 ∈ dom p and α1 , α2 ∈ F then by 3.7.6 p(α1 x1 + α2 x2 ) ≤ |α1 |p(x1 ) + |α2 |p(x2 ) < +∞ + (+∞) = +∞. Hence, (1) holds. Suppose to the contrary that (2) is false; i.e., p(x) < 0 for some x ∈ X. Observe that 0 ≤ p(x) + p(−x) < p(−x) = p(x) < 0, a contradiction. The claim of (3) is immediate from (2) and the subadditivity of p. The validity of (4) and (5) has been examined in part (cf. 3.8.8). What was left unproven follows from the Gauge Theorem. B 5.1.2. If p, q : X0 = 0 O R· are two seminorms then the inequality p ≤ q (in (R· )X ) holds if and only if Bp ⊃ Bq . C ⇒: Evidently, {q ≤ 1} ⊂ {p ≤ 1}. ⇐: In view of 5.1.1 (4), observe that p = pBp and q = pBq . Take t1 , t2 ∈ R such that t1 < t2 . If t1 < 0 then {q ≤ t1 } = ∅, and so {q ≤ t1 } ⊂ {p ≤ t2 }. If t1 ≥ 0 then t1 Bq ⊂ t1 Bp ⊂ t2 Bp . Thus, by 3.8.3, p ≤ q. B

Multinormed and Banach Spaces

61

5.1.3. Let X and Y be vector spaces, let T ⊂ X×Y be a linear correspondence, and let p : Y 0 = 0 O R· be a seminorm. If pT (x) := inf p ◦ T (x) for x ∈ X then pT : X0 = 0 O R· is a seminorm and the set BT := T −1 (Bp ) is absolutely convex. In addition, pT = pBT . C Given x1 , x2 ∈ X and α1 , α2 ∈ F, infer that pT (α1 x1 + α2 x2 ) = inf p(T (α1 x1 + α2 x2 )) ≤ inf p(α1 T (x1 ) + α2 T (x2 )) ≤ inf(|α1 |p(T (x1 )) + |α2 |p(T (x2 ))) = |α1 |pT (x1 ) + |α2 |pT (x2 ); i.e., pT is a seminorm. The absolute convexity of BT is a consequence of 5.1.1 (4) and 3.1.8. If x ∈ BT then (x, y) ∈ T for some y ∈ Bp . Hence, pT (x) ≤ p(y) ≤ 1; that is, BT ⊂ BpT . ◦

If in turn x ∈ B pT then pT (x) = inf{p(y) : (x, y) ∈ T } < 1. Thus, there is some ◦

y ∈ T (x) such that p(y) < 1. Therefore, x ∈ T −1 (B p ) ⊂ T −1 (Bp ) = BT . Finally, ◦

B pT ⊂ BT ⊂ BpT . Referring to 5.1.1 (4), conclude that pBT = pT . B 5.1.4. Definition. The seminorm pT , constructed in 5.1.3, is the inverse image or preimage of p under T . 5.1.5. Definition. Let p : X0 = 0 O R be a seminorm (by 3.4.3, this implies that dom p = X). A pair (X, p) is referred to as a seminormed space. It is convenient to take the liberty of calling X itself a seminormed space. 5.1.6. Definition. A multinorm on X is a nonempty set (a subset of RX ) of everywhere-defined seminorms. Such a multinorm is denoted by MX or simply by M, if the underlying vector space is clear from the context. A pair (X, MX ), as well as X itself, is called a multinormed space. 5.1.7. A set M in (R· )X is a multinorm if and only if (X, p) is a seminormed space for every p ∈ M. CB 5.1.8. Definition. A multinorm MX is a Hausdorff or separated multinorm whenever for all x ∈ X, x 6= 0, there is a seminorm p ∈ MX such that p(x) 6= 0. In this case X is called a Hausdorff or separated multinormed space. 5.1.9. Definition. A norm is a Hausdorff multinorm presenting a singleton. The sole element of a norm on a vector space X is also referred to as the norm on X and is denoted by k · k or (rarely) by k · kX or even k · | Xk if it is necessary to indicate the space X. A pair (X, k · k) is called a normed space; as a rule, the same term applies to X.

62

Chapter 5 5.1.10. Examples.

(1) A seminormed space (X, p) can be treated as a multinormed space (X, {p}). The same relates to a normed space. (2) If M is the set of all (everywhere-defined) seminorms on a space X then M is a Hausdorff multinorm called the finest multinorm on X. (3) Let (Y, N) be a multinormed space, and let T ⊂ X × Y be a linear correspondence such that dom T = X. By 3.4.10 and 5.1.1 (5), for every p in N the seminorm pT is defined everywhere, and hence M := {pT : p ∈ N} is a multinorm on X. The multinorm N is called the inverse image or preimage of N under the correspondence T and is (sometimes) denoted by NT . Given T ∈ L (X, Y ), set M := {p ◦ T : p ∈ N} and use the natural notation N ◦ T := M. Observe in particular the case in which X is a subspace Y0 of Y and T is the identical embedding ι : Y0 0 = 0 O . At this juncture Y0 is treated as a multinormed space with multinorm N ◦ ι. Moreover, the abuse of the phrase “N is a multinorm on Y0 ” is very convenient. (4) Each basic field F is endowed, as is well known, with the standard norm | · | : F0 = 0 O R, the modulus of a scalar. Consider a vector space X and f ∈ X # . Since f : X0 = 0 O F , it is possible to define the inverse image of the norm on F as pf (x) := |f (x)| (x ∈ X). If is some subspace of X # then the multinorm σ(X, ) := { : ∈} is the weak multinorm on X induced by . (5) Let (X, p) be a seminormed space. Assume further that X0 is a subspace of X and ϕ : X0 = 0 O /X0 is the coset mapping. The linear correspondence ϕ−1 is defined on the whole of X/X0 . Hence, the seminorm pϕ−1 appears, called the quotient seminorm of p by X0 and denoted by pX/X0 . The space (X/X0 , pX/X0 ) is called the quotient space of (X, p) by X0 . The definition of quotient space for an arbitrary multinormed space requires some subtlety and is introduced in 5.3.11. (6) Let X be a vector space and let M ⊂ (R· )X be a set of seminorms on X. In this case M can be treated as a multinorm on the space X0 := ∩{dom p : p ∈ M}. More precisely, thinking of the multinormed space (X0 , {pι : p ∈ M}), where ι is the identical embedding of X0 into X, we say: “M is a multinorm,” or “Consider the (multinormed) space specified by M.” The next example is typical: The family of seminorms 

 pα,β (f ) := sup |x ∂ f (x)| : α and β are multi-indices α

β

x∈RN

specifies the (multinormed) space of infinitely differentiable functions on RN decreasing rapidly at infinity (such functions are often called tempered, cf. 10.11.6).

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63

(7) Let (X, k·k) and (Y, k· k) be normed spaces (over the same ground field F). Given T ∈ L (X, Y ), consider the operator norm of T , i.e. the quantity kT k := sup{kT xk : x ∈ X, kxk ≤ 1} = sup x∈X

kT xk . kxk

(From now on, in analogous situations we presume 0 /0 := 0.) It is easily seen that k · k : L (X, Y )0 = 0 O R· is a seminorm. Indeed, putting BX := {k · kX ≤ 1} for T1 , T2 ∈ L (X, Y ) and α1 , α2 ∈ F, deduce that kα1 T1 + α2 T2 k = sup k · kα1 T1 +α2 T2 (BX ) = sup k · k((α1 T1 + α2 T2 )(BX )) ≤ sup kα1 T1 (BX ) + α2 T2 (BX )k ≤ |α1 | sup k · kT1 (BX ) + |α2 | sup k · kT2 (BX ) = |α1 | kT1 k + |α2 | kT2 k. The subspace B(X, Y ), the effective domain of definition of the above seminorm, is the space of bounded operators; and an element of B(X, Y ) is a bounded operator. Observe that a shorter term “operator” customarily implies a bounded operator. It is clear that B(X, Y ) is a normed space (under the operator norm). Note also that T in L (X, Y ) is bounded if and only if T maintains the normative inequality; i.e., there is a strictly positive number K such that kT xkY ≤ K kxkX

(x ∈ X).

Moreover, kT k is the greatest lower bound of the set of Ks appearing in the normative inequality. CB (8) Let X be a vector space over F and let k·k be a norm on X. Assume further that X 0 := B(X, F) is the (normed) dual of X, i.e. the space of bounded functionals f s with the dual norm kf k = sup{|f (x)| : kxk ≤ 1} = sup x∈X

|f (x)| . kxk

Consider X 00 := (X 0 )0 := B(X 0 , F), the second dual of X. Given x ∈ X and f ∈ X 0 , put x00 := ι(x) : f 7→ f (x). Undoubtedly, ι(x) ∈ (X 0 )# = L (X 0 , F ). In addition, kx00 k = kι(x)k = sup{|ι(x)(f )| : kf | X 0 ≤ 1} = sup{|f (x)| : (∀x ∈ X) |f (x)| ≤ kxkX } = sup{|f (x)| : f ∈ |∂ |(k · kX )} = kxkX . The final equality follows for instance from Theorem 3.6.5 and Lemma 3.7.9. Thus, ι(x) ∈ X 00 for every x in X. It is plain that the operator ι : X0 = 0 O 00 ,

64

Chapter 5

defined as ι : x 7→ ι(x), is linear and bounded; moreover, ι is a monomorphism and kιxk = kxk for all x ∈ X. The operator ι is referred to as the canonical embedding of X into the second dual or more suggestively the double prime mapping. As a rule, it is convenient to treat x and x00 := ιx as the same element; i.e., to consider X as a subspace of X 00 . A normed space X is reflexive if X and X 00 coincide (under the indicated embedding!). Reflexive spaces possess many advantages. Evidently, not all spaces are reflexive. Unfortunately, such is C([0, 1], F) which is irreflexive (the term “nonreflexive” is also is common parlance). CB 5.1.11. Remark. The constructions, carried out in 5.1.10 (8), show some symmetry or duality between X and X 0 . In this regard, the notation (x, f ) := hx | f i := f (x) symbolizes the action of x ∈ X on f ∈ X 0 (or the action of f on x). To achieve and ensure the utmost conformity, it is a common practice to denote elements of X 0 by symbols like x0 ; for instance, hx | x0 i = (x, x0 ) = x0 (x). 5.2.

The Uniformity and Topology of a Multinormed Space

5.2.1. Let (X, p) be a seminormed space. Given x1 , x2 ∈ X, put dp (x1 , x2 ) := p(x1 − x2 ). Then (1) dp (X 2 ) ⊂ R+ and {d ≤ 0} ⊃ IX ; (2) {dp ≤ t} = {dp ≤ t}−1 and {dp ≤ t} = t{dp ≤ 1} (t ∈ R+ \ 0); (3) {dp ≤ t1 } ◦ {dp ≤ t2 } ⊂ {dp ≤ t1 + t2 } (t1 , t2 ∈ R+ ); (4) {dp ≤ t1 } ∩ {dp ≤ t2 } ⊃ {dp ≤ t1 ∧ t2 } (t1 , t2 ∈ R+ ); (5) p is a norm ⇔ dp is a metric. CB 5.2.2. Definition. The uniformity of a seminormed space (X, p) is the filter Up := fil {{dp ≤ t} : t ∈ R+ \ 0}. 5.2.3. If Up is the uniformity of a seminormed space (X, p) then (1) Up ⊂ fil {IX }; (2) U ∈ Up ⇒ U −1 ∈ Up ; (3) (∀ U ∈ Up ) (∃ V ∈ Up ) V ◦ V ⊂ U . CB 5.2.4. Definition. Let (X, M) be a multinormed space. The filter U := sup{Up : p ∈ M} is called the uniformity of X (the other designations are UM , UX , etc.). (By virtue of 5.2.3 (1) and 1.3.13, the definition is sound.) 5.2.5. If (X, M) is a multinormed space and U is the uniformity of X then (1) U ⊂ fil {IX }; (2) U ∈ U ⇒ U −1 ∈ U ; (3) (∀ U ∈ U ) (∃ V ∈ U ) V ◦ V ⊂ U .

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C Examine (3). Given U ∈ U , by 1.2.18 and 1.3.8 there are seminorms p1 , . . . , pn ∈ M such that U = U{p1 ,... ,pn } = Up1 ∨ . . . ∨ Upn . Using 1.3.13, find sets Uk ∈ Upk satisfying U ⊃ U1 ∩ . . . ∩ Un . Applying 5.2.3 (3), choose Vk ∈ Upk so as to have Vk ◦ Vk ⊂ Uk . It is clear that (V1 ∩ . . . ∩ Vn ) ◦ (V1 ∩ . . . ∩ Vn ) ⊂ V1 ◦ V1 ∩ . . . ∩ Vn ◦ Vn ⊂ U1 ∩ . . . ∩ Un . Moreover, V1 ∩ . . . ∩ Vn ∈ Up1 ∨ . . . ∨ Upn ⊂ U . B 5.2.6. A multinorm M on X is separated if and only if so is the uniformity UM ; i.e., ∩{V : V ∈ UM } = IX . C ⇒: Let (x, y) 6∈ IX ; i.e., x 6= y. Then p(x − y) > 0 for some seminorm p in M. Hence, (x, y) 6∈ {dp ≤ 1 /2 p(x − y)}. But the last set is included in Up , and thus in UM . Consequently, X 2 \ IX ⊂ X 2 \ ∩{V : V ∈ UM }. Furthermore, IX ⊂ ∩{V : V ∈ UM }. ⇐: If p(x) = 0 for all p ∈ M then (x, 0) ∈ V for every V in UM . Hence, (x, 0) ∈ IX by hypothesis. Therefore, x = 0. B 5.2.7. Given a space X with uniformity UX , define τ (x) := {U (x) : U ∈ UX } (x ∈ X). Then τ (x) is a filter for every x ∈ X. Moreover, (1) τ (x) ⊂ fil {x}; (2) (∀ U ∈ τ (x)) (∃ V ∈ τ (x) & V ⊂ U ) (∀ y ∈ V ) V ∈ τ (y). C All is evident (cf. 4.1.8). B 5.2.8. Definition. The mapping τ : x 7→ τ (x) is called the topology of a multinormed space (X, M); a member of τ (x) is a neighborhood about x. The designation for the topology can be more detailed: τX , τM , τ (UM ), etc. 5.2.9. The following presentation holds: τX (x) = sup{τp (x) : p ∈ MX } for all x ∈ X. CB 5.2.10. If X is a multinormed space then U ∈ τ (x) ⇔ U − x ∈ τX (0) for all x ∈ X. C By 5.2.9 and 1.3.13, it suffices to consider a seminormed space (X, p). In this case for every ε > 0 the equality holds: {dp ≤ ε}(x) = εBp +x, where Bp := {p ≤ 1}. Indeed, if p(y − x) ≤ ε then y = ε(ε−1 (y − x)) + x and ε−1 (y − x) ∈ Bp . In turn, if y ∈ εBp + x, then p(y − x) = inf{t > 0 : y − x ∈ tBp } ≤ ε. B

66

Chapter 5

5.2.11. Remark. The proof of 5.2.10 demonstrates that in a seminormed space (X, p) a key role is performed by the ball with radius 1 and centered at zero. The ball bears the name of the unit ball of X and is denoted by Bp , BX , etc. 5.2.12. A multinorm MX is separated if and only if so is the topology τX ; i.e., given distinct x1 and x2 in X, there are neighborhoods U1 in τX (x1 ) and U2 in τX (x2 ) such that U1 ∩ U2 = ∅. C ⇒: Let x1 6= x2 and let ε := p(x1 −x2 ) > 0 for p ∈ MX . Put U1 := x1 + ε /3 Bp and U2 := x2 + ε /3 Bp . By 5.2.10, Uk ∈ τX (xk ). Verify that U1 ∩ U2 = ∅. Indeed, if y ∈ U1 ∩ U2 then p(x1 − y) ≤ ε /3 and p(x2 − y) ≤ ε /3 . Therefore, p(x1 − x2 ) ≤ 2 /3 ε < ε = p(x1 − x2 ), which is impossible. ⇐: If (x1 , x2 ) ∈ ∩{V : V ∈ UX } then x2 ∈ ∩{V (x1 ) : V ∈ UX }. Thus, x1 = x2 and, consequently, MX is separated by 5.2.6. B 5.2.13. Remark. The presence of a uniformity and the corresponding topology in a multinormed space readily justifies using uniform and topological concepts such as uniform continuity, completeness, continuity, openness, closure, etc. 5.2.14. Let (X, p) be a seminormed space and let X0 be a subspace of X. The quotient space (X/X0 , pX/X0 ) is separated if and only if X0 is closed. C ⇒: If x 6∈ X0 then ϕ(x) 6= 0 where, as usual, ϕ : X0 = 0 O /X0 is the coset mapping. By hypothesis, 0 6= ε := pX/X0 (ϕ(x)) = pϕ−1 (ϕ(x)) = inf{p(x + x0 ) : x0 ∈ X0 }. Hence, the ball x + ε /2 Bp does not meet X0 and x is an exterior point of X0 . Thus, X0 is closed. ⇐: Suppose that x is a nonzero point of X/X0 and x = ϕ(x) for some x in X. If pX/X0 (x) = 0 then 0 = inf{p(x − x0 ) : x0 ∈ X0 }. In other words, there is a sequence (xn ) in X0 converging to x. Consequently, by 4.1.19, x ∈ X0 and x = 0, a contradiction. B 5.2.15. The closure of a € -set is a € -set. C Given U ∈ (€ ), suppose that U 6= ∅ (otherwise there is nothing worthy of proving). By 4.1.9, for x, y ∈ cl U there are nets (xγ ) and (yγ ) in U such that xγ 0 = 0 O and yγ 0 = 0 O . If (α, β) ∈ € then αxγ + βyγ ∈ U . Appealing to 4.1.19 again, infer αx + βy = lim(αxγ + βyγ ) ∈ cl U . B 5.3. Comparison Between Topologies 5.3.1. Definition. If M and N are two multinorms on a vector space then M is said to be finer or stronger than N (in symbols, M  N) if UM ⊃ UN . If M  N and N  M simultaneously, then M and N are said to be equivalent (in symbols, M ∼ N).

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5.3.2. For multinorms M and N on a vector space X the following statements are equivalent: (1) M  N; (2) the inclusion τM (x) ⊃ τN (x) holds for all x ∈ X; (3) τM (0) ⊃ τN (0); (4) (∀ q ∈ N) (∃ p1 , . . . , pn ∈ M) (∃ ε1 , . . . , εn ∈ R+ \ 0) Bq ⊃ ε1 Bp1 ∩ . . . ∩ εn Bpn ; (5) (∀ q ∈ N) (∃ p1 , . . . , pn ∈ M) (∃ t > 0) q ≤ t(p1 ∨ . . . ∨ pn ) (with respect to the order of the K-space RX ). C The implications (1) ⇒ (2) ⇒ (3) ⇒ (4) are evident. (4) ⇒ (5): Applying the Gauge Theorem (cf. 5.1.2), infer that

q ≤ pBp1 /ε1 ∨ . . . ∨ pBpn /εn = 1 /ε1 p1 ∨ . . . ∨ 1 /εn pn

≤ 1 /ε1 ∨ . . . ∨ 1 /εn p1 ∨ . . . ∨ pn . (5) ⇒ (1): It is sufficient to check that M  {q} for a seminorm q in N. If V ∈ Uq then V ⊃ {dq ≤ ε} for some ε > 0. By hypothesis {dq ≤ ε} ⊃ {dp1 ≤ ε /t } ∩ . . . ∩ {dpn ≤ ε /t } with suitable p1 , . . . , pn ∈ M and t > 0. The right side of this inclusion is an element of Up1 ∨ . . . ∨ Upn = U{p1 ,... ,pn } ⊂ UM . Hence, V is also a member of UM . B 5.3.3. Definition. Let p, q : X0 = 0 O R be two seminorms on X. Say that p is finer or stronger than q and write p  q whenever {p}  {q}. The equivalence of seminorms p ∼ q is understood in a routine fashion. 5.3.4. p  q ⇔ (∃ t > 0) q ≤ tp ⇔ (∃ t ≥ 0) Bq ⊃ tBp ; p ∼ q ⇔ (∃ t1 , t2 > 0) t2 p ≤ q ≤ t1 p ⇔ (∃ t1 , t2 > 0) t1 Bp ⊂ Bq ⊂ t2 Bp . C Everything follows from 5.3.2 and 5.1.2. B 5.3.5. Riesz Theorem. If p, q : F N 0 = 0 O R are seminorms on the finitedimensional space F N then p  q ⇔ ker p ⊂ ker q. CB 5.3.6. Corollary. All norms in finite dimensions are equivalent. CB 5.3.7. Let (X, M) and (Y, N) be multinormed spaces, and let T be a linear operator, a member of L (X, Y ). The following statements are equivalent: (1) N ◦ T ≺ M; (2) T × (UX ) ⊃ UY and T ×−1 (UY ) ⊂ UX ; (3) x ∈ X ⇒ T (τX (x)) ⊃ τY (T x); (4) T (τX (0)) ⊃ τY (0) and τX (0) ⊃ T −1 (τY (0)); (5) (∀ q ∈ N) (∃ p1 , . . . , pn ∈ M) q ◦ T ≺ p1 ∨ . . . ∨ pn . CB

68

Chapter 5

5.3.8. Let (X, k · kX ) and (Y, k · kY ) be normed spaces and let T be a linear operator, a member of L (X, Y ). The following statements are equivalent: (1) T is bounded (that is, T ∈ B(X, Y )); (2) k · kX  k · kY ◦ T ; (3) T is uniformly continuous; (4) T is continuous; (5) T is continuous at zero. C Each of the claims is a particular case of 5.3.7. B 5.3.9. Remark. The message of 5.3.7 shows that it is sometimes convenient to substitute for M a multinorm equivalent to M but filtered upward (with respect to the relation ≥ or ). For example, we may take the multinorm M := {sup M0 : M0 is a nonempty finite subset of M}. Observe that unfiltered multinorms should be treated with due precaution. 5.3.10. Counterexample. Let X := F „ ; and let X0 comprise all constant functions; i.e., X0 := F1, where 1 : ξ 7→ 1 (ξ ∈ „). Set M := {pξ : ξ ∈ „}, with pξ (x) := |x(ξ)| (x ∈ F „ ). It is clear that M is a multinorm on X. Now let ϕ : X0 = 0 O /X0 stand for the coset mapping. Undoubtedly, Mϕ−1 consists of the sole element, zero. At the same time Mϕ−1 is separated. 5.3.11. Definition. Let (X, M) be a multinormed space and let X0 be a subspace of X. The multinorm Mϕ−1 , with ϕ : X0 = 0 O /X0 the coset mapping, is referred to as the quotient multinorm and is denoted by MX/X0 . The space (X/X0 , MX/X0 ) is called the quotient space of X by X0 . 5.3.12. The quotient space X/X0 is separated if and only if X0 is closed. CB 5.4. Metrizable and Normable Spaces 5.4.1. Definition. A multinormed space (X, M) is metrizable if there is a metric d on X such that UM = Ud . Say that X is normable if there is a norm on X equivalent to the initial multinorm M. Say that X is countably normable if there is a countable multinorm on X equivalent to the initial. 5.4.2. Metrizability Crtiterion. A multinormed space is metrizable if and only if it is countably normable and separated. C ⇒: Let UM = Ud . Passing if necessary to the multinorm M, assume that for every n in N it is possible to find a seminorm pn in M and tn > 0 such that {d ≤ 1 /n } ⊃ {dpn ≤ tn }. Put N := {pn : n ∈ N}. Clearly, M  N. If V ∈ UM then V ⊃ {d ≤ 1 /n } for some n ∈ N by the definition of metric uniformity. Hence, by construction, V ∈ Upn ⊂ UM , i.e., M ≺ N. Thus, M ∼ N. The uniformity Ud

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69

is separated, as indicated in 4.1.7. Applying 5.2.6, observe that UM and UN are both separated. ⇐: Passing if necessary to an equivalent multinorm, suppose that X, the space in question, is countably normed and separated; that is, M := {pn : n ∈ N} and M is a Hausdorff multinorm on X. Given x1 , x2 ∈ X, define ∞ X 1 pk (x1 − x2 ) d(x1 , x2 ) := 2k 1 + pk (x1 − x2 ) k=1

(the series P∞ on the right side of the above formula is dominated by the convergent series k=1 1 /2k , and so d is defined soundly). Check that d is a metric. It suffices to validate the triangle inequality. For a start, put α(t) := t(1 + t)−1 (t ∈ R+ ). It is evident that α0 (t) = (1 + t)−2 > 0. Therefore, α increases. Furthermore, α is subadditive: α(t1 + t2 ) = (t1 + t2 )(1 + t1 + t2 )−1 = t1 (1 + t1 + t2 )−1 + t2 (1 + t1 + t2 )−1 ≤ t1 (1 + t1 )−1 + t2 (1 + t2 )−1 = α(t1 ) + α(t2 ). Thus, given x, y, z ∈ X, infer that ∞ ∞ X X 1 1 d(x, y) = α(pk (x − y)) ≤ α(pk (x − z) + pk (z − y)) k 2 2k k=1

≤

∞ X k=1

k=1

1 (α(pk (x − z)) + α(pk (z − y))) = d(x, z) + d(z, y). 2k

It remains to established that Ud and UM coincide. First, show that Ud ⊂ UM . Take a cylinder, say, {d ≤ ε}; and let (x, y) ∈ {dp1 ≤ t} ∩ . . . ∩ {dpn ≤ t}. Since α is an increasing function, deduce that n ∞ X X 1 pk (x − y) 1 pk (x − y) d(x, y) = + k 2 1 + pk (x − y) 2k 1 + pk (x − y) k=1

≤

k=n+1

n ∞ X t X 1 t 1 1 + ≤ + n. k k 1+t 2 2 1+t 2 k=1

k=n+1

Since t(1 + t)−1 + 2−n tends to zero as n0 = 0 O and t0 = 0 O , for appropriate t and n observe that (x, y) ∈ {d ≤ ε}. Hence, {d ≤ ε} ∈ UM , which is required.

70

Chapter 5

Now establish that UM ⊂ Ud . To demonstrate the inclusion, given pn ∈ M and t > 0, find ε > 0 such that {dpn ≤ t} ⊃ {d ≤ ε}. For this purpose, take ε :=

1 t , 2n 1 + t

which suffices, since from the relations 1 t 1 pn (x − y) ≤ d(x, y) ≤ ε = n n 2 1 + pn (x − y) 2 1+t holding for all x, y ∈ X it follows that pn (x − y) ≤ t. B 5.4.3. Definition. A subset V of a multinormed space (X, M) is a bounded set in X if sup p(V ) < +∞ for all p ∈ M which means that the set p(V ) is bounded above in R for every seminorm p in M. 5.4.4. For a set V in (X, M) the following statements are equivalent: (1) V is bounded; (2) for every sequence (xn )n∈N in V and every sequence (λn )n∈N in F such that λn 0 = 0 O , the sequence (λn xn ) vanishes: λn xn 0 = 0 O O (i.e., p(λn xn )0 = 0 for each seminorm p in M); (3) every neighborhood of zero absorbs V . C (1) ⇒ (2): p(λn xn ) ≤ |λn |p(xn ) ≤ |λn | sup p(V )0 = 0 O . (2) ⇒ (3): Let U ∈ τX (0) and suppose that U fails to absorb V . From Definition 3.4.9, it follows that (∀ n ∈ N) (∃ xn ∈ V ) xn 6∈ nU . Thus, 1 /n xn 6∈ U for all n ∈ N; i.e., (1 /n xn ) does not converge to zero. (3) ⇒ (1): Given p ∈ M, find n ∈ N satisfying V ⊂ nBp . Obviously, sup p(V ) ≤ sup p(nBp ) = n < +∞. B 5.4.5. Kolmogorov Normability Criterion. A multinormed space is normable if and only if it is separated and has a bounded neighborhood about zero. C ⇒: It is clear. ⇐: Let V be a bounded neighborhood of zero. Without loss of generality, it may be assumed that V = Bp for some seminorm p in the given multinorm M. Undoubtedly, p ≺ M. Now if U ∈ τM (0) then nU ⊃ V for some n ∈ N. Hence, U ∈ τp (0). Using Theorem 5.3.2, observe that p  M. Thus, p ∼ M; and, therefore, p is also separated by 5.2.12. This means that p is a norm. B 5.4.6. Remark. Incidentally, 5.4.5 shows that the presence of a bounded neighborhood of zero in a multinormed space X amounts to the “seminormability” of X.

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5.5. Banach Spaces 5.5.1. Definition. A Banach space is a complete normed space. 5.5.2. Remark. The concept of Fr´echet space, complete metrizable multinormed space, serves as a natural abstraction of Banach space. It may be shown that the class of Fr´echet spaces is the least among those containing all Banach spaces and closed under the taking of countable products. CB 5.5.3. A normed space X is a Banach space if and only if every norm convergent (= absolutely convergent) series in X converges. P∞ C ⇒: Let n=1 kxn k < +∞ for some (countable) sequence (xn ). Then the sequence of partial sums sn := x1 + · · · + xn is fundamental because ksm − sk k = k

m X

xn k ≤

n=k+1

m X

kxn k0 = 0 O

n=k+1

for m > k. ⇐: Given a fundamental sequence (xn ), choose an increasing sequence (nk )k∈N such that kxn − xm k≤ 2−k as n, m ≥ nk . Then the series xn1 + (xn2 − xn1 ) + (xn3 − xn2 ) + · · · converges in norm to some x; i.e., xnk 0 = 0 O . Observe simultaneously that xn 0 = 0 O . B 5.5.4. If X is a Banach space and X0 is a closed subspace of X then the quotient space X/X0 is also a Banach space. C Let ϕ : X0 = 0 O := X/X0 be the coset mapping. Undoubtedly, P for every ∞ −1 x ∈ there is some x ∈ ϕ (x) such that 2kxk ≥ kxk ≥ kxk. Hence, given n=1 xn , a norm convergent series in , it is possible to choose xn ∈ ϕ−1 (xn ) so P that the P∞ ∞ norm series n=1 kxn k be convergent. According to 5.5.3, the sum x := n=1 xn is available. If x := ϕ(x) then kx −

n X k=1

xk k ≤ kx −

n X

xk k0 = 0 O .

k=1

Appealing to 5.5.3 again, conclude that is a Banach space. B 5.5.5. Remark. The claim of 5.5.3 may be naturally translated to seminormed spaces. In particular, if (X, p) is a complete seminormed space then the quotient space X/ ker p is a Banach space. CB 5.5.6. Theorem. If X and Y are normed spaces and X = 6 0 then B(X, Y ), the space of bounded operators, is a Banach space if and only if so is Y .

72

Chapter 5

C ⇐: Consider a Cauchy sequence (Tn ) in B(X, Y ). By the normative inequality, kTm x − Tk xk ≤ kTm − Tk k kxk0 = 0 O for all x ∈ X; i.e., (Tn x) is fundamental in Y . Thus, there is a limit T x := lim Tn x. Plainly, the so-defined operator T is linear. By virtue of the estimate |kTm k − kTk k| ≤ kTm − Tk k the sequence (kTn k) is fundamental in R and, hence, bounded; that is, supn kTn k < +∞. Therefore, passing to the limit in kTn xk ≤ supn kTn k kxk, obtain kT k < +∞. It remains to check that kTn − T k0 = 0 O . Given ε > 0, choose a number n0 such that kTm − Tn k ≤ ε /2 as m, n ≥ n0 . Further, for x ∈ BX find m ≥ n0 satisfying kTm x − T xk ≤ ε /2 . Then kTn x − T xk ≤ kTn x − Tm xk + kTm x − T xk ≤ kTn − Tm k + kTm x − T xk ≤ ε as n ≥ n0 . In other words, kTn − T k = sup{kTn x − T xk : x ∈ BX } ≤ ε for all sufficiently large n. ⇒: Let (yn ) be a Cauchy sequence in Y . By hypothesis, there is a norm-one element x in X; i.e., x has norm 1: kxk = 1. Applying 3.5.6 and 3.5.2 (1), find an element x0 ∈ |∂ |(k · k) satisfying (x, x0 ) = kxk = 1. Obviously, the rankone operator (with range of dimension 1) Tn := x0 ⊗ yn : x 7→ (x, x0 )yn belongs to B(X, Y ), since kTn k = kx0 k kyn k. Hence, kTm − Tk k = kx0 ⊗ (ym − yk )k= kx0 k kym − yk k = kym − yk k, i.e., (Tn ) is fundamental in B(X, Y ). Assign T := lim Tn . Then kT x − Tn xk = kT x − yn k ≤ kT − Tn k kxk0 = 0 O . In other words, T x is the limit of (yn ) in Y . B 5.5.7. Corollary. The dual of a normed space (furnished with the dual norm) is a Banach space. CB 5.5.8. Corollary. Let X be a normed space; and let ι : X0 = 0 O 00 , the double prime mapping, be the canonical embedding of X into the second dual X 00 . Then the closure cl ι(X) is a completion of X. C By virtue of 5.5.7, X 00 is a Banach space. By 5.1.10 (8), ι is an isometry from X into X 00 . Appealing to 4.5.16 ends the proof. B 5.5.9. Examples. (1) “Abstract” examples: a basic field, a closed subspace of a Banach space, the product of Banach spaces, and 5.5.4–5.5.8. (2) Let E be a nonempty set. Given x ∈ F E , put kxk∞ := sup |x(E )|. The space l∞ (E ) := l∞ (E , F) := dom k · k∞ is called the space of bounded functions on E . The designations B(E ) and B(E , F ) are also used. For E := N, it is customary to put m := l∞ := l∞ (E ). (3) Let a set E be infinite, i.e. not finite, and let F stand for a filter on E . By definition, x ∈ c(E , F ) ⇔ (x ∈ l∞ (E ) and x(F ) is a Cauchy filter on F). In the case E := N and F is the finite complement filter (comprising all cofinite sets each of which is the complement of a finite subset) of N, the notation c := c(E , F ) is employed, and c is called the space of convergent sequences. In c(E , F ) the

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subspace c0 (E , F ) := {x ∈ c(E , F ) : x(F )0 = 0 O } is distinguished. If F is the finite complement filter then the shorter notation c0 (E ) is used and we speak of the space of functions vanishing at infinity. Given E := N, write c0 := c0 (E ). The space c0 is referred to as the space of vanishing sequences. It is worth keeping in mind that each of these spaces without further specification is endowed with the norm taken from the corresponding space l∞ (E , F ). R (4) Let S := (E , X, ) be a system with integration. This means that X is a vector sublattice of RE , with the lattice operationsR in X coincident R R with # E O those in R , and : X0 = 0 R is a (pre)integral; i.e., ∈ X+ and xn ↓ 0 whenever xn ∈ X and xn (e) ↓ 0 for e ∈ E . Moreover, let f ∈ F E be a measurable mapping (with respect to S) (as usual, we may speak of almost everywhere finite and almost everywhere defined measurable functions). R R 1 Denote Np (f ) := ( |f |p ) /p Rfor p ≥ 1, where is the corresponding Lebesgue extension of the initial integral . (The traditional liberty is taken of using the same symbol for the original and its successor.) An element of dom N1 is an integrable or summable function. The integrability of f ∈ F E is equivalent to the integrability of its real part Re f and imaginary part Im f , both members of RE . For the sake of completeness, recall the definition  N (g) := inf sup

Z

 xn : (xn ) ⊂ X, xn ≤ xn+1 , (∀ e ∈ E ) |g(e)| = lim xn (e) n

for an arbitrary g in F E . If F = R then dom N1 obviously presents the closure of X in the normed space (dom N, N ). The H¨ older inequality is valid: N1 (f g) ≤ Np (f )Np0 (g), with p0 the conjugate exponent of p, i.e. 1 /p + 1 /p0 = 1 for p > 1. p

p0

C This is a consequence of the Young inequality, xy − x /p ≤ y /p0 for all x, y ∈ R+ , applied to |f |/Np (f ) and |g|/Np0 (g) when Np (f ) and Np0 (g) are both nonzero. If Np (f )Np0 (g) = 0 then the H¨older inequality is beyond a doubt. B The set Lp := dom Np is a vector space. C |f + g|p ≤ (|f | + |g|)p ≤ 2p (|f | ∨ |g|)p = 2p (|f |p ∨ |g|p ) ≤ 2p (|f |p + |g|p ) . The function Np is a seminorm, satisfying the Minkowski inequality: Np (f + g) ≤ Np (f ) + Np (g).

74

Chapter 5

C For p = 1, this is trivial. For p > 1 the Minkowski inequality follows from the presentation Np (f ) = sup{N1 (f g)/Np0 (g) : 0 < Np0 (g) < +∞} (f ∈ Lp ) whose right side is the upper envelope of a family of seminorms.p To prove the above presentation, using the H¨older inequality, note that g := |f | /p0 lies Rin Lpq when Np (f ) > 0; furthermore, Np (f ) = N1 (f g)/Np0 (g). Indeed, N1 (f g) = |f | /p0 +1 =
0 Np (f )p , because p /p0 +1 = p 1 − 1 /p +1 = p. Continue arguing to find Np0 (g)p = R p0 R p |g| = |f |p = Np (f )p , and so Np0 (g) = Np (f ) /p0. Finally, N1 (f g)/N (g) = Np (f ) /Np (f ) p0

p

p

/p0

p−p / 0 p

= Np (f )

= Np (f )p(1−

1

/p 0 )

= Np (f ). .

The quotient space Lp / ker Np , together with the corresponding quotient norm k · kp , is called the space of p-summable functions or Lp space with more complete R designations Lp (S), Lp (E , X, ), etc. Finally, if a system with integration S arises from inspection of measurable step functions on a measure space (Š, A , µ) then it is customary to write Lp (Š, A , µ), Lp (Š, µ) and even Lp (µ), with the unspecified parameters clear from the context. Riesz–Fisher Completeness Theorem. Each Lp space is a Banach space. P∞ C We P sketch out the proof. Consider t := k=1 Np (fk ), where fk ∈ Lp . Pn n Put σn := k=1 fk and sn := k=1 |fk |. It is seen thatR (sn ) has positive entries and increases. The same is true of (spn ). Furthermore, spn ≤ tp < +∞. Hence, by the Levy Monotone Convergence Theorem, for almost all e ∈ E there exists a limit g(e) := lim spn (e), with the resulting function g a member of L1 . Putting 1 e ∈ E . The h(e) := g /p (e), observe that h ∈ Lp and sn (e)0 = 0 O (e) for almost all P ∞ inequalities |σn | ≤ sn ≤ h imply that for almost all e ∈ E the series k=1 fk (e) converges. For the sum f0 (e) the estimate holds: |f0 (e)| ≤ h(e). Hence, it may be assumed that f0 ∈ Lp . Appealing to the Lebesgue Dominated Convergence
1 / p R Theorem, conclude that Np (σn − f0 ) = |σn − f0 |p 0 = 0 O . Thus, in the seminormed space under consideration each seminorm convergent series converges. To complete the proof, apply 5.5.3–5.5.5. B P If S is the system of conventional summation on E ; i.e., X := R P e∈E R is the direct sum of suitably many copies of the ground field R and x := e∈E x(e), then Lp comprises all p-summable families. This space is denoted by lp (E ). Further,
1 P p /p kxkp := . In the case E := N the notation lp is used and lp is e∈E |x(e)| referred to as the space of p-summable sequences. (5) Define L∞ as follows: Let X be an ordered vector space and let e ∈ X+ be a positive element. The seminorm pe associated with e is the Minkowski functional of the order interval [−e, e], i.e., pe (x) := inf{t > 0 : −te ≤ x ≤ te}.

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The effective domain of definition of pe is the space of bounded elements (with respect to e); the element e itself is referred to as the strong order-unit in Xe . An element of ker pe is said to be nonarchimedean (with respect to e). The quotient space Xe / ker pe furnished with the corresponding quotient seminorm is called the normed space of bounded elements (generated by e in X). For example, C(Q, R), the space of continuous real-valued functions on a nonempty compact set Q, presents the normed space of bounded elements with respect to 1 := 1Q : q 7→ 1 (q ∈ Q) (in itself). In RE the same element 1 generates theR
space l∞ (E ). Given a system with integration S := E , X, , assume that 1 is measurable and consider the space of functions acting from E into F and satisfying N∞ (f ) := inf{t > 0 : |f | ≤ t1} < +∞, where ≤ means “less almost everywhere than.” This space is called the space of essentially bounded functions and is labelled with L∞ . To denote the quotient space L∞ / ker N∞ and its norm the symbols L∞ and k · k∞ are in use. It is in common parlance to call the elements of L∞ (like the elements of L∞ ) essentially bounded functions. The space L∞ presents a Banach space. CB The space L∞ , as well as the spaces C(Q, F ), lp (E ), c0 (E ), c, lp , and Lp (p ≥ 1), also bears the unifying title “classical Banach space.” Nowadays a Lindenstrauss space which is a space whose dual is isometric to L1 (with respect to some system with integration) is also regarded as classical. It can be shown that a Banach space X is classical if and only if the dual X 0 is isomorphic to one of the Lp spaces with p ≥ 1. R
(6) Consider a system with integration S := E , X, and let p ≥ 1. Suppose that for every Q e in E there is a Banach space (Ye , k · kYe ). Given an arbitrary element f in e∈E Ye , define |||f ||| : e 7→ kf (e)kYe . Put Np (f ) := inf{Np (g) : g ∈ Lp , g ≥ |||f |||}. It is clear that dom Np is a vector space equipped with the seminorm Np . The sum of the family in the sense of Lp or simply the p-sum of (Ye )e∈E (with respect to the system with integration S) is the quotient space dom Np / ker Np under the corresponding (quotient) norm |||·|||p . The p-sum P∞ of a family of Banach spaces is a BanachPspace. n C If k=1 Np (fk ) < +∞ then the sequence (sn := k=1 |||fk |||) tends to some almost everywhere finite positive function g and Np (g)P < +∞. It follows that for ∞ almost all e ∈ E the sequence (sn (e)) the series k=1 kfk (e)kYe ) converges. P(i.e., ∞ By the completeness of Ye , the series k=1 fk (e) converges to some sum f0 (e) in Ye with kf0 (e)kYe ≤ g(e) for almost every e ∈ E Pn P. ∞Therefore, it may be assumed that f0 ∈ dom Np . Finally, Np ( k=1 fk − f0 ) ≤ k=n+1 Np (fk )0 = 0 O . . In the case when E := N with conventional summation, for the sum Y of a sequence of Banach spaces (Yn )n∈N (in the sense of Lp ) the following notation is often employed: Y := (Y1 ⊕ Y2 ⊕ · · · )p ,

76

Chapter 5

with p the type of summation. An element y in Y presents a sequence (yn )n∈N such that yn ∈ Yn and !1 /p ∞ X < +∞. |||y|||p := kyn kpYn k=1

In the case Ye := X for all e ∈ E , where X is some Banach space over F, put Fp := dom Np and Fp := Fp / ker Np . An element of the so-constructed space is a vector field or a X-valued function on E (having a p-summable norm). Undoubtedly, Fp is a Banach space. At the same time, if the initial system with integration contains a nonmeasurable set then extraordinary elements are plentiful in Fp (in particular, for the usual Lebesgue system with integration Fp 6= Lp ). In this connection the functions with finite range, assuming each value on a measurable set, are selected in Fp . Such a function, as well as the corresponding coset in Fp , is a simple, finitevalued or step function. The closure in Fp of the set of simple functions is denoted by Lp (or more completely Lp (X), Lp (S, X), Lp (Š, A , µ), Lp (Š, µ), etc.) and is the space of X-valued p-summable functions. Evidently, Lp (X) is a Banach space. It is in order to illustrate one of the advantages of these spaces for p = 1. First, notice that a simple function f can be written as a finite combination of characteristic functions: X f= χf −1 (x) x, x∈imf −1

where f (x) is a measurable set as x ∈ im f , with χE (e) = 1 for e ∈ E and χE (e) = 0 otherwise. Moreover, Z Z X |||f ||| = kχf −1 (x) xk x∈imf

Z =

X

χf −1 (x) kxk =

x∈imf

X

Z kxk

χf −1 (x) < +∞.

x∈imf

Next, associate with each simple function f some element in X by the rule Z X Z f := χf −1 (x) x. x∈imf

R Straightforward calculation shows that the integral defined on the subspace of simple functions is linear. Furthermore, it is bounded because Z X Z X Z k fk = k χf −1 (x) xk ≤ χf −1 (x) kxk x∈imf

Z =

x∈imf

X x∈imf

Z kxkχf −1 (x) =

|||f |||.

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77

R By virtue of 4.5.10 and 5.3.8, the operator has a unique extension to R R an element of B(L1 (X), X). This element is denoted by the same symbol, (or E , etc.), and is referred to as the Bochner integral. (7) In the case of conventional summation, the usage of the scalar theory is preserved for the Bochner integral. Namely, the common parlance favours the term “sum of a family” rather than “integral of a summable function,” and the symbols pertinent to summation are perfectly welcome. What is more important, infinite dimensions bring about significant complications. Let (xn ) be a family of elements of a Banach space. Its summability (in the sense of the Bochner integral) means the summability of the numeric family (kxn k), i.e. the norm convergence of (xn ) as a series. Consequently, (xn ) has at most countably many nonzero P∞ elements and may thus be treated as a (countable) sequence. Moreover, n=1 kxn k < +∞; i.e. P∞the series x1 + x2 + · · · converges in norm. P By 5.5.3, for the series sum x = n=1 xn observe that x = limθ sθ , where sθ := n∈θ xn is a partial sum and θ ranges over the direction of all finite subsets of N. In this case, the resulting x is sometimes called the unordered sum of (xn ), whereas the sequence P (xn ) is called unconditionally or unorderly summable to x (in symbols, x = n∈N xn ). Using these terms, observe that summability in norm implies unconditional summability (to the same sum). If dim X < +∞ then the converse holds which is the Riemann Theorem on Series. The general case is explained by the following deep and profound assertion: Dvoretzky–Rogers Theorem. In an arbitrary infinite-dimensional Banach P∞ space X, for every sequence of positive numbers (tn ) such that n=1 t2n < +∞ there is an unconditionally summable sequence of elements (xn ) with kxn k = tn for all n ∈ N. In this regard for a family of elements (xe )e∈E of an arbitrary multinormed space (X, M) the following terminology is accepted: Say that P (xe )e∈E is summable or unconditionally summable (to a sum x) and write x := e∈E xe whenever x is the limit in (X, M) of the corresponding net of partial sums sθ , with Pθ a finite subset of E ; i.e., sθ 0 = 0 O in (X, M). If for every p there is a sum e∈E p(xe ) then the family (xe )e∈E is said to be multinorm summable (or, what is more exact, fundamentally summable, or even absolutely fundamental). In conclusion, consider a Banach space Y and T ∈ B(X, Y). The operator T can be uniquely extended to an operator from L1 (X) into L1 (Y) by putting T f : e 7→ T f (e) (e ∈ E ) for an arbitrary simple R X-valued R function f . Then, given f ∈ L1 (X), observe that T f ∈ L1 (Y) and E T f = T E f . This fact is verbalized as follows: “The Bochner integral commutes with every bounded operator.” CB 5.6. The Algebra of Bounded Operators 5.6.1. Let X, Y , and Z be normed spaces. If T ∈ L (X, Y ) and S ∈ L (Y, Z)

78

Chapter 5

are linear operators then kST k ≤ kSk kT k; i.e., the operator norm is submultiplicative. C Given x ∈ X and using the normative inequality twice, infer that kST xk ≤ kSk kT xk ≤ kSk kT k kxk. . 5.6.2. Remark. In algebra, in particular, (associative) algebras are studied. An algebra (over F) is a vector space A (over F) together with some associative multiplication ◦ : (a, b) 7→ ab (a, b ∈ A). This multiplication must be distributive with respect to addition (i.e., (A, +, ◦) is an (associative) ring) and, moreover, the operation ◦ must agree with scalar multiplication in the following sense: λ(ab) = (λa)b = a(λb) for all a, b ∈ A and λ ∈ F. A displayed notation for an algebra is (A, F, +, · , ◦). However, just like on the other occasions it is customary to use the term “algebra” simply for A. 5.6.3. Definition. A normed algebra (over a ground field) is an associative algebra (over this field) together with a submultiplicative norm. A Banach algebra is a complete normed algebra. 5.6.4. Let B(X) := B(X, X) be the space of bounded endomorphisms of a normed space X. The space B(X) with the operator norm and composition as multiplication presents a normed algebra. If X 6= 0 then B(X) has a neutral element (with respect to multiplication), the identity operator IX ; i.e., B(X) is an algebra with unity. Moreover, kIX k = 1. The algebra B(X) is a Banach algebra if and only if X is a Banach space. C If X = 0 then there is nothing left to proof. Given X 6= 0, apply 5.5.6. B 5.6.5. Remark. It is usual to refer to B(X) as the algebra of bounded operators in X or even as the (bounded) endomorphism algebra of X. In connection with 5.6.4, given λ ∈ F, it is convenient to retain the same symbol λ for λIX . (In particular, 1 = I0 = 0!) For X 6= 0 this procedure may be thought as identification of F with FIX . 5.6.6. Definition. If X is a normed and T ∈ n space o B(X) then the spectral n 1 /n radius of T is the number r(T ) := inf kT k : n ∈ N . (The rationale of this term will transpire later (cf. 8.1.12).) 5.6.7. The norm of T is greater than the spectral radius of T . C Indeed, by 5.6.1, the inequality kT n k ≤ kT kn is valid. B 5.6.8. The Gelfand formula holds: r(T ) = lim

p n kT n k.

Multinormed and Banach Spaces

79 1

C Take ε > 0 and let s ∈ N satisfy kT s k /s ≤ r(T )+ε. Given n ∈ N with n ≥ s, observe the presentation n = k(n)s + l(n) with k(n), l(n) ∈ N and 0 ≤ l(n) ≤ s − 1. Hence, kT n k = kT k(n)s T l(n) k ≤ kT s kk(n) kT l(n) k
≤ 1 ∨ kT k ∨ . . . ∨ kT s−1 k kT s kk(n) = M kT s kk(n) . Consequently, 1

r(T ) ≤ kT n k ≤M

1

/n

k(n)s

(r(T ) + ε)

/n

≤M

/n

=M

1

1

/n

/n

kT s k

k(n)

/n

(r(T ) + ε)

(n−l(n))

/n

.

1 1 Since M /n 0 = 0 O and (n−l(n)) /n 0 = 0 O , find r(T ) ≤ lim supkT n k /n ≤ r(T ) + ε. 1 The inequality lim inf kT n k /n ≥ r(T ) is evident. Recall that ε is arbitrary, thus completing the proof. B

5.6.9. Neumann Series Expansion Theorem. With X a Banach space and T ∈ B(X), the following statements are equivalent: (1) the Neumann series 1+T +T 2 +· · · converges in the operator norm of B(X); (2) kT k k < 1 for some k and N; (3) r(T ) < 1. P∞ If one of the conditions (1)–(3) holds then k=0 T k = (1 − T )−1 . C (1) ⇒ (2): With the Neumann series convergent, the general term (T k ) tends to zero. (2) ⇒ (3): This is evident. (3) ⇒ (1): According to 5.6.8, given a suitable ε > 0 and a sufficiently large 1 k ∈ N, observe that r(T ) ≤ kT k k /k ≤ r(T ) + ε < 1. In other words, some tail P∞ of the series k=0 kT k k is dominated by a convergent series. The completeness P∞ k of B(X) and 5.5.3 imply that k=0 T converges in B(X). Pn P∞ Now let S := k=0 T k and Sn := k=0 T k . Then S(1 − T ) = lim Sn (1 − T ) = lim (1 + T + · · · + T n ) (1 − T ) = lim(1 − T n+1 ) = 1;
(1 − T )S = lim(1 − T )Sn = lim(1 − T )(1 + T + · · · + T n ) = lim 1 − T n+1 = 1, because T n 0 = 0 O . Thus, by 2.2.7 S = (1 − T )−1 . B

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Chapter 5

5.6.10. Corollary. If kT k < 1 then (1 − T ) is invertible (= has a bounded inverse); i.e., the inverse correspondence (1 − T )−1 is a bounded linear operator. Moreover, k(1 − T )−1 k ≤ (1 − kT k)−1 . C The Neumann series converges and ∞ ∞ X X −1 k k(1 − T ) k ≤ kT k ≤ kT kk = (1 − kT k)−1 . . k=0

k=0

5.6.11. Corollary. If k1 − T k < 1 then T is invertible and k1 − T k . k1 − T −1 k ≤ 1 − k1 − T k C By Theorem 5.6.9, ∞ ∞ X X k (1 − T )k = (1 − (1 − T ))−1 = T −1 . (1 − T ) = 1+ k=0

k=1

Hence, kT

−1

− 1k = k

∞ X k=1

k

(1 − T ) k ≤

∞ X

k

k(1 − T ) k ≤

k=1

∞ X

k1 − T kk . .

k=1

5.6.12. Banach Inversion Stability Theorem. Let X and Y be Banach spaces. The set of invertible operators Inv(X, Y ) is open. Moreover, the inversion T 7→ T −1 acting from Inv(X, Y ) to Inv(Y, X) is continuous. C Take S, T ∈ B(X, Y ) such that T −1 ∈ B(Y, X) and kT −1 k kS − T k ≤ 1 /2 . Consider the operator T −1 S in B(X). Observe that k1 − T −1 Sk = kT −1 T − T −1 Sk ≤ kT −1 k kT − Sk ≤ 1 /2 < 1. Hence, by 5.6.11, (T −1 S)−1 belongs to B(X). Put R := (T −1 S)−1 T −1 . Clearly, R ∈ B(Y, X) and, moreover, R = S −1 (T −1 )−1 T −1 = S −1 . Further, kS −1 k − kT −1 k ≤ kS −1 − T −1 k = kS −1 (T − S)T −1 k ≤ kS −1 k kT − Sk kT −1 k ≤ 1 /2 kS −1 k. This implies kS −1 k ≤ 2kT −1 k, yielding the inequalities kS −1 − T −1 k ≤ kS −1 k kT − Sk kT −1 k ≤ 2kT −1 k2 kT − Sk. . 5.6.13. Definition. If X is a Banach space over F and T ∈ B(X) then a scalar λ ∈ F is a regular or resolvent value of T whenever (λ − T )−1 ∈ B(X). In this case put R (T, λ) := (λ − T )−1 and say that R (T, λ) is the resolvent of T at λ. The set of the resolvent values of T is denoted by res(T ) and called the resolvent set of T . The mapping λ 7→ R (T, λ) from res(T ) into B(X) is naturally called the resolvent of T . The set F \ res(T ) is referred to as the spectrum of T and is denoted by Sp(T ) or σ(T ). A member of Sp(T ) is said to be a spectral value of T (which is enigmatic for the time being).

Multinormed and Banach Spaces

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5.6.14. Remark. If X = 0 then the spectrum of the only operator T = 0 in B(X) is the empty set. In this regard, in spectral analysis it is silently presumed that X 6= 0. In the case X 6= 0 for F := R the spectra of some operators can also be void, whereas for F := C it is impossible (cf. 8.1.11). CB 5.6.15. The set res(T ) is open. If λ0 ∈ res(T ) then R (T, λ) =

∞ X

(−1)k (λ − λ0 )k R (T, λ0 )k+1

k=0

in some neighborhood of λ0 . If |λ| > kT k then λ ∈ res(T ) and the expansion ∞

1 X Tk R (T, λ) = λ λk k=0

holds. Moreover, kR (T, λ)k0 = 0 O as |λ|0 = 0 O ∞. C Since k(λ−T )−(λ0 −T )k = |λ−λ0 |, the openness property of res(T ) follows from 5.6.12. Proceed along the lines λ − T = (λ − λ0 ) + (λ0 − T ) = (λ0 − T )R (T, λ0 )(λ − λ0 ) + (λ0 − T ) = (λ0 − T )((λ − λ0 )R (T, λ0 ) + 1) = (λ0 − T )(1 − ((−1)(λ − λ0 )R (T, λ0 ))). In a suitable neighborhood of λ0 , from 5.6.9 derive R (T, λ) = (λ − T )−1 ∞ X −1 −1 = (1 − ((−1)(λ − λ0 )R (T, λ0 ))) (λ0 − T ) = (−1)k (λ − λ0 )k R (T, λ0 )k+1 . k=0

According to 5.6.9, for |λ| > kT k there is an operator 1 − T /λ the sum of the Neumann series; i.e., ∞


−1 1 1 X Tk T R (T, λ) = 1 − /λ = . λ λ λk k=0

It is clear that kR (T, λ)k ≤

1 1 · .. |λ| 1 − kT k /|λ|

5.6.16. The spectrum of every operator is compact. CB


−1

presenting

82

Chapter 5

5.6.17. Remark. It is worth keeping in mind that the inequality |λ| > r(T ) is a necessary condition for the convergence of the Laurent series, P∞and ksufficient k+1 R (T, λ) = k=0 T /λ , which expands the resolvent of T at infinity. 5.6.18. An operator S commutes with an operator T if and only if S commutes with the resolvent of T . C ⇒: ST = T S ⇒ S(λ − T ) = λS − ST = λS − T S = (λ − T )S ⇒ R (T, λ)S(λ − T ) = S ⇒ R (T, λ)S = S R (T, λ) (λ ∈ res(T )). ⇐: SR (T, λ0 ) = R (T, λ0 )S ⇒ S = R (T, λ0 )S(λ0 − T ) ⇒ (λ0 − T )S = S(λ0 − T ) ⇒ T S = ST . B 5.6.19. If λ, µ ∈ res(T ) then the first resolvent equation, the Hilbert identity, holds: R (T, λ) − R (T, µ) = (µ − λ)R (T, µ)R (T, λ). C “Multiplying the equality µ − λ = (µ − T ) − (λ − T ), first, by R (T, λ) from the right and, second, by R (T, µ) from the left,” successively infer the sought identity. B 5.6.20. If λ, µ ∈ res(T ) then R (T, λ)R (T, µ) = R (T, µ)R (T, λ). CB 5.6.21. For λ ∈ res(T ) the equality holds: dk R (T, λ) = (−1)k k! R (T, λ)k+1 . /. dλk 5.6.22. Composition Spectrum Theorem. Sp(T S) may differ only by zero.

The spectra Sp(ST ) and

C It suffices to establish that 1 6∈ Sp(ST ) ⇒ 1 6∈ Sp(T S). Indeed, from λ 6∈ Sp(ST ) and λ 6= 0 it will follow that 1 6∈ 1 /λ Sp(ST ) ⇒ 1 6∈ Sp

1


/λ ST ⇒ 1 6∈ Sp

1


/λ T S ⇒ λ 6∈ Sp(T S).

Therefore, consider the case 1 6∈ Sp(ST ). The formal Neumann series expansions (1 − ST )−1 ∼ 1 + ST + (ST )(ST ) + (ST )(ST )(ST ) + · · · , T (1 − ST )−1 S ∼ T S + T ST S + T ST ST S + · · · ∼ (1 − T S)−1 − 1 lead to conjecturing that the presentation is valid: (1 − T S)−1 = 1 + T (1 − ST )−1 S

Multinormed and Banach Spaces

83

(which in turn means 1 6∈ Sp(T S)). Straightforward calculation demonstrates the above presentation, thus completing the entire proof: (1 + T (1 − ST )−1 S)(1 − T S) = 1 + T (1 − ST )−1 S − T S + T (1 − ST )−1 (−ST )S = 1 + T (1 − ST )−1 S − T S + T (1 − ST )−1 (1 − ST − 1)S = 1 + T (1 − ST )−1 S − T S + T S − T (1 − ST )−1 S = 1; (1 − T S)(1 + T (1 − ST )−1 S) = 1 − T S + T (1 − ST )−1 S + T (−ST )(1 − ST )−1 S = 1 − T S + T (1 − ST )−1 S + T (1 − ST − 1)(1 − ST )−1 S = 1 − T S + T (1 − ST )−1 S + T S − T (1 − ST )−1 S = 1. . Exercises 5.1. Prove that a normed space is finite-dimensional if and only if every linear functional on the space is bounded. 5.2. Demonstrate that it is possible to introduce a norm into each vector space. 5.3. Show that a vector space X is finite-dimensional if and only if all norms on X are equivalent to each other. 5.4. Demonstrate that all separated multinorms introduce the same topology in a finitedimensional space. 5.5. Each norm on RN is appropriate for norming the product of finitely many normed spaces, isn’t it? 5.6. Find conditions for continuity of an operator acting between multinormed spaces and having finite-dimensional range. 5.7. Describe the operator norms in the space of square matrices. When are such norms comparable? 5.8. Calculate the distance between hyperplanes in a normed space. 5.9. Find the general form of a continuous linear functional on a classical Banach space. 5.10. Study the question of reflexivity for classical Banach spaces. 5.11. Find the mutual disposition of the spaces lp and lp0 as well as Lp and Lp0 . When is the complement of one element of every pair is dense in the other? 5.12. Find the spectrum and resolvent of the Volterra operator (the taking of a primitive), a projection, and a rank-one operator. 5.13. Construct an operator whose spectrum is a prescribed nonempty compact set in C. 5.14. Prove that the identity operator (in a nonzero space) is never the commutator of any pair of operators. 5.15. Is it possible to define some reasonable spectrum for an operator in a multinormed space? 5.16. Does every Banach space over F admit an isometric embedding into the space C(Q, F), with Q a compact space?

84

Chapter 5 5.17. Find out when Lp (X)0 = Lp0 (X 0 ), with X a Banach space. 5.18. Let (Xn ) be a sequence of normed spaces and let

 X0 =

x∈

Y

Xn : kxn k0 = 0

O



n∈N

be their c0 -sum (with the norm kxk = sup{ kxn k : n ∈ N} induced from the l∞ -sum). Prove that X0 is separable if and only if so is each of the spaces Xn . 5.19. Prove that the space C (p) [0, 1] presents the sum of a finite-dimensional subspace and a space isomorphic to C[0, 1].

Chapter 6 Hilbert Spaces

6.1. Hermitian Forms and Inner Products 1. Definition. Let H be a vector space over a basic field F. A mapping f : H 2 0 = 0 O F is a hermitian form on H provided that (1) the mapping f ( · , y) : x 7→ f (x, y) belongs to H # for every y in Y ; (2) f (x, y) = f (y, x)∗ for all x, y ∈ H, where λ 7→ λ∗ is the natural involution in F; that is, the taking of the complex conjugate of a complex number. 2. Remark. It is easy to see that, for a hermitian form f and each x in H the mapping f (x, · ) : y 7→ (x, y) lies in H∗# , where H∗ is the twin of H (see 2.1.4 (2)). Consequently, in case F := R every hermitian form is bilinear, i.e., linear in each argument; and in case F := C, sesquilinear, i.e., linear in the first argument and ∗-linear in the second. 3. Every hermitian form f satisfies the polarization identity: f (x + y, x + y) − f (x − y, x − y) = 4 Re f (x, y) (x, y ∈ H). C

−

f (x + y, x + y) = f (x, x) + f (x, y) + f (y, x) + f (y, y) f (x − y, x − y) = f (x, x) − f (x, y) − f (y, x) + f (y, y) 2(f (x, y) + f (y, x)) B

4. Definition. A hermitian form f is positive or positive semidefinite provided that f (x, x) ≥ 0 for all x ∈ H. In this event, write (x, y) := hx | yi := f (x, y) (x, y ∈ H). A positive hermitian form is usually referred to as a semi-inner product on H. A semi-inner product on H is an inner product or a (positive definite) scalar product whenever (x, x) = 0 ⇒ x = 0 with x ∈ H. 5. The Cauchy–Bunyakovski˘ı–Schwarz inequality holds: |(x, y)|2 ≤ (x, x)(y, y) (x, y ∈ H).

86

Chapter 6

C If (x, x) = (y, y) = 0 then 0 ≤ (x+ty, x +ty) = t(x, y)∗ +t∗ (x, y). Letting t := −(x, y), find that −2|(x, y)|2 ≥ 0; i.e., in this case the claim is established. If, for definiteness, (y, y) 6= 0; then in view of the estimate 0 ≤ (x + ty, x + ty) = (x, x) + 2t Re(x, y) + t2 (y, y)

(t ∈ R)

conclude that Re(x, y)2 ≤ (x, x)(y, y). If (x, y) = 0 then nothing is left to proof. If (x, y) 6= 0 then let θ := |(x, y)| (x, y)−1 and x := θx. Now |θ| = 1 and, furthermore, (x, x) = (θx, θx) = θθ∗ (x, x) = |θ|2 (x, x) = (x, x); |(x, y)| = θ(x, y) = (θx, y) = (x, y) = Re(x, y). Consequently, |(x, y)|2 = Re(x, y)2 ≤ (x, x)(y, y). . 1

6. If ( · , · ) is a semi-inner product on H then the mapping k·k : x 7→ (x, x) is a seminorm on H.

/2

C It suffices to prove the triangle inequality. Applying the Cauchy–Bunyakovski˘ı–Schwarz inequality, observe that kx + yk2 = (x, x) + (y, y) + 2 Re(x, y) ≤ (x, x) + (y, y) + 2kxk kyk = (kxk + kyk)2 . . 7. Definition. A space H endowed with a semi-inner product ( · , · ) and the associate seminorm k · k is a pre-Hilbert space. A pre-Hilbert space H is a Hilbert space provided that the seminormed space (H, k · k) is a Banach space. 8. In a pre-Hilbert space H, the Parallelogram Law is effective kx + yk2 + kx − yk2 = 2(kxk2 + kyk2 )

(x, y ∈ H)

which reads: the sum of the squares of the lengths of the diagonals equals the sum of the squares of the lengths of all sides. / kx + yk2 = (x + y, x + y) = kxk2 + 2 Re(x, y) + kyk2 ; kx − yk2 = (x − y, x − y) = kxk2 − 2 Re(x, y) + kyk2 . 9. Von Neumann–Jordan Theorem. If a seminormed space (H, k · k) obeys the Parallelogram Law then H is a pre-Hilbert space; i.e., there is a unique semi1 inner product ( · , · ) on H such that kxk = (x, x) /2 for all x ∈ H.

Hilbert Spaces

87

C Considering the real carrier HR of H and x, y ∈ HR , put
(x, y)R := 1 /4 kx + yk2 − kx − yk2 . Given the mapping ( · , y)R , from the Parallelogram Law successively derive (x1 , y)R + (x2 , y)R
= 1 /4 kx1 + yk2 − kx1 − yk2 + kx2 + yk2 − kx2 − yk2


= 1 /4 kx1 + yk2 + kx2 + yk2 − kx1 − yk2 + kx2 − yk2
= 1 /4 1 /2 (k(x1 + y) + (x2 + y)k2 + kx1 − x2 k2
−1 /2 k(x1 − y) + (x2 − y)k2 + kx1 − x2 k2 )
= 1 /4 1 /2 kx1 + x2 + 2yk2 − 1 /2 kx1 + x2 − 2yk2 

2 

2

(x1 −x2 )

(x1 +x2 )

1 /2 − y = /2 /2 + y −   = 2 (x1 +x2 ) /2 , y . R

In particular, (x2 , y)R = 0 in case x2 := 0, i.e. 1 /2 (x1 , y)R = gously, given x1 := 2x1 and x2 := 2x2 , infer that

1

/2 x1 , y


R

. Analo-

(x1 + x2 , y)R = (x1 , y)R + (x2 , y)R . Since the mapping ( · , y)R is continuous for obvious reasons, conclude that ( · , y)R ∈ (HR )# . Put (x, y) := (Re−1 ( · , y)R )(x), where Re−1 is the complexifier (see 3.7.5). In case F := R it is clear that (x, y) = (x, y)R = (y, x) and (x, x) = kxk2 ; i.e., nothing is to be proven. On the other hand, if F := C then (x, y) = (x, y)R − i(ix, y)R . This entails sesquilinearity: (y, x) = (y, x)R − i(iy, x)R = (x, y)R − i(x, iy)R = (x, y)R + i(ix, y)R = (x, y)∗ , since
(x, iy)R = 1 /4 kx + iyk2 − kx − iyk2
= 1 /4 |i| ky − ixk2 − | − i| kix + yk2 = −(ix, y)R .

88

Chapter 6 Furthermore, (x, x) = (x, x)R − i(ix, x)R
= kx2 k − i /4 kix + xk2 − kix − xk2

= kxk2 1 − i /4 |1 + i|2 − |1 − i|2 = kxk2 .

The claim of uniqueness follows from 6.1.3. B 10. Examples. (1) A Hilbert space is exemplified by the L2 space (over R some system with integration), the inner P product introduced as follows (f, g) := f g ∗ for f, g ∈ L2 . In particular, (x, y) := e∈E xe ye∗ for x, y ∈ l2 (E ). (2) Assume that H is a pre-Hilbert space and ( · , · ) : H 2 0 = 0 O F is a semi-inner product on H. It is clear that the real carrier HR with the semiinner product ( · , · )R : (x, y) 7→ Re(x, y) presents a pre-Hilbert space with the norm of an element of H independent of whether it is calculated in H or in HR . The pre-Hilbert space (HR , ( · , · )R ) is the realification or decomplexification of (H, ( · , · )). In turn, if the real carrier of a seminormed space is a pre-Hilbert space then the process of complexification leads to some natural pre-Hilbert structure of the original space. (3) Assume that H is a pre-Hilbert space and H∗ is the twin vector space of H. Given x, y ∈ H∗ , put (x, y)∗ := (x, y)∗ . Clearly, ( · , · )∗ is a semiinner product on H∗ . The resulting pre-Hilbert space is the twin of H, with the denotation H∗ preserved. (4) Let H be a pre-Hilbert space and let H0 := ker k · k be the kernel of the seminorm k · k on H. Using the Cauchy–Bunyakovski˘ı–Schwarz inequality, Theorem 2.3.8 and 6.1.10 (3), observe that there is a natural inner product on the quotient space H/H0 : If x1 := ϕ(x1 ) and x2 := ϕ(x2 ), with x1 , x2 ∈ H and ϕ : H0 = 0 O /H0 the coset mapping, then (x1 , x2 ) := (x1 , x2 ). Moreover, the preHilbert space H/H0 may be considered as the quotient space of the seminormed space (H, k·k) by the kernel of the seminorm k·k. Thus, H/H0 is a Hausdorff space referred to as the Hausdorff pre-Hilbert space associated with H. Completing the normed space H/H0 , obtain a Hilbert space (for instance, by the von Neumann– Jordan Theorem). The so-constructed Hilbert space is called associated with the original pre-Hilbert space. (5) Assume that (He )e∈E is a family of Hilbert spaces and H is the 2-sum of the family; i.e., h ∈ H if and only if h := (he )e∈E , where he ∈ He for e ∈ E and !1 /2 X khk := khe k2 < +∞. e∈E

Hilbert Spaces

89

By 5.5.9 (6), H is a Banach space. Given f, g ∈ H, on successively applying the Parallelogram Law, deduce that 1

/2 kf + gk2 + kf − gk2


!

1

= /2

X

2

kfe + ge k + +

e∈E

=

X

1

X

2

kfe − ge k

e∈E


X
/2 kfe + ge k2 + kfe − ge k2 = kfe k2 + kge k2 = kf k2 + kgk2 .

e∈E

e∈E

Consequently, H is a Hilbert space by the von Neumann–Jordan Theorem. The space H, the Hilbert sum of the family (He )e∈E , is denoted by ⊕e∈E He . With E := N, it is customary to use the symbol H1 ⊕ H2 ⊕ . . . for H. (6) Let H be a Hilbert space and let S be a system with integration. The space L2 (S, H) comprising all H-valued square-integrable functions is also a Hilbert space. CB 6.2. Orthoprojections 6.2.1. Let Uε be a convex subset of the spherical layer (r + ε)BH \ rBH with r, ε > 0 in a Hilbert space H. Then the diameter of Uε vanishes as ε tends to 0. C Given x, y ∈ Uε , on considering that 1 /2 (x + y) ∈ Uε and applying the Parallelogram Law, for ε ≤ r infer that 2

2

kx − yk = 2 kxk + kyk

2






(x+y) 2 − 4 /2

≤ 4(r + ε)2 − 4r2 = 8rε + 4ε2 ≤ 12rε. . 6.2.2. Levy Projection Theorem. Let U be a nonempty closed convex set in a Hilbert space H and x ∈ H \ U . Then there is a unique element u0 of U such that kx − u0 k = inf{kx − uk : u ∈ U }. C Put Uε := {u ∈ U : kx − uk ≤ inf kU − xk + ε}. By 6.2.1, the family (Uε )ε>0 constitutes a base for a Cauchy filter in U . B 6.2.3. Definition. The element u0 appearing in 6.2.2 is the best approximation to x in U or the projection of x to U . 6.2.4. Let H0 be a closed subspace of a Hilbert space H and x ∈ H \ H0 . An element x0 of H0 is the projection of x to H0 if and only if (x − x0 , h0 ) = 0 for every h0 in H0 .

90

Chapter 6

C It suffices to consider the real carrier (H0 )R of H0 . The convex function f (h0 ) := (h0 − x, h0 − x) is defined on (H0 )R . Further, x0 in H0 serves as the projection of x to H0 if and only if 0 ∈ ∂x0 (f ). In view of 3.5.2 (4) this containment means that (x − x0 , h0 ) = 0 for every h0 in H0 , because f 0 (x0 ) = 2(x0 − x, · ). B 6.2.5. Definition. Elements x and y of H are orthogonal, in symbols x ⊥ y, if (x, y) = 0. By U ⊥ we denote the subset of H that comprises all elements orthogonal to every point of a given subset U ; i.e., U ⊥ := {y ∈ H : (∀ x ∈ U ) x ⊥ y}. The set U ⊥ is the orthogonal complement or orthocomplement of U (to H). 6.2.6. Let H0 be a closed subspace of a Hilbert space H. The orthogonal complement of H0 , the set H0⊥ , is a closed subspace and H = H0 ⊕ H0⊥ . C The closure property of H0⊥ in H is evident. It is also clear that H0 ∧ H0⊥ = H0 ∩ H0⊥ = 0. We are left with showing only that H0 ∨ H0⊥ = H0 + H0⊥ = H. Take an element h of H \ H0 . In virtue of 6.2.2 the projection h0 of H to H0 is available and, by 6.2.4, h − h0 ∈ H0⊥ . Finally, h = h0 + (h − h0 ) ∈ H0 + H0⊥ . . 6.2.7. Definition. The projection onto a (closed) subspace H0 along H0⊥ is the orthoprojection onto H0 , denoted by PH0 . 6.2.8. Pythagoras Lemma. x ⊥ y ⇒ kx + yk2 = kxk2 + kyk2 . /. 6.2.9. Corollary. The norm of an orthoprojection is at most one: (H 6= 0 & H0 6= 0) ⇒ kPH0 k = 1. /. 6.2.10. Orthoprojection Theorem. For an operator P in L (H) such that P = P , the following statements are equivalent: (1) P is the orthoprojection onto H0 := im P ; (2) khk ≤ 1 ⇒ kP hk ≤ 1; (3) (P x, P d y) = 0 for all x, y ∈ H, with P d the complement of P , i.e. P d = IH − P ; (4) (P x, y) = (x, P y) for x, y ∈ H. C (1) ⇒ (2): This is observed in 6.2.9. (2) ⇒ (3): Let H1 := ker P = im P d . Take x ∈ H1⊥ . Since x = P x + P d x and x ⊥ P d x; therefore, kxk2 ≥ kP xk2 = (x − P d x, x − P d x) = (x, x) − 2 Re(x, P d x) + (P d x, P d x) = kxk2 + kP d xk2 . Whence P d x = 0; i.e., x ∈ im P . Considering 6.2.6, from H1 = ker P and H1⊥ ⊂ im P deduce the equalities H1⊥ = im P = H0 . Thus (P x, P d y) = 0 for all x, y ∈ H, since P x ∈ H0 and P d y ∈ H1 . (3) ⇒ (4): (P x, y) = (P x, P y + P d y) = (P x, P y) = (P x, P y) + (P d x, P y) = (x, P y). 2

Hilbert Spaces

91

(4) ⇒ (1): Show first that H0 is a closed subspace. Let h0 := lim hn with hn ∈ H0 , i.e. P hn = hn . For every x in H, from the continuity property of the functionals ( · , x) and ( · , P x) successively derive (h0 , x) = lim (hn , x) = lim (P hn , x) = lim (hn , P x) = (P h0 , x). Whence (h0 − P h0 , h0 − P h0 ) = 0; i.e., h0 ∈ im P . Given x ∈ H and h0 ∈ H0 , now infer that (x − P x, h0 ) = (x − P x, P h0 ) = (P (x − P x), h0 ) = (P x − P 2 x, h0 ) = (P x − P x, h0 ) = 0. Therefore, from 6.2.4 obtain P x = PH0 x. . 6.2.11. Let P1 , and P2 be orthoprojections with P1 P2 = 0. Then P2 P1 = 0. / P1 P2 = 0 ⇒ im P2 ⊂ ker P1 ⇒ im P1 = (ker P1 )⊥ ⊂ (im P2 )⊥ = ker P2 ⇒ P2 P1 = 0 . 6.2.12. Definition. Orthoprojections P1 and P2 are orthogonal, in symbols, P1 ⊥ P2 or P2 ⊥ P1 , provided that P1 P2 = 0. 6.2.13. Theorem. Let P1 , . . . , Pn be orthoprojections. The operator P := P1 + . . . + Pn is an orthoprojection if and only if Pl ⊥ Pm for l 6= m. C ⇒: First, given an orthoprojection P0 , observe that kP0 xk2 = (P0 x, P0 x) = (P02 x, x) = (P0 x, x) by Theorem 6.2.10. Consequently, kPl xk2 + kPm xk2 n n X X 2 ≤ kPk xk = (Pk x, x) = (P x, x) = kP xk2 ≤ kxk2 k=1

k=1

for x ∈ H and l 6= m. In particular, putting x := Pl x, observe that kPl xk2 + kPm Pl xk2 kPl xk2 ⇒ kPm Pl k = 0. ⇐: Straightforward calculation shows that P is an idempotent operator. Indeed, 2

P =

n X k=1

!2 Pk

=

n n X X l=1 m=1

Pl Pm =

n X

Pk2 = P.

k=1

Furthermore, in virtue of 6.2.10 (4), (Pk x, y) = (x, Pk y) and so (P x, y) = (x, P y). It suffices to appeal to 6.2.10 (4) once again. B 6.2.14. Remark. Theorem 6.2.13 is usually referred to as the pairwise orthogonality criterion for finitely many orthoprojections.

92

Chapter 6 6.3. A Hilbert Basis

6.3.1. Definition. A family (xe )e∈E of elements of a Hilbert space H is orthogonal, if e1 6= e2 ⇒ xe1 ⊥ xe2 . By specification, a subset E of a Hilbert space H is orthogonal if so is the family (e)e∈E . 6.3.2. Pythagoras Theorem. An orthogonal family (xe )e∈E of elements of a Hilbert space is (unconditionally) summable if and only if the numeric family (kxe k2 )e∈E is summable. Moreover,

X

2 X

xe = kxe k2 .

e∈E

e∈E

2 e∈E xe , where θ is a finite subset of E . By 6.2.8, ksθ k = P C Let2 sθ := 0 e∈θ kxe k . Given a finite subset θ of E which includes θ, thus observe that X ksθ0 − sθ k2 = ksθ0 \θ k2 = kxe k2 .

P

e∈θ 0 \θ

In other words, the fundamentalness of (sθ ) amounts to the fundamentalness of the net of partial sums of the family (kxe k2 )e∈E . On using 4.5.4, complete the proof. B 6.3.3. Orthoprojection Summation Theorem. Let (Pe )e∈E be a family of pairwise orthogonal orthoprojections in a Hilbert space H. Then for every x in H the P family (Pe x)e∈E is (unconditionally) summable. Moreover, the operator P x := e∈E Pe x is the orthoprojection onto the subspace ( ) X X H := xe : xe ∈ He := im Pe , kxe k2 < +∞ . e∈E

e∈E

P C Given a finite subset θ of E , put sθ := e∈θ Pe . ByPTheorem 6.2.13, sθ is 2 2 an orthoprojection. Hence, in view of 6.2.8, ksθ xk2 = e∈θ kPe xk ≤ kxk for every x in H. Consequently, the family (kPe xk2 )e∈E is summable (the net of partial sums isPincreasing and bounded). By the Pythagoras Theorem, there 2 is a sum P x := e∈E Pe x; i.e., P x = limθ sθ x. Whence P x = limθ sθ P x = limθ sθ limθ0 sθ0 x = limθ limθ0 sθ sθ0 x = limθ limθ0 sθ∩θ0 x = limθ sθ x = P x. Finally, kP xk = k limθ sθ xk = limθ ksθ xk ≤ kxk and, moreover, P 2 = P . Appealing to 6.2.10, conclude that P is the orthoprojection onto im P . P If x ∈ im P , i.e., P x = x; then x = P e∈E e x and by the Pythagoras Theorem P 2 2 2 kxk = kP xk < +∞. Since Pe x ∈ e∈E kPe xk = P PHe (e ∈ E ); therefore, x ∈ H . 2 If xe ∈ He and e∈E kxe k < +∞ then for x := P follows from e∈E xe (existence P the same Pythagoras Theorem) observe that x = e∈E xe = e∈E Pe xe = P x; i.e., x ∈ im P . Thus, im P = H . .

Hilbert Spaces

93

6.3.4. Remark. This theorem may be treated as asserting that the space H and the Hilbert sum of the family (He )e∈E are isomorphic. The identification is clearly accomplished by the Bochner integral presenting the process of summation in this case. 6.3.5. Remark. Let h in H be a normalized or unit or norm-one element; i.e., khk = 1. Assume further that H0 := Fh is a one-dimensional subspace of H spanned over h0 . For every element x of H and every scalar λ, a member of F, observe that (x − (x, h)h, λh) = λ∗ ((x, h) − (x, h))(h, h) = 0. Therefore, by 6.2.4, PH0 = ( · , h) ⊗ h. To denote this orthoprojection, it is convenient to use the symbol hhi. Thus, hhi : x 7→ (x, h)h (x ∈ H). 6.3.6. Definition. A family of elements of a Hilbert space is called orthonormal (or orthonormalized) if, first, the family is orthogonal and, second, the norm of each member of the family equals one. Orthonormal sets are defined by specification. 6.3.7. For every orthonormal subset E of H and every element x of H, the family (heix)e∈E is (unconditionally) summable. Moreover, the Bessel inequality holds: X kxk2 ≥ |(x, e)|2 . e∈E

C It suffices to refer to the Orthoprojection Summation Theorem, for

2

2

X

X

X



2 kxk2 ≥ heix = (x, e)e = k(x, e)ek . .



e∈E

e∈E

e∈E

6.3.8. Definition. An orthonormal set E in a Hilbert space H is a Hilbert P basis (for H) if x = e∈E heix for every x in H. An orthonormal family of elements of a Hilbert space is a Hilbert basis if the range of the family is a Hilbert basis. 6.3.9. An orthonormal set E is a Hilbert basis for H if and only if lin(E ), the linear span of E , is dense in H. CB 6.3.10. Definition. A subset E of a Hilbert space is said to meet the Steklov condition if E ⊥ = 0. 6.3.11. Steklov Theorem. An orthonormal set E is a Hilbert basis if and only if E meets the Steklov condition. P P P C ⇒: Let h ∈ E ⊥ . Then h = e∈E heih = e∈E (h, e)e = e∈E 0 = 0. P ⇐: For x ∈ H, in virtue of 6.3.3 and 6.2.4, x − e∈E heix ∈ E ⊥ . .

94

Chapter 6

6.3.12. Theorem. Each Hilbert space has a Hilbert basis. C By the Kuratowski–Zorn Lemma, each Hilbert space H has an orthonormal set E maximal by inclusion. If there were some h in H \ H0 , with H0 := cl lin(E ); then the element h1 := h − PH0 h would be orthogonal to every element in E . Thus, for H 6= 0 we would have E ∪ {kh1 k−1 h1 } = E . A contradiction. In case H = 0 there is nothing left to proof. B 6.3.13. Remark. It is possible to show that two Hilbert bases for a Hilbert space H have the same cardinality. This cardinality is the Hilbert dimension of H. 6.3.14. Remark. Let (xn )n∈N be a countable sequence of linearly independent elements of a Hilbert space H. Put x0 := 0, e0 := 0 and yn := xn −

n−1 X

hek ixn ,

en :=

k=0

yn kyn k

(n ∈ N).

Evidently, (yn , ek ) = 0 for 0 ≤ k ≤ n − 1 (for instance, by 6.2.13). Also, yn 6= 0, since H is infinite-dimensional. Say that the orthonormal sequence (en )n∈N results from the sequence (xn )n∈N by the Gram–Schmidt orthogonalization process. Using the process, it is easy to prove that a Hilbert space has a countable Hilbert basis if and only if the space has a countable dense subset; i.e., whenever the space is separable. CB 6.3.15. Definition. Let E be a Hilbert basis for a space H and x ∈ H. The numeric family x b := (b xe )e∈E in F E , given by the identity x be := (x, e), is the Fourier coefficient family of x with respect to E or the Fourier transform of X (relative to E ). 6.3.16. Riesz–Fisher Isomorphism Theorem. Let E be a Hilbert basis for H. The Fourier transform F : x 7→ x b (relative to E ) is an isometric isomorphism of H onto l2 (E ). The inverse Fourier transform, the Fourier summation F −1 : P −1 l2 (E )0 = 0 O , acts by the rule F (x) := e∈E xe e for x := (xe )e∈E ∈ l2 (E ). Moreover, for all x, y ∈ H the Parseval identity holds: X (x, y) = x be ybe∗ . e∈E

C By the Pythagoras Theorem, the Fourier transform acts in l2 (E ). By Theorem 6.3.3, b is an epimorphism. By the Steklov Theorem, b is a monomorphism. −1 (x) = x for x ∈ l (E ). It is beyond a doubt that F −1 x b = x for x ∈ H and F\ 2 The equality X kxk2 = kb xe k2 = kb xk22 (x ∈ H) e∈E

Hilbert Spaces

95

follows from the Pythagoras Theorem. At the same time ! X X X X x be ybe∗0 (e, e0 ) = x be ybe∗ . . ybe e = (x, y) = x be e, e∈E

e∈E

e,e0 ∈E

e∈E

6.3.17. Remark. The Parseval identity shows that the Fourier transform preserves inner products. Therefore, the Fourier transform is a unitary operator or a Hilbert-space isomorphism; i.e., an isomorphism preserving inner products. This is why the Riesz–Fisher Theorem is sometimes referred to as the theorem on Hilbert isomorphy between Hilbert spaces (of the same Hilbert dimension). 6.4. The Adjoint of an Operator 6.4.1. Riesz Prime Theorem. Let H be a Hilbert space. Given x ∈ H, put x0 := ( · , x). Then the prime mapping x 7→ x0 presents an isometric isomorphism of H∗ onto H 0 . C It is clear that x = 0 ⇒ x0 = 0. If x 6= 0 then kx0 kH 0 = sup |(y, x)| ≤ sup kyk kxk ≤ kxk; kyk≤1

kyk≤1

kx0 kH 0 = sup |(y, x)| ≥ |(x/kxk, x)| = kxk. kyk≤1

Therefore, x 7→ x0 is an isometry of H∗ into H 0 . Check that this mapping is an epimorphism. Let l ∈ H 0 and H0 := ker l 6= H (if there no such l then nothing is to be proven). Choose a norm-one element e in H0⊥ and put grad l := l(e)∗ e. If x ∈ H0 then (grad l)0 (x) = (x, grad l) = (x, l(e)∗ e) = l(e)∗∗ (x, e) = 0. Consequently, for some α in F and all x ∈ H in virtue of 2.3.12 (grad l)0 (x) = αl(x). In particular, letting x := e, find (grad l)0 (e) = (e, grad l) = l(e)(e, e) = αl(e); i.e., α = 1. B 6.4.2. Remark. From the Riesz Prime Theorem it follows that the dual space H 0 possesses a natural structure of a Hilbert space and the prime mapping x 7→ x0 implements a Hilbert space isomorphism from H∗ onto H 0 . The inverse mapping now coincides with the gradient mapping l 7→ grad l constructed in the proof of the theorem. Implying this, the claim of 6.4.1 is referred to as the theorem on the general form of a linear functional in a Hilbert space.

96

Chapter 6

6.4.3. Each Hilbert space is reflexive. C Let ι : H0 = 0 O 00 be the double prime mapping; i.e. the canonical embedding of H into the second dual H 00 which is determined by the rule x00 (l) = ι(x)(l) = l(x), where x ∈ H and l ∈ H 0 (see 5.1.10 (8)). Check that ι is an epimorphism. Let f ∈ H 00 . Consider the mapping y 7→ f (y 0 ) for y ∈ H. It is clear that this mapping is a linear functional over H∗ and so by the Riesz Prime Theorem there is an element x ∈ H = H∗∗ such that (y, x)∗ = (x, y) = f (y 0 ) for every y in H. Observe that ι(x)(y 0 ) = y 0 (x) = (x, y) = f (y 0 ) for all y ∈ H. Since by the Riesz Prime Theorem y 7→ y 0 is a mapping onto H 0 , conclude that ι(x) = f. . 6.4.4. Let H1 and H2 be Hilbert spaces and T ∈ B(H1 , H2 ). Then there is a unique mapping T ∗ : H2 0 = 0 O 1 such that (T x, y) = (x, T ∗ y) for all x ∈ H1 and y ∈ H2 . Moreover, T ∗ ∈ B(H2 , H1 ) and kT ∗ k = kT k. C Let y ∈ H2 . The mapping x 7→ (T x, y) is the composition y 0 ◦ T ; i.e., it presents a continuous linear functional over H1 . By the Riesz Prime Theorem there is precisely one element x of H1 for which x0 = y 0 ◦ T . Put T ∗ y := x. It is clear that T ∗ ∈ L (H2 , H1 ). Furthermore, using the Cauchy–Bunyakovski˘ı– Schwarz inequality and the normative inequality, infer that |(T ∗ y, T ∗ y)| = |(T T ∗ y, y)| ≤ kT T ∗ yk kyk ≤ kT k kT ∗ yk kyk. Hence, kT ∗ yk ≤ kT k kyk for all y ∈ H2 ; i.e., kT ∗ k ≤ kT k. At the same time T = T ∗∗ := (T ∗ )∗ ; i.e., kT k = kT ∗∗ k ≤ kT ∗ k. . 6.4.5. Definition. For T ∈ B(H1 , H2 ), the operator T ∗ , the member of B(H2 , H1 ) constructed in 6.4.4, is the adjoint of T . The terms like “hermitianconjugate” and “Hilbert-space adjoint” are also in current usage. 6.4.6. Let H1 and H2 be Hilbert spaces. Assume further that S, T ∈ B(H1 , H2 ) and λ ∈ F. Then (1) T ∗∗ = T ; (2) (S + T )∗ = S ∗ + T ∗ ; (3) (λT )∗ = λ∗ T ∗ ; (4) kT ∗ T k = kT k2 . C (1)–(3) are obvious properties. If kxk ≤ 1 then kT xk2 = (T x, T x) = |(T x, T x)| = |(T ∗ T x, x)| ≤ kT ∗ T xk kxk ≤ kT ∗ T k. Furthermore, using the submultiplicativity of the operator norm and 6.4.4, infer kT ∗ T k ≤ kT ∗ k kT k = kT k2 , which proves (4). B

Hilbert Spaces

97

6.4.7. Let H1 , H2 , and H3 be three Hilbert spaces. Assume further that T ∈ B(H1 , H2 ) and S ∈ B(H2 , H3 ). Then (ST )∗ = T ∗ S ∗ . C (ST x, z) = (T x, S ∗ z) = (x, T ∗ S ∗ z)

(x ∈ H1 , z ∈ H3 ) . T

6.4.8. Definition. Consider an elementary diagram H1 −→ H2 . The diagram T∗ H1 ←− H2 is the adjoint of the initial elementary diagram. Given an arbitrary diagram composed of bounded linear mappings between Hilbert spaces, assume that each elementary subdiagram is replaced with its adjoint. Then the resulting diagram is the adjoint or, for suggestiveness, the diagram star of the initial diagram. 6.4.9. Diagram Star Principle. A diagram is commutative if and only if so is its adjoint diagram. C Follows from 6.4.7 and 6.4.6 (1). B 6.4.10. Corollary. An operator T is invertible if and only if T ∗ is invertible. Moreover, T ∗−1 = T −1∗ . CB 6.4.11. Corollary. If T ∈ B(H) then λ ∈ Sp(T ) ⇔ λ∗ ∈ Sp(T ∗ ). CB 6.4.12. Sequence Star Principle (cf. 7.6.13). A sequence ...0 = 0 O

−−−k−→ Hk −−−−→ Hk+1 0 = 0 O Tk+1

T

k−1

is exact if and only if so is the sequence star Tk∗

∗ Tk+1

. . . ← Hk−1 ←−−−− Hk ←−−−− Hk+1 ← . . . . /. 6.4.13. Definition. An involutive algebra or ∗-algebra A (over a ground field F) is an algebra with an involution ∗, i.e. with a mapping a 7→ a∗ from A to A such that (1) a∗∗ = a (a ∈ A); (2) (a + b)∗ = a∗ + b∗ (a, b ∈ A); (3) (λa)∗ = λ∗ a∗ (λ ∈ F, a ∈ A); (4) (ab)∗ = b∗ a∗ (a, b ∈ A). A Banach algebra A with involution ∗ satisfying ka∗ ak = kak2 for all a ∈ A is a C ∗ -algebra. 6.4.14. The endomorphism space B(H) of a Hilbert space H is a C ∗ -algebra (with the composition of operators as multiplication and the taking of the adjoint of an endomorphism as involution). CB

98

Chapter 6 6.5. Hermitian Operators

6.5.1. Definition. Let H be a Hilbert space over a ground field F. An element T of B(H) is a hermitian operator or selfadjoint operator in H provided that T = T ∗. 6.5.2. Rayleigh Theorem. For a hermitian operator T the equality holds: kT k = sup |(T x, x)|. kxk≤1

C Put t := sup{|(T x, x)| : kxk ≤ 1}. It is clear that |(T x, x)| ≤ kT xk kxk ≤ kT k provided kxk ≤ 1. Thus, t ≤ kT k. Since T = T ∗ ; therefore, (T x, y) = (x, T y) = (T y, x)∗ = (y, T x)∗ ; i.e., (x, y) 7→ (T x, y) is a hermitian form. Consequently, in virtue of 6.1.3 and 6.1.8 4 Re(T x, y) = (T (x + y), x + y) − (T (x − y), x − y) ≤ t(kx + yk2 + kx − yk2 ) = 2t(kxk2 + kyk2 ). If T x = 0 then it is plain that kT xk ≤ t. Assume T x 6= 0. Given kxk ≤ 1 and putting y := kT xk−1 T x, infer that 

 Tx Tx , kT xk = kT xk kT xk kT xk 

2  = (T x, y) = Re(T x, y) ≤ 1 /2 t kxk2 + T x /kT xk ≤ t; i.e., kT k = sup{kT xk : kxk ≤ 1} ≤ t. . 6.5.3. Remark. As mentioned in the proof of 6.5.2, each hermitian operator T in a Hilbert space H generates the hermitian form fT (x, y) := (T x, y). Conversely, let f be a hermitian form, with the functional f ( · , y) continuous for every y in H. Then by the Riesz Prime Theorem there is an element T y of H such that f ( · , y) = (T y)0 . Evidently, T ∈ L (H) and (x, T y) = f (x, y) = f (y, x)∗ = (y, T x)∗ = (T x, y). It is possible to show that in this case T ∈ B(H) and T = T ∗ . In addition, f = fT . Therefore, the condition T ∈ B(H) in Definition 6.5.1 can be replaced with the condition T ∈ L (H) (the Hellinger–Toeplitz Theorem). 6.5.4. Weyl Criterion. A scalar λ belongs to the spectrum of a hermitian operator T if and only if inf kλx − T xk = 0. kxk=1

C ⇒: Put t := inf{kλx − T xk : x ∈ H, kxk = 1} > 0. Demonstrate that λ∈ / Sp(T ). Given an x in H, observe that kλx − T xk ≥ tkxk. Thus, first, (λ − T )

Hilbert Spaces

99

is a monomorphism; second, H0 := im(λ − T ) is a closed subspace (because k(λ − T )xm − (λ − T )xk k ≥ tkxm − xk k; i.e., the inverse image of a Cauchy sequence is a Cauchy sequence); and, third, which is final, (λ − T )−1 ∈ B(H) whenever H = H0 (in such situation kR (T, λ)k ≤ t−1 ). Suppose to the contrary that H 6= H0 . Then there is some y in H0⊥ satisfying kyk = 1. For all x ∈ H, note that 0 = (λx − T x, y) = (x, λ∗ y − T y); i.e., λ∗ y = T y. Further, λ∗ = (T y, y)/(y, y) and the hermiticity of T guarantees λ∗ ∈ R. Whence λ∗ = λ and y ∈ ker(λ − T ). We arrive at a contradiction: 1 = kyk = k0k = 0. ⇐: If λ ∈ / Sp(T ) then the resolvent of T at λ, the member R (T, λ) of B(H), is available. Hence, inf{kλx − T xk : kxk = 1} ≥ kR (T, λ)k−1 . . 6.5.5. Spectral Endpoint Theorem. Let T be a hermitian operator in a Hilbert space. Put mT := inf (T x, x), kxk=1

MT := sup (T x, x). kxk=1

Then Sp(T ) ⊂ [mT , MT ] and mT , MT ∈ Sp(T ). C Considering that the operator T − Re λ is hermitian in the space H under study, from the identity kλx − T xk2 = | Im λ|2 kxk2 + kT x − Re λxk2 infer the inclusion Sp(T ) ⊂ R by 6.5.4. Given a norm-one element x of H and invoking the Cauchy–Bunyakovski˘ı–Schwarz inequality, in case λ < mT deduce that kλx − T xk = kλx − T xk kxk ≥ |(λx − T x, x)| = |λ − (T x, x)| = (T x, x) − λ ≥ mT − λ > 0. On appealing to 6.5.4, find λ ∈ res(T ). In case λ > MT , similarly infer that kλx − T xk ≥ |(λx − T x, x)| = |λ − (T x, x)| = λ − (T x, x) ≥ λ − MT > 0. Once again λ ∈ res(T ). Finally, Sp(T ) ⊂ [mT , MT ]. Since (T x, x) ∈ R for x ∈ H; therefore, in virtue of 6.5.2 kT k = sup{|(T x, x)| : kxk ≤ 1} = sup{(T x, x) ∨ (−(T x, x)) : kxk ≤ 1} = MT ∨ (−mT ). Assume first that λ := kT k = MT . If kxk = 1 then kλx − T xk2 = λ2 − 2λ(T x, x) + kT xk2 ≤ 2kT k2 − 2kT k(T x, x).

100

Chapter 6

In other words, the next estimate holds: inf kλx − T xk2 ≤ 2kT k inf (kT k − (T x, x)) = 0.

kxk=1

kxk=1

Using 6.5.4, conclude that λ ∈ Sp(T ). Now consider the operator S := T −mT . It is clear that MS = MT −mT ≥ 0 and mS = mT − mT = 0. Therefore, kSk = MS and in view of the above MS ∈ Sp(S). Whence it follows that MT belongs to Sp(T ), since T = S+mT and MT = MS +mT . It suffices to observe that mT = −M−T and Sp(T ) = − Sp(−T ). . 6.5.6. Corollary. The norm of a hermitian operator equals the radius of its spectrum (and the spectral radius). CB 6.5.7. Corollary. A hermitian operator is zero if and only if its spectrum consists of zero. CB 6.6. Compact Hermitian Operators 6.6.1. Definition. Let X and Y be Banach spaces. An operator T , a member of L (X, Y ), is called compact (in symbols, T ∈ K (X, Y )) if the image T (BX ) of the unit ball BX of X is relatively compact in Y . 6.6.2. Remark. Detailed study of compact operators in Banach spaces is the purpose of the Riesz–Schauder theory to be exposed in Chapter 8. 6.6.3. Let T be a compact hermitian operator. If 0 6= λ ∈ Sp(T ) then λ is an eigenvalue of T ; i.e., ker(λ − T ) 6= 0. C By the Weyl Criterion, λxn − T xn 0 = 0 O for some sequence (xn ) such that kxn k = 1. Without loss of generality, assume that the sequence (T xn ) converges to y := lim T xn . Then from the identity λxn = (λxn − T xn ) + T xn obtain that there is a limit (λxn ) and y = lim λxn . Consequently, T y = T (lim λxn ) = λ lim T xn = λy. Since kyk = |λ|, conclude that y is an eigenvector of T , i.e. y ∈ ker(λ − T ). B 6.6.4. Let λ1 and λ2 be distinct eigenvalues of a hermitian operator T . Assume further that x1 and x2 are eigenvectors with eigenvalues λ1 and λ2 (i.e., xs ∈ ker(λs − T ), s := 1, 2). Then x1 and x2 are orthogonal. / (x1 , x2 ) = λ11 (T x1 , x2 ) = λ11 (x1 , T x2 ) = λλ21 (x1 , x2 ) . 6.6.5. For whatever strictly positive ε, there are only finitely many eigenvalues of a compact hermitian operator beyond the interval [−ε, ε]. C Let (λn )n∈N be a sequence of pairwise distinct eigenvalues of T satisfying |λn | > ε. Further, let xn be an eigenvector corresponding to λn and such that kxn k = 1. By virtue of 6.6.4 (xk , xm ) = 0 for m 6= k. Consequently, kT xm − T xk k2 = kT xm k2 + kT xk k2 = λ2m + λ2k ≥ 2ε2 ; i.e., the sequence (T xn )n∈N is relatively compact. We arrive at a contradiction to the compactness property of T. .

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101

6.6.6. Spectral Decomposition Lemma. Let T be a compact hermitian operator in a Hilbert space H and 0 6= λ ∈ Sp(T ). Put Hλ := ker(λ − T ). Then Hλ is finite-dimensional and the decomposition H = Hλ ⊕ Hλ⊥ reduces T . Moreover, the matrix presentation holds   λ 0 T ∼ , 0 Tλ where the operator Tλ , the part of T in Hλ⊥ , is hermitian and compact, with Sp(Tλ ) = Sp(T ) \ {λ}. C The subspace Hλ is finite-dimensional in view of the compactness of T . Furthermore, Hλ is invariant under T . Consequently, the orthogonal complement Hλ⊥ of Hλ is an invariant subspace of T ∗ (coincident with T ), since h ∈ Hλ⊥ ⇒ (∀ x ∈ Hλ ) x ⊥ h ⇒ (∀ x ∈ Hλ ) 0 = (h, T x) = (T ∗ h, x) ⇒ T ∗ h ∈ Hλ⊥ . The part of T in Hλ is clearly λ. The part Tλ of T in Hλ⊥ is undoubtedly compact and hermitian. Obviously, for µ 6= λ, the operator   µ−λ 0 µ−T ∼ 0 µ − Tλ is invertible if and only if so is µ − Tλ . It is also clear that λ is not an eigenvalue of Tλ . . 6.6.7. Hilbert–Schmidt Theorem. Let H be a Hilbert space and let T be a compact hermitian operator in H. Assume further that Pλ is the orthoprojection onto ker(λ − T ) for λ ∈ Sp(T ). Then T =

X

λPλ .

λ∈Sp(T )

C Using 6.5.6 and 6.6.6 as many times as need be, for every finite subset θ of Sp(T ) obtain the equality



X

λPλ = sup{|λ| : λ ∈ (Sp(T ) ∪ 0) \ θ}.

T −

λ∈θ

It suffices to refer to 6.6.5. B 6.6.8. Remark. The Hilbert–Schmidt Theorem provides essentially new information, as compared with the case of finite dimensions, only if the operator T has infinite-rank, that is, the dimension of its range is infinite or, which is the same, H0⊥ is an infinite-dimensional space, where H0 := ker T . In fact, if the operator T

102

Chapter 6

has finite rank (i.e., its range is finite-dimensional) then, since the subspace H0⊥ is isomorphic with the range of T , observe that T =

n X

λk hek i =

k=1

n X

λk e0k ⊗ ek ,

k=1

where λ1 , . . . , λn are nonzero points of Sp(T ) counted with multiplicity, and {e1 , . . . , en } is a properly-chosen orthonormal basis for H0⊥ . The Hilbert–Schmidt Theorem shows that, to within substitution of series for sum, an infinite-rank compact hermitian operator looks like a finite-rank operator. Indeed, for λ 6= µ, where λ and µ are nonzero points of Sp(T ), the eigenspaces Hλ and Hµ are finite-dimensional and orthogonal. Moreover, the Hilbert sum ⊕λ∈Sp(T )\0 Hλ equals H0⊥ = cl im T , because H0 = (im T )⊥ . Successively selecting a basis for each finite-dimensional space Hλ by enumerating the eigenvalues in decreasing order of magnitudes with multiplicity counted; i.e., putting λ1 := λ2 := . . . := λdim Hλ1 := λ1 , λdim Hλ1 +1 := . . . := λdim Hλ1 +dim Hλ2 := λ2 , etc., obtain the decomposition H = H0 ⊕ Hλ1 ⊕ Hλ2 ⊕ . . . and the presentation T =

∞ X

λk hek i =

k=1

∞ X

λk e0k ⊗ ek ,

k=1

where the series is summed in operator norm. CB 6.6.9. Theorem. Let T in K (H1 , H2 ) be an infinite-rank compact operator from a Hilbert space H1 to a Hilbert space H2 . There are orthonormal families (ek )k∈N in H1 , (fk )k∈N in H2 , and a numeric family (µk )k∈N in R+ \ 0, µk ↓ 0, such that the following presentation holds: T =

∞ X

µk e0k ⊗ fk .

k=1

C Put S := T ∗ T . It is clear that S ∈ B(H1 ) and S is compact. Furthermore, (Sx, x) = (T ∗ T x, x) = (T x, T x) = kT xk2 . Consequently, in virtue of 6.4.6, S is hermitian and H0 := ker S = ker T . Observe also that Sp(S) ⊂ R+ by Theorem 6.5.5. Let (ek )k∈N be an orthonormal basis for H0⊥ comprising all eigenvalues of S and let (λk )k∈N be a corresponding decreasing sequence of strictly positive eigenvalues λk > 0, k ∈ N (cf. 6.6.8). Then every element x ∈ H1 expands into the Fourier series ∞ X x − PH 0 x = (x, ek )ek . k=1

Hilbert Spaces

103 √

Therefore, considering that T PH0 = 0 and assigning µk := λk and fk := µ−1 k T ek , find ∞ ∞ ∞ X X X µk Tx = (x, ek )T ek = µk (x, ek )fk . (x, ek ) T ek = µk k=1

k=1

k=1

The family (fk )k∈N is orthonormal because 



1 (T en , T em ) µn µm 1 1 = (T ∗ T en , em ) = (Sen , em ) µn µm µn µm 1 µn = (λn en , em ) = (en , em ). µn , µm µm

(fn , fm ) =

T en T em , µn µm

=

Successively using the Pythagoras Theorem and the Bessel inequality, argue as follows:

2

! 2 ∞ n

X

X



µk (x, ek )fk µk e0k ⊗ fk x =

T−



k=n+1

k=1

=

∞ X

µ2k |(x, ek )|2 ≤ λn+1

k=n+1

∞ X

|(x, ek )|2 ≤ λn+1 kxk2 .

k=n+1

Since λk ↓ 0, finally deduce that

n

X

µk e0k ⊗ fk ≤ µn+1 0 = 0 O . .

T −

k=1

6.6.10. Remark. Theorem 6.6.9 means in particular that a compact operator (and only a compact operator) is an adherent point of the set of finite-rank operators. This fact is also expressed as follows: “Every Hilbert space possesses the approximation property.” Exercises 6.1. Describe the extreme points of the unit ball of a Hilbert space. 6.2. Find out which classical Banach spaces are Hilbert spaces and which are not. 6.3. Is a quotient space of a Hilbert space also a Hilbert space? 6.4. Is it true that each Banach space may be embedded into a Hilbert space? 6.5. Is it possible that the (bounded) endomorphism space of a Hilbert space presents a Hilbert space?

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Chapter 6

6.6. 6.7. 6.8. at most n

Describe the second (= repeated) orthogonal complement of a set in a Hilbert space. Prove that no Hilbert basis for an infinite-dimensional Hilbert space is a Hamel basis. Find the best approximation to a polynomial of degree n + 1 by polynomials of degree in the L2 space on an interval.

6.9. Prove that x ⊥ y if and only if kx + yk2 = kxk2 + kyk2 and kx + iyk2 = kxk2 + kyk2 . 6.10. Given a bounded operator T , prove that (ker T )⊥ = cl im T ∗ ,

(im T )⊥ = ker T ∗ .

6.11. Reveal the interplay between hermitian forms and hermitian operators (cf. 6.5.3). 6.12. Find the adjoint of a shift operator, a multiplier, and a finite-rank operator. 6.13. Prove that an operator between Hilbert spaces is compact if and only if so is its adjoint. 6.14. Assume that an operator T is an isometry. Is T ∗ an isometry too? 6.15. A partial isometry is an operator isometric on the orthogonal complement of its kernel. What is the structure of the adjoint of a partial isometry? 6.16. What are the extreme points of the unit ball of the endomorphism space of a Hilbert space? 6.17. Prove that the weak topology of a separable Hilbert space becomes metrizable if restricted onto the unit ball. 6.18. Show that an idempotent operator P in a Hilbert space is an orthoprojection if and only if P commutes with P ∗ . 6.19. Let (akl )k,l∈N be an infinite matrix such that akl ≥ 0 for all k and l. Assume further that there are also pk and β, γ > 0 satisfying ∞ X k=1

akl pk ≤ βpl ;

∞ X

akl pl ≤ γpk

(k, l ∈ N).

l=1

Then there is some T in B(l2 ) such that (ek , el ) = akl and kT k = characteristic function of k, a member of N.

p

βγ, where ek is the

Chapter 7 Principles of Banach Spaces

7.1. Banach’s Fundamental Principle 7.1.1. Lemma. Let U be a convex set with nonempty interior in a multinormed space: int U 6= ∅. Then (1) 0 ≤ α < 1 ⇒ α cl U + (1 − α) int U ⊂ int U ; (2) core U = int U ; (3) cl U = cl int U ; (4) int cl U = int U . C (1): For u0 ∈ int U , the set int U − u0 is an open neighborhood of zero in virtue of 5.2.10. Whence, given 0 ≤ α < 1, obtain α cl U = cl αU ⊂ αU + (1 − α)(int U − u0 ) = αU + (1 − α) int U − (1 − α)u0 ⊂ αU + (1 − α)U − (1 − α)u0 ⊂ U − (1 − α)u0 . Thus, (1 − α)u0 + α cl U ⊂ U and so U includes (1 − α) int U + α cl U . The last set is open as presenting the sum of α cl U and (1 − α) int U , an open set. (2): Undoubtedly, int U ⊂ core U . If u0 ∈ int U and u ∈ core U then u = αu0 + (1 − α)u1 for some u1 in U and 0 < α < 1. Since u1 ∈ cl U , from (1) deduce that u ∈ int U . (3): Clearly, cl int U ⊂ cl U for int U ⊂ U . If, in turn, u ∈ cl U ; then, choosing u0 in the set int U and putting uα := αu0 +(1−α)u, infer that uα 0 = 0 O as α0 = 0 O and uα ∈ int U when 0 < α < 1. Thus, by construction u ∈ cl int U . (4): From the inclusions int U ⊂ U ⊂ cl U it follows that int U ⊂ int cl U . If now u ∈ int cl U then, in virtue of (2), u ∈ core cl U . Consequently, taking u0 in the set int U again, find u1 ∈ cl U and 0 < α < 1 satisfying u = αu0 + (1 − α)u1 . Using (1), finally infer that u ∈ int U . B

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7.1.2. Remark. In the case of finite dimensions, the condition int U 6= ∅ may be omitted in 7.1.1 (2) and 7.1.1 (4). In the opposite case, the presence of an interior point is an essential requirement, as shown by numerous examples. For instance, take U := Bc0 ∩ X, where c0 is the space of vanishing sequences and X is the subspace of terminating sequences in c0 , the direct sum of countably many copies of a basic field. Evidently, core U = ∅ and at the same time cl U = Bc0 . CB 7.1.3. Definition. A subset U of a (multi)normed space X is an ideally convex set in X, if U is closed under the taking of countable convex combinations. More precisely, P∞ U is ideally convex if for all sequences (αn )n∈N P∞and (un )n∈N , with αn ∈ R+ , n=1 αn = 1 and un ∈ U such that the series n=1 αn un converges in X to some u, the containment holds: u ∈ U . 7.1.4. Examples. (1) Translation (by a vector) preserves ideal convexity. CB (2) Every closed convex set is ideally convex. CB (3) Every open convex set is ideally convex. C Take an open and convex U . If U = ∅ then nothing is left to proof. If U 6= ∅ then by 7.1.4 (1) it may be assumed that 0 ∈ U and, consequently, U = {pU < 1}, where pU is the MinkowskiP functional of U . Let (un )n∈N and (αnP )n∈N be sequences ∞ ∞ in U and in R+ such that n=1 αn = 1, with the element u := n=1 αn un failing to lie in U . By virtue of 7.1.4 (2), u belongs to cl (u) = 1. PU∞ = {pU ≤ 1} and so pUP ∞ On the other hand, it P is clear that pU (u) ≤ α p (u ) ≤ 1 = n=1 n=1 αn P∞ n U n ∞ (cf. 7.2.1). Thus, 0 = n=1 (αn − αn pU (un )) = n=1 αn (1 − pU (un )). Whence αn = 0 for all n ∈ N. We arrive at a contradiction. B (4) The intersection of a family of ideally convex sets is ideally convex. CB (5) Every convex subset of a finite-dimensional space is ideally convex. CB 7.1.5. Banach’s Fundamental Principle. In a Banach space, each ideally convex set with absorbing closure is a neighborhood of zero. C Let U be such set in a Banach space X. By hypothesis X = ∪n∈N n cl U . By the Baire Category Theorem, X is nonmeager and so there is some n in N such that int n cl U 6= ∅. Therefore, int cl U = 1 /n int n cl U 6= ∅. By hypothesis 0 ∈ core cl U . Consequently, from 7.1.1 it follows that 0 ∈ int cl U . In other words, there is a δ > 0 such that cl U ⊃ δBX . Consequently, ε > 0 ⇒ cl 1 /ε U ⊃ δ /ε BX . Using the above implication, show that U ⊃ δ /2 BX .

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Let x0 ∈ δ /2 BX . Putting ε := 2, choose y1 ∈ 1 /ε U from ky1 −

the condition

x0 k ≤ 1 /2ε δ. Thus obtain an element u1 of U such that 1 /2 u1 − x0 ≤ 1 /2ε δ = 1 /4 δ. Now putting x0 := −1 /2 u1 + x0 and ε := 4 and applying of the

1 the argument 1

preceding paragraph, find an element u2 of U satisfying /4 u2 + /2 u1 − x0 ≤ 1 /2ε δ = 1 /8 δ. Proceeding by induction, construct a sequence P∞ 1 (un )n∈N of the points n of U which possesses the property that the series n=1 /2 un converges to x0 . P∞ 1 n Since n=1 /2 = 1 and the set U is ideally convex, deduce x0 ∈ U. . 7.1.6. For every ideally convex set U in a Banach space the following four sets coincide: the core of U , the interior of U , the core of the closure of U and the interior of the closure of U . C It is clear that int U ⊂ core U ⊂ core cl U . If u ∈ core cl U then cl(U − u) equal to cl U − u is an absorbing set. An ideally convex set translates into an ideally convex set (cf. 7.1.4 (1)). Consequently, U −u is a neighborhood of zero by Banach’s Fundamental Principle. By virtue of 5.2.10, u belongs to int U . Thus, int U = core U = core cl U . Using 7.1.1, conclude that int cl U = int U. . 7.1.7. The core and the interior of a closed convex set in a Banach space coincide. C A closed convex set is ideally convex. B 7.1.8. Remark. Inspection of the proof of 7.1.5 shows that the condition for the ambient space to be a Banach space in 7.1.7 is not utilized to a full extent. There are examples of incomplete normed spaces in which the core and interior of each closed convex set coincide. A space with this property is called barreled. The concept of barreledness is seen to make sense also in multinormed spaces. Barreled multinormed spaces are plentiful. In particular, such are all Fr´echet spaces. 7.1.9. Counterexample. Each infinite-dimensional Banach space contains absolutely convex, absorbing and not ideally convex sets. C Using, for instance, a Hamel basis, take a discontinuous linear functional f . Then the set {|f | ≤ 1} is what was sought. B 7.2. Boundedness Principles 7.2.1. Let p : X0 = 0 O R be a sublinear functional on a normed space (X, k·k). The following conditions are equivalent: (1) p is uniformly continuous; (2) p is continuous; (3) p is continuous at zero; (4) {p ≤ 1} is a neighborhood of zero; (5) kpk := sup{|p(x)| : kxk ≤ 1} < +∞; i.e., p is bounded.

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C The implications (1) ⇒ (2) ⇒ (3) ⇒ (4) are immediate. (4) ⇒ (5): There is some t > 0 such that t−1 BX ⊂ {p ≤ 1}. Given kxk ≤ 1, thus find p(x) ≤ t. In addition, the inequality −p(−x) ≤ p(x) implies that −p(x) ≤ t for x ∈ BX . Finally, kpk ≤ t < +∞. (5) ⇒ (1): From the subadditivity of p, given x, y ∈ X, observe that p(x) − p(y) ≤ p(x − y);

p(y) − p(x) ≤ p(y − x).

Whence |p(x) − p(y)| ≤ p(x − y) ∨ p(y − x) ≤ kpk kx − yk. . 7.2.2. Gelfand Theorem. Every lower semicontinuous sublinear functional with domain a Banach space is continuous. C Let p be such functional. Then the set {p ≤ 1} is closed (cf. 4.3.8). Since dom p is the whole space; therefore, by 3.8.8, {p ≤ 1} is an absorbing set. By Banach’s Fundamental Principle {p ≤ 1} is a neighborhood of zero. Application to 7.2.1 completes the proof. B 7.2.3. Remark. The Gelfand Theorem is stated amply as follows: “If X is a Banach space then each of the equivalent conditions 7.2.1 (1)–7.2.1 (5) amounts to the statement: ‘p is lower semicontinuous on X.’ ” Observe immediately that the requirement dom p = X may be slightly relaxed by assuming dom p to be a nonmeager linear set and withdrawing the condition for X to be a Banach space. 7.2.4. Equicontinuity Principle. Suppose that X is a Banach space and Y is a (semi)normed space. For every nonempty set E of continuous linear operators from X to Y the following statements are equivalent: (1) E is pointwise bounded; i.e., for all x ∈ X the set {T x : T ∈ E } is bounded in Y ; (2) E is equicontinuous. C (1) ⇒ (2): Put q(x) := sup{p(T x) : T ∈ E }, with p the (semi)norm of Y . Evidently, q is a lower semicontinuous sublinear functional and so by the Gelfand Theorem kqk < +∞; i.e., p(T (x − y)) ≤ kqk kx − yk for all T ∈ E . Consequently, T ×−1 ({dp ≤ ε}) ⊃ {dk·k ≤ ε/kqk} for every T in E , where ε > 0 is taken arbitrarily. This means the equicontinuity property of E . (2) ⇒ (1): Straightforward. B 7.2.5. Uniform Boundedness Pronciple. Let X be a Banach space and let Y be a normed space. For every nonempty family (Tξ )ξ∈„ of bounded operators the following statements are equivalent: (1) x ∈ X ⇒ supξ∈„ kTξ xk < +∞; (2) supξ∈„ kTξ k < +∞. C It suffices to observe that 7.2.5 (2) is another expression for 7.2.4 (2). B

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7.2.6. Let X be a Banach space and let U be a subset of X 0 . Then the following statements are equivalent: (1) U is bounded in X 0 ; (2) for every x in X the numeric set {hx | x0 i : x0 ∈ U } is bounded in F. C This is a particular case of 7.2.5. B 7.2.7. Let X be a normed space and let U be a subset of X. Then the following statements are equivalent: (1) U is bounded in X; (2) for every x0 in X 0 the numeric set {hx | x0 i : x ∈ U } is bounded in F. C Only (2) ⇒ (1) needs examining. Observe that X 0 is a Banach space (cf. 5.5.7) and X is isometrically embedded into X 00 by the double prime mapping (cf. 5.1.10 (8)). So, the claim follows from 7.2.6. B 7.2.8. Remark. The message of 7.2.7 (2) may be reformulated as “U is bounded in the space (X, σ(X, X 0 ))” or, in view of 5.1.10 (4), as “U is weakly bounded.” The duality between 7.2.6 and 7.2.7 is perfectly revealed in 10.4.6. 7.2.9. Banach–Steinhaus Theorem. Let X and Y be Banach spaces. Assume further that (Tn )n∈N , Tn ∈ B(X, Y ), is a sequence of bounded operators. Put E := {x ∈ X : ∃ lim Tn x}. The following conditions are equivalent: (1) E = X; (2) supn∈N kTn k < +∞ and E is dense in X. Under either (and, hence, both) of the conditions (1) and (2) the mapping T0 : X0 = 0 O , defined as T0 x := lim Tn x, presents a bounded linear operator and kT0 k ≤ lim inf kTn k. C If E = X then, of course, cl E = X. In addition, for every x in X the sequence (Tn x)n∈N is bounded in Y (for, it converges). Consequently, by the Uniform Boundedness Principle supn∈N kTn k < +∞ and (1) ⇒ (2) is proven. If (2) holds and x ∈ X then, given x ∈ E and m, k ∈ N, infer that kTm x − Tk xk = kTm x − Tm x + Tm x − Tk x + Tk x − Tk xk ≤ kTm x − Tm xk + kTm x − Tk xk + kTk x − Tk xk ≤ kTm k kx − xk + kTm x − Tk xk + kTk k kx − xk ≤ 2 sup kTn k kx − xk + kTm x − Tk xk. n∈N

Take ε > 0 and choose, first, x ∈ E such that 2 supn kTn k kx − xk ≤ ε /2 , and, second, n ∈ N such that kTm x − Tk xk ≤ ε /2 for m, k ≥ n. By virtue of what was

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proven kTm x − Tk xk ≤ ε; i.e., (Tn x)n∈N is a Cauchy sequence in Y . Since Y is a Banach space, conclude that x ∈ E. Thus, (2) ⇒ (1) is proven. To complete the proof it suffices to observe that kT0 xk = lim kTn xk ≤ lim inf kTn k kxk for all x ∈ X, because every norm is a continuous function. B 7.2.10. Remark. Under the hypotheses of the Banach–Steinhaus Theorem, the validity of either of the equivalent items 7.2.9 (1) and 7.2.9 (2) implies that (Tn ) converges to T0 compactly on X (= uniformly on every compact subset of X). In other words, sup kTn x − T0 xk0 = 0 O x∈Q

for every (nonempty) compact set Q in X. C Indeed, it follows from the Gelfand Theorem that the sublinear functional pn (x) := sup{kTm x − T0 xk : m ≥ n} is continuous. Moreover, pn (x) ≥ pn+1 (x) and pn (x)0 = 0 O for all x ∈ X. Consequently, the claim follows from the Dini Theorem: “Each decreasing sequence of continuous real functions which converges pointwise to a continuous function on a compact set converges uniformly.” B 7.2.11. Singularity Fixation Principle. Let X be a Banach space and let Y be a normed space. If (Tn )n∈N is a sequence of operators, Tn ∈ B(X, Y ) and supn kTn k = +∞ then there is a point x of X satisfying supn kTn xk = +∞. The set of such points “fixing a singularity” is residual. C The first part of the assertion is contained in the Uniform Boundedness Principle. The second requires referring to 7.2.3 and 4.7.4. B 7.2.12. Singularity Condensation Principle. Let X be a Banach space and let Y be a normed space. If (Tn,m )n,m∈N is a family of operators, Tn ∈ B(X, Y ), such that supn kTn,m k = +∞ for every m ∈ N then there is a point x of X satisfying supn kTn,m xk = +∞ for all m ∈ N. CB 7.3. The Ideal Correspondence Principle 7.3.1. Let X and Y be vector spaces. A correspondence F ⊂ X × Y is convex if and only if for x1 , x2 ∈ X and α1 , α2 ∈ R+ such that α1 + α2 = 1, the inclusion holds: F (α1 x1 + α2 x2 ) ⊃ α1 F (x1 ) + α2 F (x2 ). C ⇐: If (x1 , y1 ), (x2 , y2 ) ∈ F and α1 , α2 ≥ 0, α1 +α2 = 1, then α1 y +α2 y2 ∈ F (α1 x1 + α2 x2 ) since y1 ∈ F (x1 ) and y2 ∈ F (x2 ). ⇒: If either x1 or x2 fails to enter in dom F then there is nothing to prove. If y1 ∈ F (x1 ) and y2 ∈ F (x2 ) with x1 , x2 ∈ dom F then α1 (x1 , y1 ) + α2 (x2 , y2 ) ∈ F for α1 , α2 ≥ 0, α1 + α2 = 1 (cf. 3.1.2 (8)). B

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7.3.2. Remark. Let X and Y be Banach spaces. It is clear that there are many ways for furnishing the space X × Y with a norm so that the norm topology be coincident with the product of the topologies τX and τY . For instance, it is possible to put k(x, y)k := kxkX + kykY ; i.e., to define the norm on X × Y as that of the 1-sum of X and Y . Observe immediately that the concept of ideally convex set has a linear topological character: the class of objects distinguished in a space is independent of the way of introducing the space topology; in particular, the class remains invariant under passage to an equivalent (multi)norm. In this connection the next definition is sound. 7.3.3. Definition. A correspondence F ⊂ X × Y , with X and Y Banach spaces, is called ideally convex or, briefly, ideal if F is an ideally convex set. 7.3.4. Ideal Correspondence Lemma. The image of a bounded ideally convex set under an ideal correspondence is an ideally convex set. C Let F ⊂ X × Y be an ideal correspondence and let U be a bounded ideally convex set in X. If U ∩ dom F = ∅ then F (U ) = ∅ and nothing is left unproven. Let now (yn )n∈N ⊂ F (U ); i.e., yn ∈ F (xn ), where xnP∈ U and n ∈ N. Let, finally, ∞ (αn ) be a sequence of positive P numbers such that n=1 αn = 1 and, moreover, ∞ there is a sum of the series y := n=1 αn yn in Y . It is beyond a doubt that ∞ X n=1

kαn xn k =

∞ X n=1

αn kxn k ≤

∞ X

αn sup kU k = sup kU k < +∞

n=1

in view of the boundedness property of U .P Since X is complete, from 5.5.3 it ∞ follows that X contains the element x := n=1 αn xn . Consequently, (x, y) = P∞ α (x , y ) in the space X × Y . Successively using the ideal convexity of F n n=1 n n and U , infer that (x, y) ∈ F and x ∈ U . Thus, y ∈ F (U ). . 7.3.5. Ideal Correspondence Principle. Let X and Y be Banach spaces. Assume further that F ⊂ X × Y is an ideal correspondence and (x, y) ∈ F . A correspondence F carries each neighborhood of x onto a neighborhood about y if and only if y ∈ core F (X). C ⇒: This is obvious. ⇐: On account of 7.1.4 it may be assumed that x = 0 and y = 0. Since each neighborhood of zero U includes εBX for some ε > 0, it suffices to settle the case U := BX . Since U is bounded; by 7.3.4, F (U ) is ideally convex. To complete the proof, it suffices to show that F (U ) is an absorbing set and to cite 7.1.6. Take an arbitrary element y of Y . Since by hypothesis 0 ∈ core F (X), there is a real α in R+ such that αy ∈ F (X). In other words, αy ∈ F (X) for some x in X.

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If kxk ≤ 1 then there is nothing to prove. If kxk > 1 then λ := kxk−1 < 1. Whence, using 7.3.1, infer that αλy = (1 − λ)0 + λαy ∈ (1 − λ)F (0) + λF (x) ⊂ F ((1 − λ)0 + λx) = F (λx) ⊂ F (BX ) = F (U ). Here use was made of the fact that kλxk = 1; i.e., λx ∈ BX . B 7.3.6. Remark. The property of F , described in 7.3.5, is referred to as the openness of F at (x, y). 7.3.7. Remark. The Ideal Correspondence Principle is formally weaker that Banach’s Fundamental Principle. Nevertheless, the gap is tiny and can be easily filled in. Namely, the conclusion of 7.3.5 remains valid if we suppose that y ∈ core cl F (X), on additionally requiring ideal convexity from F (X). The requirement is not too stringent and certainly valid provided that the domain of F is bounded in virtue of 7.3.4. As a result of this slight modification, 7.1.5 becomes a particular case of 7.3.5. In this connection the claim of 7.3.5 is often referred to as Banach’s Fundamental Principle for a Correspondence. 7.3.8. Definition. Let X and Y be Banach spaces and let F ⊂ X × Y be a correspondence. Then F is called closed if F is a closed set in X × Y . 7.3.9. Remark. For obvious reasons, a closed correspondence is often referred to as a closed-graph correspondence. 7.3.10. A correspondence F is closed if and only if for all sequences (xn ) in X and (yn ) in Y such that xn ∈ dom F, yn ∈ F (xn ) and xn 0 = 0 O , yn 0 = 0 O , it follows that x ∈ dom F and y ∈ F (x). CB 7.3.11. Assume that X and Y are Banach spaces and F ⊂ X × Y is a closed convex correspondence. Further, let (x, y) ∈ F and y ∈ core im F . Then F carries each neighborhood of x onto a neighborhood about y. C A closed convex set is ideally convex and so all follows from 7.3.5. B 7.3.12. Definition. A correspondence F ⊂ X × Y is called open if the image of each open set in X is an open set in Y . 7.3.13. Open Correspondence Principle. Let X and Y be Banach spaces and let F ⊂ X × Y be an ideal correspondence, with im F an open set. Then F is an open correspondence. C Let U be an open set in X. If y ∈ F (U ) then there is some x in U such that (x, y) ∈ F . It is clear that y ∈ core im F . By the criterion of 7.3.5 F (U ) is a neighborhood of y because U is a neighborhood of x. This means that F (U ) is an open set. B

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7.4. Open Mapping and Closed Graph Theorems 7.4.1. Definition. A member T of L (X, Y ) is a homomorphism, if T ∈ B(X, Y ) and T is an open correspondence. 7.4.2. Assume that X is a Banach space, Y is a normed space and T is a homomorphism from X to Y . Then im T = Y and Y is a Banach space. C It is obvious that im T = Y . Presuming T to be a monomorphism, observe that T −1 ∈ L (Y, X). Since T is open, T −1 belongs to B(Y, X), which ensures the completeness of Y (the inverse image of a Cauchy sequence in a subset is a Cauchy sequence in the inverse image of the subset). In the general case, consider the coimage coim T := X/ ker T endowed with the quotient norm. In virtue of 5.5.4, coim T is a Banach space. In addition, by 2.3.11 there is a unique quotient T of T by coim T , the monoquotient of T . Taking account of the definition of quotient norm and 5.1.3, conclude that T is a homomorphism. Furthermore, T is a monomorphism by definition. It remains to observe that im T = im T = Y. . 7.4.3. Remark. As regards the monoquotient T : coim T 0 = 0 O of T , it may be asserted that kT k = kT k. CB 7.4.4. Banach Homomorphism Theorem. Every bounded epimorphism from one Banach space onto the other is a homomorphism. C Let T ∈ B(X, Y ) and im T = Y . On applying the Open Correspondence Principle to T , complete the proof. B 7.4.5. Banach Isomorphism Theorem. Let X and Y be Banach spaces and T ∈ B(X, Y ). If T is an isomorphism of the vector spaces X and Y , i.e. ker T = 0 and im T = Y ; then T −1 ∈ B(Y, X). C A particular case of 7.4.4. B 7.4.6. Remark. The Banach Homomorphism Theorem is often referred to as the Open Mapping Theorem for understandable reasons. Theorem 7.4.5 is briefly formulated as follows: “A continuous (algebraic) isomorphism of Banach spaces is a topological isomorphism.” It is also worth observing that the theorem is sometimes referred to as the Well-Posedness Principle and verbalized as follows: “If an equation T x = y, with T ∈ B(X, Y ) and X and Y Banach spaces, is uniquely solvable given an arbitrary right side; then the solution x depends continuously on the right side y.” 7.4.7. Banach Closed Graph Theorem. Let X and Y be Banach spaces and let T in L (X, Y ) be a closed linear operator. Then T is continuous. C The correspondence T −1 is ideal and T −1 (Y ) = X. B

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7.4.8. Corollary. Suppose that X and Y are Banach spaces and T is a linear operator from X to Y . The following conditions are equivalent: (1) T ∈ B(X, Y ); (2) for every sequence (xn )n∈N in X, together with some x in X and y in Y satisfying xn 0 = 0 O and T xn 0 = 0 O , it happens that y = T x. C (2): This is a reformulation of the closure property of T . B 7.4.9. Definition. A subspace X1 of a Banach space X is complemented (rarely, a topologically complemented), if X1 is closed and, moreover, there is a closed subspace X2 such that X = X1 ⊕ X2 (i.e., X1 ∧ X2 = 0 and X1 ∨ X2 = X). These X1 and X2 are called complementary to one another. 7.4.10. Complementation Principle. For a subspace X1 of some Banach space X one of the following conditions amounts to the other: (1) X1 is complemented; (2) X1 is the range of a bounded projection; i.e., there is a member P of B(X) such that P 2 = P and im P = X1 . C (1) ⇒ (2): Let P be the projection of X onto X1 along X2 (cf. 2.2.9 (4)). Let (xn )n∈N be a sequence in X with xn 0 = 0 O and P xn 0 = 0 O . It is clear that P xn ∈ X1 for n ∈ N. Since X1 is closed, by 4.1.19 y ∈ X1 . Similarly, the condition (xn − P xn ∈ X2 for n ∈ N) implies that x − y ∈ X2 . Consequently, P (x − y) = 0. Furthermore, y = P y; i.e., y = P x. It remains to refer to 7.4.8. (2) ⇒ (1): It needs showing only that X1 equal to im P is closed. Take a sequence (xn )n∈N in X1 such that xn 0 = 0 O in X. Then P xn 0 = 0 O x in view of the boundedness of P . Obtain P xn = xn , because xn ∈ im P and P is an idempotent. Finally, x = P x, i.e. x ∈ X1 , what was required. B 7.4.11. Examples. (1) Every finite-dimensional subspace is complemented. CB (2) The space c0 is not complemented in l∞ . C It turns out more convenient to work with X := l∞ (Q) and Y := c0 (Q), where Q is the set of rational numbers. Given t ∈ R, choose a sequence (tn ) of pairwise distinct rational numbers other than t and such that tn 0 = 0 O . Let Qt := {tn : n ∈ N}. Observe that Qt0 ∩ Qt00 is a finite set if t0 6= t00 . Let χt be the coset containing the characteristic function of Qt in the quotient space X/Y and V := {χt : t ∈ R}. Since χt0 6= χt00 for t0 6= t00 , the set V is uncountable. Take f ∈ (X/Y )0 and put Vf := {v ∈ V : f (v) 6= 0}. It is evident that Vf = ∪n∈N Vf (n), where Vf (n) := {v ∈ V : |f (v)| ≥ 1 /n }. Given P m ∈ N and n pairwise distinct v1 , . . . , vm in Vf (n), putPαk := |f (vk )|/f (vkP ) and x := k=1 αk vk m m to find kxk ≤ 1 and kf k ≥ |f (x)| = | k=1 αk f (vk )| = | k=1 |f (vk )|| ≥ m /n .

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Hence, Vf (n) is a finite set. Consequently, Vf is countable. Whence it follows that for every countable subset F of (X/Y )0 there is an element v of V satisfying (∀ f ∈ F ) f (v) = 0. At the same time the countable set of the coordinate projections δq : x 7→ x(q) (q ∈ Q) is total over l∞ (Q); i.e., (∀ q ∈ Q) δq (x) = 0 ⇒ x = 0 for x ∈ l∞ (Q). It remains to compare the above observations. B (3) Every closed subspace of a Hilbert space is complemented in virtue of 6.2.6. Conversely, if, in an arbitrary Banach space X with dim X ≥ 3, each closed subspace is the range of some projection P with kP k ≤ 1; then X is isometrically isomorphic to a Hilbert space (this is the Kakutani Theorem). The next fact is much deeper: Lindenstrauss–TzafririTheorem. A Banach space having every closed subspace complemented is topologically isomorphic to a Hilbert space. 7.4.12. Sard Theorem. Suppose that X, Y, and Z are Banach spaces. Take A ∈ B(X, Y ) and B ∈ B(Y, Z). Suppose further that im A is a complemented subspace in Y. The diagram A - Y X @ X B@@ R ? Z is commutative for some X in B(Y, Z) if and only if ker A ⊂ ker B. C Only ⇐ needs examining. Moreover, in the case im A = Y the sole member X0 of L (Y, Z) such that X0 A = B is continuous. Indeed, X0−1 (U ) = A(B −1 (U )) for every open set U in Z. The set B −1 (U ) is open in virtue of the boundedness property of B, and A(B −1 (U )) is open by the Banach Homomorphism Theorem. In the general case, construct X0 ∈ B(im A, Z) and take as X the operator X0 P , where P is some continuous projection of Y onto im A. (Such a projection is available by the Complementation Principle.) B 7.4.13. Phillips Theorem. Suppose that X, Y, and Z are Banach spaces. Take A ∈ B(Y, X) and B ∈ B(Z, X). Suppose further that ker A is a complemented subspace of Y . The diagram A X  Y 6 @ I @ X B @ Z is commutative for some X in B(Z, Y ) if and only if im A ⊃ im B.

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C Once again only ⇐ needs examining. Using the definition of complemented subspace, express Y as the direct sum of ker A and Y0 , where Y0 is a closed subspace. By 5.5.9 (1), Y0 is a Banach space. Consider the part A0 of A in Y0 . Undoubtedly, im A0 = im A ⊃ im B. Consequently, by 2.3.13 and 2.3.14 the equation A0 X0 = B has a unique solution X0 := A−1 0 . It suffices to prove that the operator X0 , treated as a member of L (Z, Y0 ), is bounded. O The operator X0 is closed. Indeed (cf. 7.4.8), if zn 0 = 0 O and A−1 0 Bzn 0 = 0 O then Bzn 0 = 0 z, since B is bounded. In addition, by the continuity of A0 , the correspondence A−1 ⊂ X × Y0 is closed; and so 7.3.10 yields the equality 0 y = A−1 Bz. B 0 7.4.14. Remark. We use neither the completeness of Z in proving the Sard Theorem nor the completeness of X in proving the Phillips Theorem. 7.4.15. Remark. The Sard Theorem and the Phillips Theorem are in “formal duality”; i.e., one results from the other by reversing arrows and inclusions and substituting ranges for kernels (cf. 2.3.15). 7.4.16. Two Norm Principle. Let a vector space be complete in each of two comparable norms. Then the norms are equivalent. C For definiteness, assume that k · k2  k · k1 in X. Consider the diagram IX (X, k · k1 )  (X, k · k2 ) @ I @ IX @

6 X (X, k · k1 )

By the Phillips Theorem some continuous operator X makes the diagram commutative. Such an operator is unique: it is IX . B 7.4.17. Graph Norm Principle. Let X and Y be Banach spaces and let T in L (X, Y ) be a closed operator. Given x ∈ X, define the graph norm of x as kxkgr T := kxkX + kT xkY . Then k · kgr T ∼ k · kX . C Observe that the space (X, k · kgr T ) is complete. Further, k · kgr T ≥ k · kX . It remains to refer to the Two Norm Principle. B 7.4.18. Definition. A normed space X is a Banach range, if X is the range of some bounded operator given on some Banach space. 7.4.19. Kato Criterion. Let X be a Banach space and X = X1 ⊕ X2 , where X1 , X2 ∈ Lat(X). The subspaces X1 and X2 are closed if and only if each of them is a Banach range.

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C ⇒: A corollary to the Complementation Principle. ⇐: Let Z be a some Banach range, i.e. Z = T (Y ) for some Banach space Y and T ∈ B(Y, Z). Passing, if need be, to the monoquotient of T , we may assume that T is an isomorphism. Put kzk0 := kT −1 zkY . It is clear that (Z, k · k0 ) is a Banach space and kzk = kT T −1 zk ≤ kT kkT −1 zk = kT kkzk0 ; i.e., k · k0  k · kZ . Applying this construction to X1 and X2 , obtain Banach spaces (X1 , k · k1 ) and (X2 , k · k2 ). Now k · kk  k · kX on Xk for k := 1, 2. Given x1 ∈ X1 and x2 ∈ X2 , put kx1 + x2 k0 := kx1 k1 + kx2 k2 . Thereby we introduce in X some norm k · k that is stronger than the initial norm k · kX . By construction (X, k · k0 ) is a Banach space. It remains to refer to 7.4.16. B 7.5. The Automatic Continuity Principle 7.5.1. Lemma. Let f : X0 = 0 O R· be a convex function on a (multi)normed space X. The following statements are equivalent: (1) U := int dom f 6= ∅ and f |U is a continuous function; (2) there is a nonempty open set V such that sup f (V ) < +∞. C (1) ⇒ (2): This is obvious. (2) ⇒ (1): It is clear that U 6= ∅. Using 7.1.1, observe that each point u of U has a neighborhood W in which f is bounded above, i.e. t := sup f (W ) < +∞. Without loss of generality, it may be assumed that u := 0, f (u) := 0 and W is an absolutely convex set. From the convexity of f , for every α ∈ R+ such that α ≤ 1 and for an arbitrary v in W , obtain f (αv) = f (αv + (1 − α)0) ≤ αf (v) + (1 − α)f (0) = αf (v); f (αv) + αf (−v) ≥ f (αv) + f (α(−v))
= 2(1 /2 f αv) + 1 /2 f (−αv) ≥ 2f (0) = 0. Therefore, |f (αW )| ≤ αt, which implies that f is continuous at zero. B 7.5.2. Corollary. If x ∈ int dom f and f is continuous at x then the subdifferential ∂x (f ) contains only continuous functionals. C If l ∈ ∂x (f ) then (∀ x ∈ X) l(x) ≤ l(x) + f (x) − f (x) and so l is bounded above on some neighborhood about x. Consequently, l is continuous at x by 7.5.1. From 5.3.7 derive that l is continuous. B 7.5.3. Corollary. Every convex function on a finite-dimensional space is continuous on the interior of its domain. CB 7.5.4. Definition. A function f : X0 = 0 O R· is called ideally convex if 0 = 0 OO is an ideal correspondence.

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7.5.5. Automatic Continuity Principle Every ideally convex function on a Banach space is continuous on the core of its domain. C Let f be such function. If core dom f = ∅ then there is nothing to prove. If x ∈ core dom f then put t := f (x) and F := (0 = 0 OO )−1 ⊂ R × X. Applying the Ideal Correspondence Principle, find a δ > 0 from the condition F (t + BR ) ⊃ x + δBX . Whence, in particular, infer the estimate f (x + δBX ) ≤ t + 1. In virtue of 7.5.1, f is continuous on int dom f . Since x ∈ int dom f ; therefore, by Lemma 7.1.1, core dom f = int dom f . B 7.5.6. Remark. Using 7.3.6, it is possible to prove that an ideally convex function f , defined in a Banach space on a subset with nonempty core, is locally Lipschitz on int dom f . In other words, given x0 ∈ int dom f , there are a positive number L and a neighborhood U about x0 such that kf (x) − f (x0 )k ≤ Lkx − x0 k whenever x ∈ U . CB 7.5.7. Corollary. Let f : X0 = 0 O R· be an ideally convex function on a Banach space X and x ∈ core dom f . Then the directional derivative f 0 (x) is a continuous sublinear functional and ∂x (f ) ⊂ X 0 . C Apply the Automatic Continuity Principle twice. B 7.5.8. Remark. In view of 7.5.7, in studying a Banach space X, only continuous functionals on X are usually admitted into the subdifferential of a function f : X0 = 0 O R· at a point x; i.e., we agree to define ∂x (f ) := ∂x (f ) ∩ X 0 . Proceed likewise in (multi)normed spaces. If a need is felt to distinguish the “old” (wider) subdifferential, a subset of X # , from the “new” (narrower) subdifferential, a subset of X 0 ; then the first is called algebraic, whereas the second is called topological. With this in mind, we refer to the facts indicated in 7.5.2 and 7.5.7 as to the Coincidence Principle for algebraic and topological subdifferentials. Observe finally that if f := p is a seminorm on X then, for similar reasons, it is customary to put |∂|(p) := |∂|(p) ∩ X 0 . 7.5.9. Ideal Hahn–Banach Theorem. Let f : Y 0 = 0 O R· be an ideally convex function on a Banach space Y . Further, let X be a normed space and T ∈ B(X, Y ). If a point x in X is such that T x ∈ core dom f then ∂x (f ◦ T ) = ∂ T x (f ) ◦ T. C The right side of the sought formula is included into its left side for obvious reasons. If l in X 0 belongs to ∂x (f ◦ T ), then by the Hahn–Banach Theorem there is an element l1 of the algebraic subdifferential of f at T x for which l = l1 ◦ T . It suffices to observe that, in virtue of 7.5.7, l1 is an element of Y 0 and so it is a member of the topological subdifferential ∂ T x (f ). B

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7.5.10. Balanced Hahn–Banach Theorem. Suppose that X and Y are normed spaces. Given T ∈ B(X, Y ), let p : Y 0 = 0 O R be a continuous seminorm. Then |∂|(p ◦ T ) = |∂|(p) ◦ T. C If l ∈ |∂|(p ◦ T ) then l = l1 ◦ T for some l1 in the algebraic balanced subdifferential of p (cf. 3.7.11). From 7.5.2 it follows that l1 is continuous. Thus, |∂|(p ◦ T ) ⊂ |∂|(p) ◦ T . The reverse inclusion raises no doubts. B 7.5.11. Continuous Extension Principle. Let X0 be a subspace of X and let l0 be a continuous linear functional on X0 . Then there is a continuous linear functional l on X extending l0 and such that klk = kl0 k. C Take p := kl0 k k · k, and consider the identical embedding ι : X0 0 = 0 O . On account of 7.5.10, l0 ∈ |∂|(p ◦ ι) = |∂|(p) ◦ ι = kl0 k |∂|(k · k) ◦ ι. It suffices to observe that |∂|(k · kX ) = BX 0 . B 7.5.12. Topological Separation Theorem. Let U be a convex set with nonempty interior in a space X. If L is an affine variety in X and L ∩ int U = ∅ then there is a closed hyperplane H in X such that H ⊃ L and H ∩ int U = ∅. CB 7.5.13. Remark. When applying Theorem 7.5.12, it is useful to bear in mind that a closed hyperplane is precisely a level set of a nonzero continuous linear functional. CB 7.5.14. Corollary. Let X0 be a subspace of X. Then cl X0 = ∩ {ker f : f ∈ X 0 , ker f ⊃ X0 }. C It is clear that (f ∈ X 0 & ker f ⊃ X0 ) ⇒ ker f ⊃ cl X0 . If x0 ∈ / cl X0 then there is an open convex neighborhood about x0 disjoint from cl X0 . In virtue of 7.5.12 and 7.5.13 there is a functional f0 , a member of (XR )0 , such that ker f0 ⊃ cl X0 and f0 (x0 ) = 1. From the properties of the complexifier infer that the functional Re−1 f0 vanishes on X0 and differs from zero at the point x0 . It is also beyond a doubt that the functional is continuous. B 7.6. Prime Principles 7.6.1. Let X and Y be (multi)normed vector spaces (over the same ground field F). Assume further that X 0 and Y 0 are the duals of X and Y respectively. Take a continuous linear operator T from X to Y . Then y 0 ◦ T ∈ X 0 for y 0 ∈ Y 0 and the mapping y 0 7→ y 0 ◦ T is a linear operator. CB 7.6.2. Definition. The operator T 0 : Y 0 0 = 0 O 0 , constructed in 7.6.1, is the dual or transpose of T : X0 = 0 O .

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7.6.3. Theorem. The prime mapping T 7→ T 0 implements a linear isometry of the space B(X, Y ) into the space B(Y 0 , X 0 ). C The prime mapping is clearly a linear operator from B(X, Y ) to L (Y 0 , X 0 ). Furthermore, since kyk = sup{|l(y) : l ∈ |∂|(k · k)}; therefore, kT 0 k = sup{kT 0 y 0 k : ky 0 k ≤ 1} = sup{|y 0 (T x)| : ky 0 k ≤ 1, kxk ≤ 1} = sup{kT xk : kxk ≤ 1} = kT k, what was required. B 7.6.4. Examples. (1) Let X and Y be Hilbert spaces. Take T ∈ B(X, Y ). Observe first that, in a plain sense, T ∈ B(X, Y ) ⇔ T ∈ B(X∗ , Y∗ ). Denote the prime mapping of X by (·)0X : X∗ 0 = 0 O 0 , i.e., x 7→ x0 := ( · , x); and denote the prime mapping of Y by (·)0Y : Y∗ 0 = 0 O 0 , i.e., y 7→ y 0 := ( · , y). The adjoint of T , the member T ∗ of B(Y, X), and the dual of T , the member T 0 of B(Y 0 , X 0 ), are related by the commutative diagram: T∗

X∗ ←−Y∗ 0 ↓ (·)0Y (·)X ↓ T0

X 0 ←−Y 0 C Indeed, it is necessary to show the equality T 0 y 0 = (T ∗ y)0 for y ∈ Y . Given x ∈ X, by definition observe that T 0 y 0 (x) = y 0 (T x) = (T x, y) = (x, T ∗ y) = (T ∗ y)0 (x). Since x is arbitrary, the proof is complete. B (2) Let ι : X0 0 = 0 O be the identical embedding of X0 into X. Then 0 ι : X0 = 0 O 00 . Moreover, ι0 (x0 )(x0 ) = x0 (x0 ) for all x0 ∈ X0 and x0 ∈ X 0 and ι0 is ι0 an epimorphism; i.e., X 0 −→ X 0 0 = 0 O is an exact sequence. CB 0

T

7.6.5. Definition. Let an elementary diagram X −→ Y be given. The diaT0

gram Y 0 −→ X 0 is referred to as resulting from setting primes or as the diagram prime of the original diagram or as the dual diagram. If primes are set in every elementary subdiagram in an arbitrary diagram composed of bounded linear operators in Banach spaces, then the so-obtained diagram is referred to as dual or resulting from setting primes in the original diagram. The term “diagram prime” is used for suggestiveness.

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7.6.6. Double Prime Lemma. Let X 00 −→ Y 00 be the diagram that results T from setting primes in the diagram X −→ Y twice. Then the following diagram commutes: T X −→Y 00 ↓ ↓00 T 00

X 00 −→Y 00 Here 00 : X0 = 0 O 00 and 00 : Y 0 = 0 O 00 are the respective double prime mappings; i.e. the canonical embeddings of X into X 00 and of Y into Y 00 (cf. 5.1.10 (8)). C Let x ∈ X. We have to prove that T 00 x00 = (T x)00 . Take y 0 ∈ Y 0 . Then T 00 x00 (y 0 ) = x00 (T 0 y 0 ) = T 0 y 0 (x) = y 0 (T x) = (T x)00 (y 0 ). Since y 0 ∈ Y 0 is arbitrary, the proof is complete. B 7.6.7. Diagram Prime Principle. A diagram is commutative if and only if so is its diagram prime. C It suffices to convince oneself that the triangles T

X −→ Y R& .S Z

T0

X 0 ←− Y 0 0 0 R - %S 0 Z

are commutative or not simultaneously. Since R = ST ⇒ R0 = (ST )0 = T 0 S 0 ; therefore, the commutativity of the triangle on the left entails the commutativity of the triangle on the right. If the latter commutes then by what was already proven R00 = S 00 T 00 . Using 7.6.6, argue as follows: (Rx)00 = R00 x00 = S 00 T 00 x00 = S 00 (T 00 x00 ) = S 00 (T x)00 = (ST x)00 for all x ∈ X. Consequently, R = ST . B 7.6.8. Definition. Let X0 be a subspace of X and let X0 be a subspace of X 0 . Put X0⊥ := {f ∈ X 0 : ker f ⊃ X0 } = |∂|(δ(X0 )); ⊥

X0 := {x ∈ X : f ∈ X0 ⇒ f (x) = 0} = ∩ {ker f : f ∈ X0 }.

The subspace X0⊥ is the (direct) polar of X0 , and the subspace polar of X0 . A less exact term “annihilator” is also in use.

⊥

X0 is the reverse

7.6.9. Definition. Let X and Y be Banach spaces. An arbitrary element T of B(X, Y ) is a normally solvable operator, if im T is a closed subspace of Y . The natural term “closed range operator” is also in common parlance.

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7.6.10. An operator T , a member of B(X, Y ), is normally solvable if and only if T is a homomorphism, when regarded as acting from X to im T . C ⇒: The Banach Homomorphism Theorem. ⇐: Refer to 7.4.2. B 7.6.11. Polar Lemma. Let T ∈ B(X, Y ). Then (1) (im T )⊥ = ker(T 0 ); (2) if T is normally solvable then im T = ⊥ ker(T 0 ),

(ker T )⊥ = im(T 0 ).

C (1): y 0 ∈ ker(T 0 ) ⇔ T 0 y 0 = 0 ⇔ (∀ x ∈ X) T 0 y 0 (x) = 0 ⇔ (∀ x ∈ X) y 0 (T x) = 0 ⇔ y 0 ∈ (im T )⊥ . (2): The equality cl im T = ⊥ ker(T 0 ) follows from 7.5.13. Furthermore, by hypothesis im T is closed. If x0 = T 0 y 0 and T x = 0 then x0 (x) = T 0 y 0 (x) = y 0 (T x) = 0, which means that x0 ∈ (ker T )⊥ . Consequently, im(T 0 ) ⊂ (ker T )⊥ . Now take x0 ∈ (ker T )⊥ . Considering the operator T acting onto im T , apply the Sard Theorem to the left side of the diagram T X −→ im T −→ Y x0 & ↓ y00 . y 0 F As a result of this, obtain y 00 in (im T )0 such that y 00 ◦T = x0 . By the Continuous Extension Principle there is an element y 0 of Y 0 satisfying y 0 ⊃ y 00 . Thus, x0 = T 0 y 0 , i.e. x0 ∈ im(T 0 ). B 7.6.12. Hausdorff Theorem. Let X and Y be Banach spaces. Assume further that T ∈ B(X, Y ). Then T is normally solvable if and only if T 0 is normally solvable. C ⇒: In virtue of 7.6.11 (2), im(T 0 ) = (ker T )⊥ . Evidently, the subspace (ker T )⊥ is closed. ⇐: To begin with, suppose cl im T = Y . It is clear that 0 = Y ⊥ = (cl im T )⊥ = (im T )⊥ = ker(T 0 ) in virtue of 7.6.11. By the Banach Isomorphism Theorem there is some S ∈ B(im(T 0 ), Y 0 ) such that ST 0 = IY 0 . The case r := kSk = 0 is trivial. Therefore, it may be assumed that kT 0 y 0 k ≥ 1 /r ky 0 k for all y 0 ∈ Y 0 . Show now that cl T (BX ) ⊃ 1 /2r BY . With this at hand, from the ideal convexity of T (BX ) it is possible to infer that T (BX ) ⊃ 1 /4r BY . The last inclusion implies that T is a homomorphism. Let y ∈ / cl T (BX ). Then y does not belong to some open convex set including T (BX ). Passing, if need be, to the real carriers of X and Y , assume that F := R.

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Applying the Topological Separation Theorem, find some nonzero y 0 in Y 0 satisfying ky 0 k kyk ≥ y 0 (y) ≥ sup y 0 (T x) = kT 0 y 0 k ≥ 1 /r ky 0 k. kxk≤1

Whence kyk ≥ 1 /r > 1 /2r . Therefore, the sought inclusion is established and the operator T is normally solvable under the above supposition. Finally, address the general case. Put Y0 := cl im T and let ι : Y0 0 = 0 O be the identical embedding. Then T = ιT , where T : X0 = 0 O 0 is the operator acting by the rule T x = T x for x ∈ X. In addition, im(T 0 ) = im(T 0 ι0 ) = T 0 (im(ι0 )) = T 0 (Y00 ), because ι0 (Y 0 ) = Y00 (cf. 7.6.4 (2)). Thus, T 0 is a normally solvable operator. By what was proven, T is normally solvable. It remains to observe that im T = im T . B 7.6.13. Sequence Prime Principle. A sequence . . . 0 = 0 O k−1 −−−k−→ Xk −−−−→ Xk+1 0 = 0 O is exact if and only if so is the sequence prime Tk+1

T

... ←

0 Xk−1

Tk0

←−−−−

0 Tk+1

Xk0

0 ←−−−− Xk+1 ← ... .

C ⇒: Since im Tk+1 = ker Tk+2 ; therefore, Tk+1 is normally solvable. Using the Polar Lemma, conclude that 0 ker(Tk0 ) = (im Tk )⊥ = (ker Tk+1 )⊥ = im(Tk+1 ).

⇒: By the Hausdorff Theorem Tk+1 is normally solvable. Once again appealing to 7.6.11 (2), infer that 0 (im Tk )⊥ = ker(Tk0 ) = im(Tk+1 ) = (ker Tk+1 )⊥ .

Since Tk is normally solvable by Theorem 7.6.12; therefore, im Tk is a closed subspace. Using 7.5.14, observe that im T k = ⊥ ((im Tk )⊥ ) = ⊥ ((ker Tk+1 )⊥ ) = ker Tk+1 . Here account was taken of the fact that ker Tk+1 itself is a closed subspace. B 7.6.14. Corollary. For a normally solvable operator T , the following isomorphisms hold: (ker T )0 ' coker(T 0 ) and (coker T )0 ' ker(T 0 ). C By virtue of 2.3.5 (6) the sequence 00 = 0 O T 0 = 0 O −→ Y 0 = 0 O T 0 = 0 O T

is exact. From 7.6.13 obtain that the sequence 0

T 00 = 0 O coker T )0 0 = 0 O 0 −→ X 0 0 = 0 O ker T )0 0 = 0 O

is exact. B 7.6.15. Corollary. T is an isomorphism ⇔ T 0 is an isomorphism. CB 7.6.16. Corollary. Sp(T ) = Sp(T 0 ). CB

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7.1. Find out which linear operators are ideal. 7.2. Establish that a separately continuous bilinear form on a Banach space is jointly continuous. 7.3. Is a family of lower semicontinuous sublinear functionals on a Banach space uniformly bounded on the unit ball? 7.4. Let X and Y be Banach spaces and T : X0 = 0 O . Prove that kT xkY ≥ tkxkX for some strictly positive t and all x ∈ X if and only if ker T = 0 and im T is a complete set. 7.5. Find conditions for normal solvability of the operator of multiplication by a function in the space of continuous functions on a compact set. 7.6. Let T be a bounded epimorphism of a Banach space X onto l1 (E ). Show that ker T is complemented. 7.7. Establish that a uniformly closed subspace of C([a, b]) composed of continuously differentiable functions (i.e., elements of C (1) ([a, b])) is finite-dimensional. 7.8. Let X and Y be different Banach spaces, with X continuously embedded into Y . Establish that X is a meager subset of Y . 7.9. Let X1 and X2 be nonzero closed subspaces of a Banach space and X1 ∩ X2 = 0. Prove that the sum X1 + X2 is closed if and only if the next quantity inf{kx1 − x2 k/kx1 k : x1 6= 0, x1 ∈ X1 , x2 ∈ X2 } is strictly positive. 7.10. Let (amn ) be a double countable sequence such that there is a sequence (x(m) ) of P∞ (m) elements of l1 for which all the series n=1 P∞amn xn fail to converge in norm. Prove that there is a sequence x in l1 such that the series a x fail to converge in norm for all m ∈ N. n=1 mn n 7.11. Let T be an endomorphism of a Hilbert space H which satisfies hT x | yi = hx | T yi for all x, y ∈ H. Establish that T is bounded. 7.12. Let a closed cone X + in a Banach space X be reproducing: X = X + − X + . Prove that there is a constant t > 0 such that for all x ∈ X and each presentation x = x1 − x2 with x1 , x2 ∈ X + , the estimates hold: kx1 k ≤ tkxk and kx2 k ≤ tkxk. 7.13. Let lower semicontinuous sublinear functionals p and q on a Banach space X be such that the cones dom p and dom q are closed and the subspace dom p − dom q = dom q − dom p is complemented in X. Prove that ∂ (p + q) = ∂ (p) + ∂ (q) in the case of topological subdifferentials (cf. Exercise 3.10). 7.14. Let p be a continuous sublinear functional defined on a normed space X and let T be a continuous endomorphism of X. Assume further that the dual T 0 of T takes the subdifferential ∂ (p) into itself. Establish that ∂ (p) contains a fixed point of T 0 . 7.15. Given a function f : X0 = 0 O R· on a (multi)normed space X, put f ∗ (x0 ) := sup{hx | x0 i − f (x) : x ∈ dom f }

(x0 ∈ X 0 );

f ∗∗ (x) := sup{hx | x0 i − f ∗ (x0 ) : x0 ∈ dom(f ∗ )} Find conditions for f to satisfy f = f ∗∗ .

(x ∈ X).

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7.16. Establish that l∞ is complemented in each ambient Banach space. 7.17. A Banach space X is called primary, if each of its infinite-dimensional complemented subspaces is isomorphic to X. Verify that c0 and lp (1 ≤ p ≤ +∞) are primary. 7.18. Let X and Y be Banach spaces. Take an operator T , a member of B(X, Y ), such that im T is nonmeager. Prove that T is normally solvable. 7.19. Let X0 be a closed subspace of a normed space X. Assume further that X0 and X/X0 are Banach spaces. Show that X itself is a Banach space.

Chapter 8 Operators in Banach Spaces

8.1. Holomorphic Functions and Contour Integrals 8.1.1. Definition. Let X be a Banach space. A subset ƒ of the ball BX 0 in the dual space X 0 is called norming (for X) if kxk = sup{|l(x)| : l ∈ ƒ} for all x ∈ X. If each subset U of X satisfies sup kU k < +∞ on condition that sup{|l(u)| : u ∈ U } < +∞ for all l ∈ ƒ, then ƒ is a fully norming set. 8.1.2. Examples. (1) The ball BX 0 is a fully norming set in virtue of 5.1.10 (8) and 7.2.7. (2) If ƒ0 is a (fully) norming set and ƒ0 ⊂ ƒ1 ⊂ BX 0 then ƒ1 itself is a (fully) norming set. (3) The set ext BX 0 of the extreme points of BX 0 is norming in virtue of the Kre˘ın–Milman Theorem in subdifferential form and the obvious equality BX 0 = |∂|(k · kX ) which has already been used many times. However, ext BX 0 can fail to be fully norming (in particular, the possibility is realized in the space C([0, 1], R)). CB (4) Let X and Y be Banach spaces (over the same ground field F) and let ƒY be a norming set for Y . Put ƒB := {δ(y,0 x) : y 0 ∈ ƒY , x ∈ BX }, where δ(y,0 x) (T ) := y 0 (T x) for y 0 ∈ Y, x ∈ X and T ∈ B(X, Y ). It is clear that |δ(y,0 x) (T )| = |y 0 (T x)| ≤ ky 0 k kT xk ≤ ky 0 k kT k kxk; i.e., δ(y,0 x) ∈ B(X, Y )0 . Furthermore, given T ∈ B(X, Y ), infer that kT k = sup{kT xk : kxk ≤ 1} = sup{|y 0 (T x)| : y 0 ∈ ƒY , kxk ≤ 1} = sup{|δ(y,0 x) (T )| : δ(y,0 x) ∈ ƒB }.

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Therefore, ƒB is a norming set for B(X, Y ). If ƒY is a fully norming set then ƒB is also a fully norming set. Indeed, if U is such that the numeric set {|y 0 (T x)| : T ∈ U } is bounded in R for all x ∈ BX and y 0 ∈ ƒY , then by hypothesis the set {T x : T ∈ U } is bounded in Y for every x in X. By virtue of the Uniform Boundedness Principle it means that sup kU k < +∞. 8.1.3. Dunford–Hille Theorem. Let X be a complex Banach space and let ƒ be a fully norming set for X. Assume further that f : D0 = 0 O is an X-valued function with domain D an open (in C R ' R2 ) subset of C. The following statements are equivalent: (1) for every z0 in D there is a limit limO

z0=0

0

f (z) − f (z0 ) ; z − z0

(2) for all z0 ∈ D and l ∈ ƒ there is a limit limO

z0=0

0

l ◦ f (z) − l ◦ f (z0 ) ; z − z0

i.e., the function l ◦ f : D0 = 0 O C is holomorphic for l ∈ ƒ. C (1) ⇒ (2): This is obvious. (2) ⇒ (1): For the sake of simplicity assume that z0 = 0 and f (z0 ) = 0. Consider the disk of radius 2ε centered at zero and included in D; i.e., 2εD ⊂ D, where D := BC := {z ∈ C : |z| ≤ 1} is the unit disk. As is customary in complex analysis, treat the disk εD as an (oriented) compact manifold with boundary εT, where T is the (properly oriented) unit circle T := {z ∈ C : |z| = 1} in the complex plane C R . Take z1 , z2 ∈ εD \ 0 and the holomorphic function l ◦ f (the functional l lies in ƒ). Specifying the Cauchy Integral Formula, observe that Z l ◦ f (zk ) 1 l ◦ f (z) = dz (k := 1, 2). zk 2πi z(z − zk ) 2εT

If now z1 6= z2 then, using the condition |z − zk | ≥ ε (k := 1, 2) for z ∈ 2εT and the continuity property of the function l ◦ f on D, find    f (z1 ) f (z2 ) 1 l − z1 − z2 z1 z2   Z 1 1 1 1 l ◦ f (z) dz = · − z(z − z1 ) z(z − z2 ) z1 − z2 2πi 2εT Z 1 1 dz ≤ M sup |l ◦ f (z)| < +∞ = l ◦ f (z) 2π z(z − z1 )(z − z2 ) z∈2εT 2εT

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for a suitable M > 0. Since ƒ is a fully norming set, conclude that

f (z1 ) f (z2 ) 1

sup

z1 − z2 < +∞. z 6=z ;z ,z 6=0 |z1 − z2 | 1

2

1

2

|z1 |≤ε,|z2 |≤ε

This final inequality ensures that the sought limit exists. B 8.1.4. Definition. A mapping f : D0 = 0 O satisfying 8.1.3 (1) (or, which is the same, 8.1.3 (2) for some fully norming set ƒ) is called holomorphic. 8.1.5. Remark. A meticulous terminology is used sometimes. Namely, if f satisfies 8.1.3 (1) then f is called a strongly holomorphic function. If f satisfies 8.1.3 (2) with ƒ := BX 0 then f is called weakly holomorphic. Under the hypotheses of 8.1.3 (2) and 8.1.2 (4), i.e. for f : D0 = 0 O (X, Y ), ƒY := BY 0 and the corresponding ƒ := ƒB , the expression, “f is weakly operator holomorphic,” is employed. With regard to this terminology, the Dunford–Hille Theorem is often referred to as the Holomorphy Theorem and verbalized as follows: “A weakly holomorphic function is strongly holomorphic.” 8.1.6. Remark. It is convenient in the sequel to use the integrals of the simplest smooth X-valued forms f (z)dz over the simplest oriented manifolds, the boundaries of elementary planar compacta (cf. 4.8.5) which are composed of finitely many disjoint simple loops. An obvious meaning is ascribed to the integrals: Namely, given a loop γ, choose an appropriate (smooth) parametrization ‰ : T0 = 0 O (with orientation accounted for) and put Z Z f (z)dz := f ◦ ‰d‰, γ

T

with the integral treated for instance as a suitable Bochner integral (cf. 5.5.9 (6)). The soundness of the definition is beyond a doubt, since the needed Bochner integral exists independently of the choice of the parametrization ‰. 8.1.7. Cauchy–Wiener Integral Theorem. Let D be an nonempty open subset of the complex plane and let f : D0 = 0 O be a holomorphic X-valued function, with X a Banach space. Assume further that F is a rough draft for the pair (∅, D). Then Z f (z)dz = 0. ∂F

Moreover, 1 f (z0 ) = 2πi

Z ∂F

for z0 ∈ int F .

f (z) dz z − z0

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C By virtue of 8.1.3 the Bochner integrals exist. The sought equalities follow readily from the validity of their scalar versions basing on the Cauchy Integral Formula, the message of 8.1.2 (1) and the fact that the Bochner integral commutes with every bounded functional as mentioned in 5.5.9 (6). B 8.1.8. Remark. The Cauchy–Wiener Integral Theorem enables us to infer analogs of the theorems of classical complex analysis for X-valued holomorphic functions on following the familiar patterns. 8.1.9. Taylor Series Expansion Theorem. Let f : D0 = 0 O be a holomorphic X-valued function, with X a Banach space, and take z0 ∈ D. In every open disk U := {z ∈ C : |z − z0 | < ε} such that cl U lies in D, the Taylor series expansion holds (in a compactly convergent power series, cf. 7.2.10): f (z) =

∞ X

cn (z − z0 )n ,

n=0

where the coefficients cn , members of X, are calculated by the formulas: 1 cn = 2πi

Z

f (z) 1 dn f dz = (z0 ). (z − z0 )n+1 n! dz n

∂U

C The proof results from a standard argument: Expand the kernel u 7→ (u − z)−1 of the formula 1 f (z) = 2πi

Z

f (u) du (z ∈ cl U ) u−z

∂U 0

in the powers of z − z0 ; i.e., 1 1  = u−z (u − z0 ) 1 −

∞ X (z − z0 )n = . n+1 z−z0 (u − z ) 0 n=0

u−z0

The last series converges uniformly in u ∈ ∂U 0 . (Here U 0 = U + qD for some q > 0 such that cl U 0 ⊂ D.) Taking it into account that sup kf (∂U 0 )k < +∞ and integrating, arrive at the sought presentation of f (z) for z ∈ cl U . Applying this to U 0 and using 8.1.7, observe that the power series under study converges in norm at every point of U 0 . This yields uniform convergence on every compact subset of U 0 , and so on U . B

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Chapter 8

8.1.10. Liouville Theorem. If an X-valued function f : C0 = 0 O , with X a Banach space, is holomorphic and sup kf (C)k < +∞ then f is a constant mapping. C Considering the disk εD with ε > 0 and taking note of 8.1.9, infer that kcn k ≤ sup kf (z)k · ε−n ≤ sup kf (C )k · ε−n z∈εT

for all n ∈ N and ε > 0. Therefore, cn = 0 for n ∈ N. B 8.1.11. Each bounded endomorphism of a nonzero complex Banach space has a nonempty spectrum. C Let T be such an endomorphism. If Sp(T ) = ∅ then the resolvent R (T, · ) is holomorphic on the entire complex plane C, for instance, by 5.6.21. Furthermore, by 5.6.15, kR (T, λ)k0 = 0 O as |λ|0 = 0 O ∞. By virtue of 8.1.10 conclude that R (T, · ) = 0. At the same time, using 5.6.15, observe that R (T, λ)(λ − T ) = 1 for |λ| > kT k. A contradiction. B 8.1.12. The Beurling–Gelfand formula holds: r(T ) = sup{|λ| : λ ∈ Sp(T )} for all T ∈ B(X), with X a complex Banach space; i.e., the spectral radius of an operator T coincides with the radius of the spectrum of T . C It is an easy matter that the spectral radius r(T ) is greater than the radius of the spectrum of T . So, there is nothing to prove if r(T ) = 0. Assume now that r(T ) > 0. Take λ ∈ C so that |λ| > sup{|µ| : µ ∈ Sp(T )}. Then the disk of radius |λ|−1 lies entirely in the domain of the holomorphic function (cf. 5.6.15)  f (z) :=

R (T, z −1 ), for z 6= 0 and z −1 ∈ res(T ) 0,

for z = 0.

Using 8.1.9 and 5.6.17, conclude that |λ|−1 < r(T )−1 . Consequently, |λ| > r(T ). B 8.1.13. Let K be a nonempty compact subset of C. Denote by H(K) the set of all functions holomorphic in a neighborhood of K (i.e., f ∈ H(K) ⇔ f : dom f 0 = 0 O C is a holomorphic function with dom f ⊃ K). Given f1 , f2 ∈ H(K), let the notation f1 ∼ f2 mean that it is possible to find an open subset D of dom f1 ∩dom f2 satisfying K ⊂ D and f1 |D = f2 |D . Then ∼ is an equivalence in H(K). CB 8.1.14. Definition. Under the hypotheses of 8.1.13, put H (K) := H(K)/∼. The element f in H (K) containing a function f in H(K) is the germ of f on K.

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8.1.15. Let f , g ∈ H (K). Take f1 , f2 ∈ f and g1 , g2 ∈ g. Put x ∈ dom f1 ∩ dom g1 ⇒ ϕ1 (x) := f1 (x) + g1 (x), x ∈ dom f2 ∩ dom g2 ⇒ ϕ2 (x) := f2 (x) + g2 (x). Then ϕ1 , ϕ2 ∈ H(K) and ϕ1 = ϕ2 . C Choose open sets D1 and D2 such that K ⊂ D1 ⊂ dom f1 ∩ dom f2 and K ⊂ D2 ⊂ dom g1 ∩ dom g2 , with f1 |D1 = f2 |D1 and g1 |D2 = g2 |D2 . Observe now that ϕ1 and ϕ2 agree on D1 ∩ D2 . B 8.1.16. Definition. The coset, introduced in 8.1.15, is the sum of the germs f 1 and f 2 . It is denoted by f 1 + f 2 . The product of germs and multiplication of a germ by a complex number are introduced by analogy. 8.1.17. The set H (K) with operations defined in 8.1.16 is an algebra. CB 8.1.18. Definition. The algebra H (K) is the algebra of germs of holomorphic functions on a compact set K. 8.1.19. Let K be a compact subset of C, and let R : C \ K0 = 0 O be an X-valued holomorphic function, with X a Banach space. Further, take f ∈ H (K) and f1 , f2 ∈ f . If F1 is a rough draft for the pair (K, dom f1 ) and F2 is a rough draft for the pair (K, dom f2 ) then Z

Z f1 (z)R (z)dz =

∂F1

f2 (z)R (z)dz. ∂F2

C Let K ⊂ D ⊂ int F1 ∩ int F2 , with D open and f1 |D = f2 |D . Choose a rough draft F for the pair (K, D). Since f1 R is holomorphic on dom f1 \ K and f2 R is holomorphic on dom f2 \ K, infer the equalities

Z

Z f1 (z)R (z)dz =

f1 (z)R (z)dz,

∂F

∂F1

Z

Z f2 (z)R (z)dz =

∂F

f2 (z)R (z)dz ∂F2

(from the nontrivial fact of the validity of their scalar analogs). Since f1 and f2 agree on D, the proof is complete. B

132

Chapter 8

8.1.20. Definition. Under the hypotheses of 8.1.19, given an element h in H (K), define the contour integral of h with kernel R as the element I

Z h(z)R (z)dz :=

f (z)R (z)dz, ∂F

where h = f and F is a rough draft for the pair (K, dom f ). 8.1.21. Remark. The notation h(z) in 8.1.20 is far from being ad hoc. It is well justified by the fact that w := f1 (z) = f2 (z) for every point z in K and every two members f1 and f2 of a germ h. In this connection the element w is said to be the value of h at z, which is expressed in writing as h(z) = w. It is also worth noting that the function R in 8.1.2 may be assumed to be given only in U \ K, where int U ⊃ K. 8.2. The Holomorphic Functional Calculus 8.2.1. Definition. Let X be a (nonzero) complex Banach space and let T be a bounded endomorphism of X; i.e., T ∈ B(X). For h ∈ H (Sp(T )), the contour integral with kernel the resolvent R (T, · ) of T is denoted by 1 RT h := 2πi

I h(z)R (T, z)dz

and called the Riesz–Dunford integral (of the germ h). If f is a function holomorphic in a neighborhood about Sp(T ) then put f (T ) := RT f := RT f . We also use more suggestive designations like 1 f (T ) = 2πi

I

f (z) dz. z−T

8.2.2. Remark. In algebra, in particular, various representations of mathematical objects are under research. It is convenient to use the primary notions of representation theory for the most “algebraic” objects, namely, algebras. Recall the simplest of them. Let A1 and A2 be two algebras (over the same field). A morphism from A1 to A2 or a representation of A1 in A2 (rarely, over A2 ) is a multiplicative linear operator R, i.e. a member R of L (A1 , A2 ) such that R(ab) = R(a)R(b) for all a, b ∈ A1 . The expression, “T represents A1 in A2 ,” is also in common parlance. A representation R is called faithful if ker R = 0. The presence of a faithful representation R : A1 0 = 0 O 2 makes it possible to treat A1 as a subalgebra of A2 .

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If A2 is a (sub)algebra of the endomorphism algebra L (X) of some vector space X (over the same field), then a morphism of A1 in A2 is referred to as a (linear) representation of A1 on X or as an operator representation of A1 . The space X is then called the representation space for the algebra A1 .

Given a representation R, suppose that the representation space X for A has a subspace X1 invariant under all operators R(a), a ∈ A. Then the representation R1 : A0 = 0 O L(X1 ) arises naturally, acting by the rule R1 (a)x1 = R(a)x1 for x1 ∈ X1 and a ∈ A and called a subrepresentation of R (induced in X1 ). If X = X1 ⊕ X2 and the decomposition reduces each operator R(a) for a ∈ A, then it is said that the representation R reduces to the direct sum of subrepresentations R1 and R2 (induced in X1 and X2 ). We mention in passing the significance of studying arbitrary irreducible representations, each of which has only trivial subrepresentations by definition.

8.2.3. Gelfand–Dunford Theorem. Let T be a bounded endomorphism of a Banach space X. The Riesz–Dunford integral RT represents the algebra of germs of holomorphic functions on the spectrum of T on the space X. Moreover, P∞ P∞ if f (z) = n=0 cn z n (in a neighborhood about Sp(T )) then f (T ) = n=0 cn T n (summation is understood in the operator norm of B(X)).

C There is no doubt that RT is a linear operator. Check the multiplicativity property of RT . To this end, take f 1 , f 2 ∈ H (Sp(T )) and choose rough drafts F1 and F2 such that Sp(T ) ⊂ int F1 ⊂ F1 ⊂ int F2 ⊂ F2 ⊂ D, with the functions f1 in f 1 and f2 in f 2 holomorphic on D. Using the obvious properties of the Bochner integral, the Cauchy Integral For-

134

Chapter 8

mula and the Hilbert identity, successively infer the chain of equalities Z Z 1 1 f1 (z1 ) f2 (z2 ) RT f 1 ◦ RT f 2 = f1 (T )f2 (T ) = dz1 ◦ dz2 2πi 2πi z1 − T z2 − T ∂F1 ∂F2   Z Z 1 1  = f1 (z1 )R (T, z1 )dz1  f2 (z2 )R (T, z2 )dz2 2πi 2πi ∂F2 ∂F Z Z1 1 1 f1 (z1 )f2 (z2 )R (T, z1 )R (T, z2 )dz2 dz1 = 2πi 2πi ∂F1 ∂F2 Z Z 1 1 R (T, z1 ) − R (T, z2 ) = f1 (z1 )f2 (z2 ) dz2 dz1 2πi 2πi z2 − z1 ∂F1 ∂F2   Z Z 1 f2 (z2 ) 1 f1 (z1 )  = dz2  R (T, z1 )dz1 2πi 2πi z2 − z1 ∂F1 ∂F2   Z Z 1 f1 (z1 ) 1 − f2 (z2 )  dz1  R (T, z2 )dz2 2πi 2πi z2 − z1 ∂F2 ∂F1 Z 1 f1 (z1 )f2 (z1 )R (T, z1 )dz1 − 0 = f1 f2 (T ) = RT (f 1 f 2 ). = 2πi γ

Choose a circle γ := εT that lies in res(T ) as well as in the (open) disk of P∞ convergence of the series f (z) = n=0 cn z n . From 5.6.16 and 5.5.9 (6) derive 1 f (T ) = 2πi

Z f (z) γ

∞ X

z −n−1 T n dz

n=0



 Z ∞ Z ∞ ∞ X X 1 1 f (z)  n X −n−1 n  = f (z)z T dz = dz T = cn T n n+1 2πi n=0 2πi z n=0 n=0 γ

γ

in virtue of 8.1.9. B 8.2.4. Remark. Theorem 8.2.3 is often referred to as the Principal Theorem of the holomorphic functional calculus. 8.2.5. Spectral Mapping Theorem. For every function f holomorphic in a neighborhood about the spectrum of an operator T in B(X), the equality holds: f (Sp(T )) = Sp(f (T )).

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135

C Assume first that λ ∈ Sp(f (T )) and f −1 (λ) ∩ Sp(T ) = ∅. Given z ∈ (C \ f −1 (λ)) ∩ dom f , put g(z) := (λ − f (z))−1 . Then g is a holomorphic function on a neighborhood of Sp(T ), satisfying g(λ − f ) = (λ − f )g = 1C . Using 8.2.3, observe that λ ∈ res(f (T )), a contradiction. Consequently, f −1 (λ) ∩ Sp(T ) 6= ∅; i.e., Sp(f (T )) ⊂ f (Sp(T )). Now take λ ∈ Sp(T ). Put λ 6= z ⇒ g(z) :=

f (λ) − f (z) ; λ−z

g(λ) := f 0 (λ).

Clearly, g is a holomorphic function (the singularity is “removed”). From 8.2.3 obtain g(T )(λ − T ) = (λ − T )g(T ) = f (λ) − f (T ). Consequently, if f (λ) ∈ res(f (T )) then the operator R (f (T ), f (λ))g(T ) is inverse to λ − T . In other words, λ ∈ res(T ), which is a contradiction. Thus, f (λ) ∈ C \ res(f (T )) = Sp(f (T )); i.e., f (Sp(T )) ⊂ Sp(f (T )). B 8.2.6. Let K be a nonempty compact subset of C and let g : dom g0 = 0 O C ◦ be a holomorphic function with dom g ⊃ K. Given f ∈ H(g(K)), put g(f ) := f ◦ g. ◦ Then g is a representation of the algebra H (g(K)) in the algebra H (K). CB 8.2.7. Dunford Theorem. For every function g : dom g0 = 0 O C holomorphic in the neighborhood dom g about the spectrum Sp(T ) of an endomorphism T of a Banach space X, the following diagram of representations commutes: ◦

g H (Sp(T ))  H (Sp(g(T ))) @ @ Rg(T ) RT @ @ R ? @ B(X) C Let f ∈ H (g(Sp(T ))) with f : D0 = 0 O C such that f ∈ f and D ⊃ g(Sp(T )) = Sp(g(T )). Let F1 be a rough draft for the pair (Sp(g(T )), D) and let F2 be a rough draft for the pair (Sp(T ), g −1 (int F1 )). It is clear that now g(∂F2 ) ⊂ int F1 and, moreover, the function z2 7→ (z1 − g(z2 ))−1 is defined and holomorphic on int F2 for z1 ∈ ∂F1 . Therefore, by 8.2.3 Z 1 R (T, z2 ) R (g(T ), z1 ) = dz2 (z1 ∈ ∂F1 ). 2πi z1 − g(z2 ) ∂F2

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Chapter 8

From this equality, successively derive  Rg(T ) f =

1 2πi

Z

Z

 Z

f (z1 ) 1 1 R (T, z2 ) dz1 = f (z1 )  dz2  dz1 z1 − g(T ) 2πi 2πi z1 − g(z2 ) ∂F1 ∂F1 ∂F2   Z Z f (z1 ) 1 1  dz1  R (T, z2 )dz2 . = 2πi 2πi z1 − g(z2 ) ∂F1

∂F2

Since g(z2 ) ∈ int F1 for z2 ∈ ∂F2 by construction; therefore, the Cauchy Integral Formula yields the equality 1 f (g(z2 )) = 2πi

Z

f (z1 ) dz1 . z1 − g(z2 )

∂F1

Consequently, 1 Rg(T ) f = 2πi

Z

◦

f (g(z2 ))R (T, z2 )dz2 = RT g(f ).

∂F2

8.2.8. Remark. The Dunford Theorem is often referred to as the Composite Function Theorem and written down symbolically as f ◦ g(T ) = f (g(T )). 8.2.9. Definition. A subset σ of Sp(T ) is a clopen or isolated part or rarely an exclave of Sp(T ), if σ and its complement σ 0 := Sp(T ) \ σ are closed. 8.2.10. Let σ be a clopen part of Sp(T ) and let κσ be some function that equals 1 in an open neighborhood of σ and 0 in an open neighborhood of σ 0 . Further, assign I 1 κσ (z) Pσ := κσ (T ) := dz. 2πi z−T Then Pσ is a projection in X and the (closed) subspace Xσ := im Pσ is invariant under T . C Since κσ2 = κσ , it follows from 8.2.3 that κσ (T )2 = κσ (T ). Furthermore, T = RT IC , where IC : z 7→ z. Hence T Pσ = Pσ T (because I C κ σ = κ σ I C ). Consequently, in virtue of 2.2.9 (4), Xσ is invariant under T . B 8.2.11. Definition. The projection Pσ is the Riesz projection or the Riesz idempotent corresponding to σ.

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137

8.2.12. Spectral Decomposition Theorem. Let σ be a clopen part of the spectrum of an operator T in B(X). Then X splits into the direct sum decomposition of the invariant subspaces X = Xσ ⊕ Xσ0 which reduces T to matrix form   Tσ 0 T ∼ , 0 Tσ0 with the part Tσ of T in Xσ and the part Tσ0 of T in Xσ0 satisfying Sp(Tσ0 ) = σ 0 .

Sp(Tσ ) = σ,

C Since κ σ + κ σ0 = κSp(T ) = 1C , in view of 8.2.3 and 8.2.10 it suffices to establish the claim about the spectrum of Tσ . From 8.2.5 and 8.2.3 obtain σ ∪ 0 = κσ IC (Sp(T )) = Sp(κσ IC (T )) = Sp(RT (κσ IC )) = Sp(RT κσ ◦ RT IC ) = Sp(Pσ T ). Moreover, in matrix form 

 Tσ 0 Pσ T ∼ . 0 0 Let λ be a nonzero complex number. Then   λ − Tσ 0 λ − Pσ T ∼ ; 0 λ i.e., the operator λ − Pσ T is not invertible if and only if the same is true of the operator λ − Tσ . Thus, Sp(Tσ ) \ 0 ⊂ Sp(Pσ T ) \ 0 = (σ ∪ 0) \ 0 ⊂ σ. Suppose that 0 ∈ Sp(Tσ ) and 0 ∈ / σ. Choose disjoint open sets Dσ and Dσ0 so that σ ⊂ Dσ , 0 ∈ / Dσ and σ 0 ⊂ Dσ0 , and put 1 ; z ⇒ h(z) := 0.

z ∈ Dσ ⇒ h(z) := z ∈ Dσ 0

By 8.2.3, h(T )T = T h(T ) = Pσ . Moreover, since hκ σ = κ σ h, the decomposition X = Xσ ⊕Xσ0 reduces h(T ) and h(T )σ Tσ = Tσ h(T )σ = 1 for the part h(T )σ of h(T ) in Xσ . So, Tσ is invertible; i.e., 0 ∈ / Sp(Tσ ). We thus arrive at a contradiction which implies that 0 ∈ σ. In other words, Sp(Tσ ) ⊂ σ. Observe now that res(T ) = res(Tσ ) ∩ res(Tσ0 ). Consequently, by the above Sp(T ) = C \ res(T ) = C \ (res(Tσ ) ∩ res(Tσ0 )) = (C \ res(Tσ )) ∪ (C \ res(Tσ0 )) = Sp(Tσ ) ∪ Sp(Tσ0 ) ⊂ σ ∪ σ 0 = Sp(T ). Considering that σ ∩ σ 0 = ∅, complete the proof. B

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Chapter 8

8.2.13. Riesz–Dunford Integral Decomposition Let σ be a clopen part of Sp(T ) for an endomorphism T of a Banach space X. The direct sum decomposition X = Xσ ⊕ Xσ0 reduces the representation RT of the algebra H (Sp(T )) in X to the direct sum of the representations Rσ and Rσ0 . Moreover, the following diagrams of representations commute: κ σH (Sp(T )) H (σ) @ @ RTσ @ Rσ @ R ? @ B(Xσ )

H (Sp(T )) @ @

κ σ0-

H (σ 0 )

RTσ0 @ Rσ0 @ R ? @ B(Xσ0 )

Here κ σ (f ) := κσ f and κ σ0 (f ) := κσ0 f for f ∈ H(Sp(T )) are the representations induced by restricting f onto σ and σ 0 . CB 8.3. The Approximation Property 8.3.1. Let X and Y be Banach spaces. For K ∈ L (X, Y ) the following statements are equivalent: (1) the operator K is compact: K ∈ K (X, Y ); (2) there are a neighborhood of zero U in X and a compact subset V of Y such that K(U ) ⊂ V ; (3) the image under K of every bounded set in X is relatively compact in Y ; (4) the image under K of every bounded set in X is totally bounded in Y ; (5) for each sequence (xn )n∈N of points of the unit ball BX , the sequence (Kxn )n∈N has a Cauchy subsequence. CB 8.3.2. Theorem. Let X and Y be Banach spaces over a basic field F. Then (1) K (X, Y ) is a closed subspace of B(X, Y ); (2) for all Banach spaces W and Z, it holds that B(Y, Z) ◦ K (X, Y ) ◦ B(W, X) ⊂ K (W, Z); i.e., if S ∈ B(W, X), T ∈ B(Y, Z) and K ∈ K (X, Y ) then T KS ∈ K (W, Z); (3) IF ∈ K (F) := K (F, F ).

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C That K (X, Y ) is a subspace of B(X, Y ) follows from 8.3.1. If Kn ∈ K (X, Y ) and Kn 0 = 0 O ; then, given ε > 0, for n sufficiently large observe that kKx − Kn xk ≤ kK − Kn k kxk ≤ ε whenever x ∈ BX . Therefore, Kn (BX ) serves as an ε-net (= Bε -net) for K(BX ). It remains to refer to 4.6.4 and thus complete the proof of the closure property of K (X, Y ). The other claims are evident. B 8.3.3. Remark. Theorem 8.3.2 is often verbalized as follows: “The class of all compact operators is an operator ideal.” Behind this lies a conspicuous analogy with the fact that K (X) := K (X, X) presents a closed bilateral (two-sided) ideal in the (bounded) endomorphism algebra B(X); i.e., K (X) ◦ B(X) ⊂ K (X) and B(X) ◦ K (X) ⊂ K (X). 8.3.4. Calkin Theorem. The ideals 0, K (l2 ), and B(l2 ) exhaust the list of closed bilateral ideals in the endomorphism algebra B(l2 ) of the Hilbert space l2 . 8.3.5. Remark. In view of 8.3.4 it is clear that a distinguishable role in operator theory should be performed by the algebra B(X)/K (X) called the Calkin algebra (on X). The performance is partly delivered in 8.5. 8.3.6. Definition. An operator T , a member of L (X, Y ), is called a finiterank operator provided that T ∈ B(X, Y ) and im T is a finite-dimensional subspace of Y . In notation: T ∈ F (X, Y ). A hasty term “finite-dimensional operator” would abuse consistency since T as a subspace of X × Y is usually infinite-dimensional. 8.3.7. The linear span of the set of (bounded) rank-one operators comprises all finite-rank operators: T ∈ F (X, Y ) ⇔

(∃ x01 , . . .

, x0n

0

∈ X ) (∃ y1 , . . . , yn ∈ Y ) T =

n X

x0k ⊗ yk . /.

k=1

8.3.8. Definition. Let Q be a (nonempty) compact set in X. Given T ∈ B(X, Y ), put kT kQ := sup kT (Q)k. The collection of all seminorms B(X, Y ) of type k · kQ is the Arens multinorm in B(X, Y ), denoted by κB(X,Y ) . The corresponding topology is the topology of uniform convergence on compact sets or the compact-open topology (cf. 7.2.10). 8.3.9. Grothendieck Theorem. Let X be a Banach space. The following conditions are equivalent: (1) for every ε > 0 and every compact set Q in X there is a finite-rank endomorphism T of X, a member of F (X) := F (X, X), such that kT x − xk ≤ ε for all x ∈ Q;

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(2) for every Banach space W , the subspace F (W, X) is dense in the space B(W, X) with respect to the Arens multinorm κB(W,X) ; (3) for every Banach space Y , the subspace F (X, Y ) is dense in the space B(X, Y ) with respect to the Arens multinorm κB(X,Y ) . C It is clear that (2) ⇒ (1) and (3) ⇒ (1). Therefore, we are to show only that (1) ⇒ (2) and (1) ⇒ (3). (1) ⇒ (2): If T ∈ B(W, X) and Q is a nonempty compact set in W then, in view of the Weierstrass Theorem, T (Q) is a nonempty compact set in X. So, for ε > 0, by hypothesis there is a member T0 of F (X) such that kT0 − IX kT (Q) = kT0 T − T kQ ≤ ε. Undoubtedly, T0 T ∈ F (W, X). (1) ⇒ (3): Let T ∈ B(X, Y ). If T = 0 then there is nothing to be proven. Let T 6= 0, ε > 0 and Q be a nonempty compact set in X. By hypothesis there is a member T0 of F (X) such that kT0 − IX kQ ≤ εkT k−1 . Then kT T0 − T kQ ≤ kT k kT0 − IX kQ ≤ ε. Furthermore, T T0 ∈ F (X, Y ). B 8.3.10. Definition. A Banach space satisfying one (and hence all) of the equivalent conditions 8.3.9 (1)–8.3.9 (3) is said to possess the approximation property. 8.3.11. Grothendieck Criterion. A Banach space X possesses the approximation property if and only if, for every Banach space W , the equality holds: cl F (W, X) = K (W, X), with the closure taken in operator norm. 8.3.12. Remark. For a long time there was an unwavering (and yet unprovable) belief that every Banach space possesses the approximation property. Therefore, P. Enflo’s rather sophisticated example of a Banach space lacking the approximation property was acclaimed as sensational in the late seventies of the current century. Now similar counterexamples are in plenty: 8.3.13. Szankowski Counerexample. The space B(l2 ) lacks the approximation property. 8.3.14. Davis–Figiel–Szankowski Couterexample. The spaces c0 and lp with p 6= 2 have closed subspaces lacking the approximation property. 8.4. The Riesz–Schauder Theory 8.4.1. ε-Perpendicular Lemma. Let X0 be a closed subspace of a Banach space X and X 6= X0 . Given an ε > 0, there is an ε-perpendicular to X0 in X; i.e., an element xε in X such that kxε k = 1 and d(xε , X0 ) := inf dk·k ({xε }×X0 ) ≥ 1−ε. C Take 1 > ε and x ∈ X \ X0 . It is clear that d := d(x, X0 ) > 0. Find x0 in the subspace X0 satisfying kx−x0 k ≤ d/(1−ε), which is possible because d/(1−ε) > d.

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Put xε := (x − x0 )kx − x0 k−1 . Then kxε k = 1. Finally, for x0 ∈ X0 , observe that

x − x0

kx0 − xε k = x0 − kx − x0 k 1 d(x, X0 ) = 0 k(kx − x0 kx0 + x0 ) − xk ≥ ≥ 1 − ε. . kx − xk kx0 − xk 8.4.2. Riesz Criterion. Let X be a Banach space. The identity operator in X is compact if and only if X is finite-dimensional. C Only the implication ⇒ needs proving. If X fails to be finite-dimensional, then select a sequence of finite-dimensional subspaces X1 ⊂ X2 ⊂ . . . in X such that Xn+1 6= Xn for all n ∈ N. In virtue of 8.4.1 there is a sequence (xn ) satisfying xn+1 ∈ Xn+1 , kxn+1 k = 1 and d(xn+1 , Xn ) ≥ 1 /2 , namely, some sequence of 1 /2 perpendiculars to Xn in Xn+1 . It is clear that d(xm , xk ) ≥ 1 /2 for m 6= k. In other words, the sequence (xn ) lacks Cauchy subsequences. Consequently, by 8.3.1 the operator IX is not compact. B 8.4.3. Let T ∈ K (X, Y ), with X and Y Banach spaces. The operator T is normally solvable if and only if T has finite rank. C Only the implication ⇒ needs examining. Let Y0 := im T be a closed subspace in Y . By the Banach Homomorphism Theorem, the image T (BX ) of the unit ball of X is a neighborhood of zero in Y0 . Furthermore, in virtue of the compactness property of T , the set T (BX ) is relatively compact in Y0 . It remains to apply 8.4.2 to Y0 . B 8.4.4. Let X be a Banach space and K ∈ K (X). Then the operator 1 − K is normally solvable. C Put T := 1 − K and X1 := ker T . Undoubtedly, X1 is finite-dimensional by 8.4.2. In accordance with 7.4.11 (1) a finite-dimensional subspace is complemented. Denote a topological complement of X1 to X by X2 . Considering that X2 is a Banach space and T (X) = T (X2 ), it suffices to verify that kT xk ≥ tkxk for some t > 0 and all x ∈ X2 . In the opposite case, there is a sequence (xn ) such that kxn k = 1, xn ∈ X2 and T xn 0 = 0 O . Using the compactness property of K, we may assume that (Kxn ) converges. Put y := lim Kxn . Then the sequence (xn ) converges to y, because y = lim(T xn + Kxn ) = lim xn . Moreover, T y = lim T xn = 0; i.e., y ∈ X1 . It is beyond a doubt that y ∈ X2 . Thus, y ∈ X1 ∩ X2 ; i.e., y = 0. We arrive at a contradiction: kyk = lim kxn k = 1. B 8.4.5. For whatever strictly positive ε, there are only finitely many eigenvalues of a compact operator beyond the disk centered at zero and having radius ε.

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C Suppose by way of contradiction that there is a sequence (λn )n∈N of pairwise distinct eigenvalues of a compact operator K, with |λn | ≥ ε for all n ∈ N and ε > 0. Suppose further that xn satisfying 0 6= xn ∈ ker(λn − K) is an eigenvector with eigenvalue λn . Establish first that the set {xn : n ∈ N} is linearly independent.PTo this end, assume the set {x1 , . . . , xn } linearly In case Pindependent. n n xn+1 = k=1 αk xk , we would have 0 = (λn+1 − K)xn+1 = k=1 αk (λn+1 − λk )xk . Consequently, αk = 0 for k := 1, . . . , n. Whence the false equality xn+1 = 0 would ensue. Put Xn := lin({x1 , . . . , xn }). By definition X1 ⊂ X2 ⊂ . . . ; moreover, as was proven, Xn+1 6= Xn for n ∈ N. By virtue of 8.4.1 there is a sequence (xn ) such that xn+1 ∈ Xn+1 , kxn+1 k = 1 and d(xn+1 , Xn ) ≥ 1 /2 . For m > k, straightforward calculation shows that z := (λm+1 − K)xm+1 ∈ Xm and z + Kxk ∈ Xm + Xk ⊂ Xm . Consequently, kKxm+1 − Kxk k = k − λm+1 xm+1 + Kxm+1 + λm+1 xm+1 − Kxk k = kλm+1 xm+1 − (z + Kxk )k ≥ |λm+1 |d(xm+1 , Xm ) ≥ ε /2 . In other words, the sequence (Kxn ) has no Cauchy subsequences. B 8.4.6. Schauder Theorem. Let X and Y be Banach spaces (over the same ground field F). Then K ∈ K (X, Y ) ⇔ K 0 ∈ K (Y 0 , X 0 ). C ⇒: Observe first of all that the restriction mapping x0 7→ x0 |BX implements an isometry of X 0 into l∞ (BX ). Therefore, to check that K 0 (BY 0 ) is relatively compact we are to show the same for the set V := {K 0 y 0 |BX : y 0 ∈ BY 0 }. Since K 0 y 0 |BX (x) = y 0 ◦ K|BX (x) = y 0 (Kx) for x ∈ BX and y 0 ∈ BY 0 , consider the ◦ compact set Q := cl K(BX ) and the mapping K : C(Q, F)0 = 0 O ∞ (BX ) defined ◦

◦

by the rule Kg : x 7→ g(Kx). Undoubtedly, the operator K is bounded and, hence, continuous. Now put S := {y 0 |Q : y 0 ∈ BY 0 }. It is clear that S is simultaneously an equicontinuous and bounded subset of C(Q, F ). Consequently, by the Ascoli– Arzel` a Theorem, S is relatively compact. From the Weierstrass Theorem derive that ◦

◦

K(S) is a relatively compact set too. It remains to observe that Ky 0 |Q = K 0 y 0 |BX ◦

for y 0 ∈ BY 0 ; i.e., K(S) = V . ⇐: If K 0 ∈ K (Y 0 , X 0 ) then, as is proven, K 00 ∈ K (X 00 , Y 00 ). By the Double Prime Lemma, K 00 |X = K. Whence it follows that the operator K is compact. B 8.4.7. Every nonzero point of the spectrum of a compact operator is isolated (i.e., such point constitutes a clopen part of the spectrum).

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C Taking note of 8.4.4 and the Sequence Prime Principle, observe that each nonzero point of the spectrum of a compact operator K is either an eigenvalue of K or an eigenvalue of the dual of K. Using 8.4.5 and 8.4.6, conclude that, for each strictly positive ε, there are only finitely many points of Sp(K) beyond the disk centered at zero and having radius ε. B 8.4.8. Riesz–Schauder Theorem. The spectrum of a compact operator K in an infinite-dimensional space contains zero. Each nonzero point of the spectrum of K is isolated and presents an eigenvalue of K with the corresponding eigenspace finite-dimensional. C Considering K, a compact endomorphism of a Banach space X, we must only demonstrate the implication 0 6= λ ∈ Sp(K) ⇒ ker(λ − K) 6= 0. First, settle the case F := C. Note that {λ} is a clopen part of Sp(K). Putting g(z) := 1 /z in some neighborhood about λ and g(z) := 0 for z in a suitable neighborhood about {λ}0 , observe that κ{λ} = gI C . Thus, by 8.2.3 and 8.2.10, P{λ} = g(K)K. By virtue of 8.3.2 (2), P{λ} ∈ K (X). From 8.4.3 it follows that im P{λ} is a finite-dimensional space. It remains to invoke the Spectral Decomposition Theorem. In the case of the reals, F := R, implement the process of complexification. Namely, furnish the space X 2 with multiplication by an element of C which is introduced by the rule i(x, y) := (−y, x). The resulting complex vector space is denoted by X ⊕ iX. Define the operator K(x, y) := (Kx, Ky) in the space X ⊕ iX. Equipping X ⊕ iX with an appropriate norm (cf. 7.3.2), observe that the operator K is compact and λ ∈ Sp(K). Consequently, λ is an eigenvalue of K by what was proven. Whence it follows that λ is an eigenvalue of K. B 8.4.9. Theorem. Let X be a complex Banach space. Given T ∈ B(X), assume further that f : C0 = 0 O C is a holomorphic function vanishing only at zero and such that f (T ) ∈ K (X). Then every nonzero point λ of the spectrum of T is isolated and the Riesz projection P{λ} is compact. C Suppose the contrary; i.e., find a sequence (λn )n∈N of distinct points of Sp(T ) such that λn 0 = 0 O 6= 0 (in particular, X is infinite-dimensional). Then f (λn )0 = 0 O (λ) and f (λ) 6= 0 by hypothesis. By the Spectral Mapping Theorem, Sp(f (T )) = f (Sp(T )). Thus, by 8.4.8, f (λn ) = f (λ) for all sufficiently large n. Whence it follows that f (z) = f (λ) for all z ∈ C and so f (T ) = f (λ). By the Riesz Criterion in this case X is finite-dimensional. We come to a contradiction meaning that λ is an isolated point of Sp(T ). Letting g(z) := f (z)−1 in some neighborhood about λ disjoint from zero, infer that gf = κ{λ} . Consequently, by the Gelfand–Dunford Theorem, P{λ} = g(T )f (T ); i.e., in virtue of 8.3.2 (2) the Riesz projection P{λ} is compact. B

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8.4.10. Remark. Theorem 8.4.9 is sometimes referred to as the Generalized Riesz–Schauder Theorem. 8.5. Fredholm Operators 8.5.1. Definition. Let X and Y be Banach spaces (over the same ground field F). An operator T , a member of B(X, Y ), is a Fredholm operator (in symbols, T ∈ F r(X, Y )) if ker T := T −1 (0) and coker T := Y / im T are finite-dimensional; i.e., if the following quantities, called the nullity and the deficiency of T , are finite: α(T ) := nul T := dim ker T ;

β(T ) := def T := dim coker T.

The integer ind T := α(T )−β(T ), a member of Z, is the index or, fully, the Fredholm index of T . 8.5.2. Remark. In the Russian literature, a Fredholm operator is usually called a Noether operator, whereas the term “Fredholm operator” is applied only to an index-zero Fredholm operator. 8.5.3. Every Fredholm operator is normally solvable. C Immediate from the Kato Criterion. B 8.5.4. For T ∈ B(X, Y ), the equivalence holds: T ∈ F r(X, Y ) ⇔ T 0 ∈ F r(Y 0 , X 0 ). Moreover, ind T = − ind T 0 . C By virtue of 2.3.5 (6), 8.5.3, 5.5.4 and the Sequence Prime Principle, the next pairs of sequences are exact simultaneously: T 0 → ker T 0 = 0 O −→ Y → coker T → 0; T0

0 ← (ker T )0 ← X 0 ←− Y 0 ← (coker T )0 ← 0; T0

0 → ker(T 0 ) → Y 0 −→ X 0 → coker(T 0 ) → 0; T

0 ← (ker(T 0 ))0 ← Y ←− X ← (coker(T 0 ))0 ← 0. Moreover, α(T ) = β(T 0 ) and β(T ) = α(T 0 ) (cf. 7.6.14). B 8.5.5. An operator T is an index-zero Fredholm operator if and only if so is the dual of T . C This is a particular case of 8.5.4. B

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8.5.6. Fredholm Alternative. For an index-zero Fredholm operator T either of the following mutually exclusive events takes place: (1) The homogeneous equation T x = 0 has a sole solution, zero. The homogeneous conjugate equation T 0 y 0 = 0 has a sole solution, zero. The equation T x = y is solvable and has a unique solution given an arbitrary right side. The conjugate equation T 0 y 0 = x0 is solvable and has a unique solution given an arbitrary right side. (2) The homogeneous equation T x = 0 has a nonzero solution. The homogeneous conjugate equation T 0 y 0 = 0 has a nonzero solution. The homogeneous equation T x = 0 has finitely many linearly independent solutions x1 , . . . , xn . The homogeneous conjugate equation T 0 y 0 = 0 has finitely many linearly independent solutions y 01 , . . . , y 0n . The equation T x = y is solvable if and only if y 01 (y) = . . . = y 0n (y) = 0. Moreover, the general solution x is the sum of a particular solution x0 and the general solution of the homogeneous equation; i.e., it has the form x = x0 +

n X

λk xk

(λk ∈ F ).

k=1

The conjugate equation T 0 y 0 = x0 is solvable if and only if x0 (x1 ) = . . . = x (xn ) = 0. Moreover, the general solution y 0 is the sum of a particular solution y 00 and the general solution of the homogeneous equation; i.e., it has the form 0

y 0 = y 00 +

n X

µk y 0k

(µk ∈ F ).

k=1

C This is a reformulation of 8.5.5 with account taken of the Polar Lemma. B 8.5.7. Examples. (1) If T is invertible then T is an index-zero Fredholm operator. (2) Let T ∈ L (F n , F m ). Let rank T := dim im T be the rank of T . Then α(T ) = n − rank T and β(T ) = m − rank T . Consequently, T ∈ F r(F n , F m ) and ind T = n − m. (3) Let T ∈ B(X) and X = X1 ⊕ X2 . Assume that this direct sum decomposition of X reduces T to matrix form   T1 0 T ∼ . 0 T2 Undoubtedly, T is a Fredholm operator if and only if its parts are Fredholm operators. Moreover, α(T ) = α(T1 )+α(T2 ) and β(T ) = β(T1 )+β(T2 ); i.e., ind T = ind T1 + ind T2 . CB

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Chapter 8

8.5.8. Fredholm Theorem. Let K ∈ K (X). Then 1 − K is an index-zero Fredholm operator. C First, settle the case F := C. If 1 ∈ / Sp(K) then 1 − K is invertible and ind (1 − K) = 0. If 1 ∈ Sp(K) then in virtue of the Riesz–Schauder Theorem and the Spectral Decomposition Theorem there is a decomposition X = X1 ⊕ X2 such that X1 is finite-dimensional and 1 ∈ / Sp(K2 ), with K2 the part of K in X2 . Furthermore,   1 − K1 0 1−K ∼ . 0 1 − K2 By 8.5.7 (2), ind (1 − K1 ) = 0 and, by 8.5.7 (3), ind (1 − K) = ind (1 − K1 ) + ind (1 − K2 ) = 0. In the case of the reals, F := R, proceed by way of complexification as in the proof of 8.4.8. Namely, consider the operator K(x, y) := (Kx, Ky) in the space X ⊕ iX. By above, ind (1 − K) = 0. Considering the difference between R and C, observe that α(1 − K) = α(1 − K) and β(1 − K) = β(1 − K). Finally, ind (1 − K) = 0. B 8.5.9. Definition. Let T ∈ B(X, Y ). An operator L, a member of B(Y, X), is a left approximate inverse of T if LT − 1 ∈ K (X). An operator R, a member of B(Y, X), is a right approximate inverse of T if T R − 1 ∈ K (Y ). An operator S, a member of B(Y, X), is an approximate inverse of T if S is simultaneously a left and right approximate inverse of T . If an operator T has an approximate inverse S then T is called approximately invertible. The terms “regularizer” and “parametrix” are all current in this context with regard to S. 8.5.10. Let L and R be a left approximate inverse and a right approximate inverse of T , respectively. Then L − R ∈ K (Y, X). / LT = 1 + KX (KX ∈ K (X)) ⇒ LT R = R + KX R; T R = 1 + KY (KY ∈ K (Y )) ⇒ LT R = L + LKY . 8.5.11. If L is a left approximate inverse of T and K ∈ K (Y, X) then L + K is also a left approximate inverse of T . / (L + K)T − 1 = (LT − 1) + KT ∈ K (X) . 8.5.12. An operator is approximately invertible if and only if it has a right approximate inverse and a left approximate inverse. C Only the implication ⇐ needs examining. Let L and R be a left approximate inverse and a right approximate inverse of T , respectively. By 8.5.10, K := L − R ∈ K (Y, X). Consequently, by 8.5.11, R = L − K is a left approximate inverse of T . Thus, R is an approximate inverse of T . B

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8.5.13. Remark. The above shows that, in the case X = Y , an operator S is an approximate inverse of T if and only if ϕ(S)ϕ(T ) = ϕ(T )ϕ(S) = 1, where ϕ : B(X)0 = 0 O (X)/K (X) is the coset mapping to the Calkin algebra. In other words, a left approximate inverse is the inverse image of a left inverse in the Calkin algebra, etc. 8.5.14. Noether Criterion. An operator is a Fredholm operator if and only if it is approximately invertible. C ⇒: Let T ∈ F r(X, Y ). Using the Complementation Principle, consider the decompositions X = ker T ⊕ X1 and Y = im T ⊕ Y1 and the respective finite-rank projections P which carries X onto ker T along X1 and Q which carries Y onto Y1 along im T . It is clear that the restriction T1 := T |X1 is an invertible operator T1 : X1 0 = 0 O T . Put S := T1−1 (1 − Q). The operator S may be viewed as a member of B(Y, X). Moreover, it is beyond a doubt that ST + P = 1 and T S + Q = 1. ⇐: Let S be an approximate inverse of T ; i.e., ST = 1 + KX and T S = 1 + KY for appropriate compact operators KX and KY . Consequently, ker T ⊂ ker(1+KX ); i.e., ker T is finite-dimensional since so is ker(1+KX ) in virtue of 8.5.8. Furthermore, im T ⊃ im(1 + KY ); and so the range of T is of finite codimension because 1 + KY is an index-zero Fredholm operator. B 8.5.15. Corollary. If T ∈ F r(X, Y ) and S is an approximate inverse of T then S ∈ F r(Y, X). CB 8.5.16. Corollary. The product of Fredholm operators is itself a Fredholm operator. C The composition of approximate inverses (taken in due succession) is an approximate inverse to the composition of the originals. B 8.5.17. Consider an exact sequence 00 = 0 O 1 0 = 0 O 2 0 = 0 O 0 = 0 O

n−1 0

= 0 O n0 = 0 O

of finite-dimensional vector spaces. Then the Euler identity holds: n X

(−1)k dim Xk = 0.

k=1

C For n = 1 the exactness of the sequence 00 = 0 O 1 0 = 0 O means that X1 = 0, and for n = 2 the exactness of 00 = 0 O 1 0 = 0 O 2 0 = 0 O amounts to isomorphy between X1 and X2 (cf. 2.3.5 (4)). Therefore, the Euler identity is beyond a doubt for n := 1, 2.

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Suppose now that for m ≤ n − 1, where n > 2, the desired identity is already established. The exact sequence Tn−2 Tn−1 0 → X1 0 = 0 O 2 0 = 0 O → Xn−2 −−−−→ Xn−1 −−−−→ Xn → 0

reduces to the exact sequence Tn−2

0 → X1 → X2 → . . . → Xn−2 −−−→ ker Tn−1 → 0. By hypothesis, n−2 X

(−1)k dim Xk + (−1)n−1 dim ker Tn−1 = 0.

k=1

Furthermore, since Tn−1 is an epimorphism, dim Xn−1 = dim ker Tn−1 + dim Xn . Finally, 0=

n−2 X

(−1)k dim Xk + (−1)n−1 (dim Xn−1 − dim Xn )

k=1

=

n X

(−1)k dim Xk . .

k=1

8.5.18. Atkinson Theorem. The index of the product of Fredholm operators equals the sum of the indices of the factors. C Let T ∈ F r(X, Y ) and S ∈ F r(Y, Z). By virtue of 8.5.16, ST ∈ F r(X, Z). Using the Snowflake Lemma, obtain the exact sequence of finite-dimensional spaces 00 = 0 O T 0 = 0 O ST 0 = 0 O S0 = 0 O T 0 = 0 O ST 0 = 0 O S0 = 0 O . Applying 8.5.17, infer that α(T ) − α(ST ) + α(S) − β(T ) + β(ST ) − β(S) = 0; whence ind (ST ) = ind S + ind T . B 8.5.19. Corollary. Let T be a Fredholm operator and let S be an approximate inverse of T . Then ind T = − ind S. C ind (ST ) = ind (1 + K) for some compact operator K. By Theorem 8.5.8, 1 + K is an index-zero Fredholm operator. B

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8.5.20. Compact Index Stability Theorem. The property of being a Fredholm operator and the index of a Fredholm operator are preserved under compact perturbations: if T ∈ F r(X, Y ) and K ∈ K (X, Y ) then T + K ∈ F r(X, Y ) and ind (T + K) = ind T . C Let S be an approximate inverse to T ; i.e., ST = 1 + KX ;

T S = 1 + KY

with some KX ∈ K (X) and KY ∈ K (Y ) (that S exists is ensured by 8.5.14). It is clear that S(T + K) = ST + SK = 1 + KX + SK ∈ 1 + K (X); (T + K)S = T S + KS = 1 + KY + KS ∈ 1 + K (Y ); i.e., S is an approximate inverse of T + K. By virtue of 8.5.14, T + K ∈ F r(X, Y ). Finally, from 8.5.19 infer the equalities ind (T +K) = − ind S and ind T = − ind S. B 8.5.21. Bounded Index Stability Theorem. The property of being a Fredholm operator and the index of a Fredholm operator are preserved under sufficiently small bounded perturbations: the set F r(X, Y ) is open in the space of bounded operators, and the index of a Fredholm operator ind : F r(X, Y )0 = 0 O Z is a continuous function. C Let T ∈ F r(X, Y ). By 8.5.14 there are operators S ∈ B(Y, X), KX ∈ K (X) and KY ∈ K (Y ) such that ST = 1 + KX ;

T S = 1 + KY .

If S = 0 then the spaces X and Y are finite-dimensional by the Riesz Criterion, i.e., nothing is left to proof: it suffices to refer to 8.5.7 (2). If S 6= 0 then for all V ∈ B(X, Y ) with kV k < 1/kSk, from the inequality of 5.6.1 it follows: kSV k < 1 and kV Sk < 1. Consequently, in virtue of 5.6.10 the operators 1 + SV and 1 + V S are invertible in B(X) and in B(Y ), respectively. Observe that (1 + SV )−1 S(T + V ) = (1 + SV )−1 (1 + KX + SV ) = 1 + (1 + SV )−1 KX ∈ 1 + K (X); i.e., (1 + SV )−1 S is a left approximate inverse of T + V . By analogy, show that S(1 + V S)−1 is a right approximate inverse of T + V . Indeed, (T + V )S(1 + V S)−1 = (1 + KY + V S)(1 + V S)−1 = 1 + KY (1 + V S)−1 ∈ 1 + K (Y ).

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Chapter 8

By 8.5.12, T +V is approximately invertible. In virtue of 8.5.14, T +V ∈ F r(X, Y ). This proves the openness property of F r(X, Y ). When left and right approximate inverses of a Fredholm operator W exist, each of them is an approximate inverse to W (cf. 8.5.12), Therefore, from 8.5.19 and 8.5.18 obtain ind (T + V ) = − ind ((1 + SV )−1 S) = − ind (1 + SV )−1 − ind S = − ind S = ind T (because (1 + SV )−1 is a Fredholm operator by 8.5.7 (1)). This means that the Fredholm index is continuous. B 8.5.22. Nikol 0ski˘ı Criterion. An operator is an index-zero Fredholm operator if and only if it is the sum of an invertible operator and a compact operator. C ⇒: Let T ∈ F r(X, Y ) and ind T = 0. Consider the direct sum decompositions X = X1 ⊕ ker T and Y = im T ⊕ Y1 . It is beyond a doubt that the operator T1 , the restriction of T to X1 , implements an isomorphism between X1 and im T . Furthermore, in virtue of 8.5.5, dim Y1 = β(T ) = α(T ); i.e., there is a natural isomorphism Id : ker T 0 = 0 O 1 . Therefore, T admits the matrix presentation       T1 0 T1 0 0 0 T ∼ = + . 0 0 0 Id 0 − Id ⇐: If T := S + K with K ∈ K (X, Y ) and S −1 ∈ B(Y, X) then, by 8.5.20 and 8.5.7 (1), ind T = ind (S + K) = ind S = 0. B 8.5.23. Remark. Let Inv(X, Y ) stand as before for the set of all invertible operators from X to Y (this set is open by Theorem 5.6.12). Denote by F (X, Z) the set of all Fredholm operators acting from X to Y and having index zero. The Nikol0 ski˘ı Criterion may now be written down as F (X, Y ) = Inv(X, Y ) + K (X, Y ). As is seen from the proof of 8.5.22, it may also be asserted that F (X, Y ) = Inv(X, Y ) + F (X, Y ), where, as usual, F (X, Y ) is the subspace of B(X, Y ) comprising all finite-rank operators. CB Exercises 8.1. Study the Riesz–Dunford integral in finite dimensions. 8.2. Describe the kernel of the Riesz–Dunford integral. 8.3. Given n ∈ N, let fn be a function holomorphic in a neighborhood U about the spectrum of an operator T . Prove that the uniform convergence of (fn ) to zero on U follows from the convergence of (fn (T )) to zero in operator norm.

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8.4. Let σ be an isolated part of the spectrum of an operator T . Assume that the part σ 0 := Sp(T ) \ σ is separated from σ by some open disk with center a and radius r so that σ ⊂ {z ∈ C : |z − a| < r}. Considering the Riesz projection Pσ , prove that Pσ = lim (1 − z −n (T − a)n )−1 ; n

1

x ∈ im(Pσ ) ⇔ lim sup k(a − T )n xk

/n

< r.

n

8.5. Find conditions for a projection to be a compact operator. 8.6. Prove that every closed subspace lying in the range of a compact operator in a Banach space is finite-dimensional. 8.7. Prove that a linear operator carries each closed linear subspace onto a closed set if and only if the operator is normally solvable and its kernel is finite-dimensional or finite-codimensional (the latter means that the kernel has a finite-dimensional algebraic complement). 8.8. Let 1 ≤ p < r < +∞. Prove that every bounded operator from lr to lp or from c0 to lp is compact. 8.9. Let H be a separable Hilbert space. Given an operator T in B(H) and a Hilbert basis (en ) for H, define the Hilbert–Schmidt norm as

kT k2 :=

∞ X

!1 /2 kT en k2

.

n=1

(Examine soundness!) An operator with finite Hilbert–Schmidt norm is a Hilbert–Schmidt operator. Demonstrate that an operator T is a Hilbert–Schmidt operator if and only if T is compact P∞ 2 1 and λ < +∞, where (λn ) ranges over the eigenvalues of some operator (T ∗ T ) /2 (define n=1 n the latter!). 8.10. Let T be an endomorphism. Then im(T 0 ) ⊃ im(T 1 ) ⊃ im(T 2 ) ⊃ . . . . If there is a number n satisfying im(T n ) = im(T n+1 ) then say that T has finite descent. The least number n with which stabilization begins is the descent of T , denoted by d(T ). By analogy, considering the kernels ker(T 0 ) ⊂ ker(T 1 ) ⊂ ker(T 2 ) ⊂ . . . , introduce the concept of ascent and the denotation a(T ). Demonstrate that, for an operator T with finite descent and finite ascent, the two quantities, a(T ) and d(T ), coincide. 8.11. An operator T is a Riesz–Schauder operator, if T is a Fredholm operator and has finite descent and finite ascent. Prove that an operator T is a Riesz–Schauder operator if and only if T is of the form T = U + V , where U is invertible and V is of finite rank (or compact) and commutes with U . 8.12. Let T be a bounded endomorphism of a Banach space X which has finite descent and finite ascent, r := a(T ) = d(T ). Prove that the subspaces im(T r ) and ker(T r ) are closed, the decomposition X = ker(T r ) ⊕ im(T r ) reduces T , and the restriction of T to im(T r ) is invertible.

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Chapter 8 8.13. Let T be a normally solvable operator. If either of the next quantities is finite α(T ) := dim ker T,

β(T ) := dim coker T,

then T is called semi-Fredholm. Put Ф+ (X) := {T ∈ B(X) : im T ∈ Cl(X), α(T ) < +∞}; Ф− (X) := {T ∈ B(X) : im T ∈ Cl(X), β(T ) < +∞}. Prove that T ∈ Ф+ (X) ⇔ T 0 ∈ Ф− (X 0 ); T ∈ Ф− (X) ⇔ T 0 ∈ Ф+ (X 0 ). 8.14. Let T be a bounded endomorphism. Prove that T belongs to Ф+ (X) if and only if for every bounded but not totally bounded set U , the image T (U ) is not a totally bounded set in X. 8.15. A bounded endomorphism T in a Banach space is a Riesz operator, if for every nonzero complex λ the operator (λ − T ) is a Fredholm operator. Prove that T is a Riesz operator if and only if for all λ ∈ C, λ 6= 0, the following conditions are fulfilled: (1) the operator (λ − T ) has finite descent and finite ascent; (2) the kernel of (λ − T )k is finite-dimensional for every k ∈ N; (3) the range of (λ − T )k has finite deficiency for k ∈ N, and, moreover, all nonzero points of the spectrum of T are eigenvalues, with zero serving as the only admissible limit point (that is, for whatever strictly positive ε, there are only finitely many points of Sp(T ) beyond the disk centered at zero with radius ε). 8.16. Establish the isometric isomorphisms: (X/Y )0 ' Y ⊥ and X 0 /Y ⊥ ' Y 0 for Banach spaces X and Y such that Y is embedded into X. 8.17. Prove that for a normal operator T in a Hilbert space and a holomorphic function f , a member of H(Sp(T )), the operator f (T ) is normal. (An operator is normal if it commutes with its adjoint, cf. 11.7.1.) 8.18. Show that a continuous endomorphism T of a Hilbert space is a Riesz operator if and only if T is the sum of a compact operator and a quasinilpotent operator. (Quasinilpotency of an operator means triviality of its spectral radius.) 8.19. Given two Fredholm operators S and T , members of Fr(X, Y ), with ind S = ind T , demonstrate that there is a Jordan arc joining S and T within Fr(X, Y ).

Chapter 9 An Excursus into General Topology

9.1. Pretopologies and Topologies 9.1.1. Definition. Let X be a set. A mapping τ : X0 = 0 O P (P(X)) is a pretopology on X if (1) x ∈ X ⇒ τ (x) is a filter on X; (2) x ∈ X ⇒ τ (x) ⊂ fil{x}. A member of τ (x) is a (pre)neighborhood about x or of x. The pair (X, τ ), as well as the set X itself, is called a pretopological space. 9.1.2. Definition. Let T (X) be the collection of all pretopologies on X. If τ1 , τ2 ∈ T (X) then τ1 is said to be stronger than τ2 or finer than τ2 (in symbols, τ1 ≥ τ2 ) provided that x ∈ X ⇒ τ1 (x) ⊃ τ2 (x). Of course, τ2 is weaker or coarser than τ1 . 9.1.3. The set T (X) with the relation “to be stronger” presents a complete lattice. C If X = ∅ then T (X) = {∅} and there is nothing to be proven. If X 6= ∅ then refer to 1.3.13. B 9.1.4. Definition. A subset G of X is an open set in X, if G is a (pre)neighborhood of its every point (in symbols, G ∈ Op(τ ) ⇔ (∀ x ∈ G) (G ∈ τ (x))). A subset F of X is a closed set in X if the complement of F to X is open; that is, F ∈ Cl(τ ) ⇔ X \ F ∈ Op(τ ). 9.1.5. The union of a family of open sets and the intersection of a finite family of open sets are open. The intersection of a family of closed sets and the union of a finite family of closed sets are closed. CB 9.1.6. Let (X, τ ) be a pretopological space. Given x ∈ X, put U ∈ t(τ )(x) ⇔ (∃ V ∈ Op(τ )) x ∈ V & U ⊃ V. The mapping t(τ ) : x 7→ t(τ )(x) is a pretopology on X. CB

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Chapter 9

9.1.7. Definition. A pretopology τ on X is a topology if τ = t(τ ). The pair (X, τ ), as well as the underlying set X itself, is then called a topological space. The set of all topologies on X is denoted by the symbol T(X). 9.1.8. Examples. (1) A metric topology. (2) The topology of a multinormed space. (3) Let τ◦ := inf T (X). It is clear that τ◦ (x) = {X} for x ∈ X. Consequently, Op(τ◦ ) = {∅, X} and so τ◦ = t(τ◦ ); i.e., τ◦ is a topology. This topology is called trivial, or antidiscrete, or even indiscrete. (4) Let τ ◦ := sup T (X). It is clear that τ ◦ (x) = fil {x} for x ∈ X. Consequently, Op(τ ◦ ) = 2X and so τ ◦ = t(τ ◦ ); i.e., τ ◦ is a topology. This topology is called discrete. (5) Let Op be a collection of subsets in X which is stable under the taking of the union of each of its subfamilies and the intersection of each of its finite subfamilies. Then there is a unique topology τ on X such that Op(τ ) = Op. C Put τ (x) := fil {V ∈ Op : x ∈ V } for x ∈ X (in case X = ∅ there is nothing to prove). Observe that τ (x) 6= ∅ since the intersection of the empty family equals X (cf. inf ∅ = +∞). From the construction derive that t(τ ) = τ and Op ⊂ Op(τ ). If G ∈ Op(τ ) then G = ∪{V : V ∈ Op, V ⊂ G} and so G ∈ Op by hypothesis. The claim of uniqueness raises no doubts. B 9.1.9. Let the mapping t : T (X)0 = 0 O T (X) act by the rule t : τ 7→ t(τ ). Then (1) im t = T(X); i.e., τ ∈ T (X) ⇒ t(τ ) ∈ T(X); (2) τ1 ≤ τ2 ⇒ t(τ1 ) ≤ t(τ2 ) (τ1 , τ2 ∈ T (X)); (3) t ◦ t = t; (4) τ ∈ T (X) ⇒ t(τ ) ≤ τ ; (5) Op(τ ) = Op(t(τ )) (τ ∈ T (X)). C The inclusion Op(τ ) ⊃ Op(t(τ )) holds because it is easier to be open in τ . The reverse inclusion Op(τ ) ⊂ Op(t(τ )) follows from the definition of t(τ ). The equality Op(τ ) = Op(t(τ )) makes everything evident. B 9.1.10. A pretopology τ on X is a topology if and only if (∀ U ∈ τ (x))(∃ V ∈ τ (x) & V ⊂ U )(∀ y) (y ∈ V ⇒ V ∈ τ (y)) for x ∈ X. C Straightforward from 9.1.9 (5). B

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9.1.11. Let τ1 , τ2 ∈ T(X). The following statements are equivalent: (1) τ1 ≥ τ2 ; (2) Op(τ1 ) ⊃ Op(τ2 ); (3) Cl(τ1 ) ⊃ Cl(τ2 ). CB 9.1.12. Remark. As follows from 9.1.8 (5) and 9.1.11, the topology of a space is uniquely determined from the collection of its open sets. Therefore, the set Op(X) itself is legitimately called the topology of the space X. In particular, the collection of open sets of a pretopological space (X, τ ) makes X into the topological space (X, t(τ )) with the same open sets in stock. Therefore, given a pretopology τ , the topology t(τ ) is usually called the topology associated with τ . 9.1.13. Theorem. The set T(X) of all topologies on X with the relation “to be stronger” presents a complete lattice. Moreover, for every subset E of T(X) the equality holds: supT(X) E = supT (X) E . C Evidently, t(supT (X) E ) ≥ supT (X) t(E ) ≥ supT (X) E ≥ t(supT (X) E ). Thus, τ := supT (X) E belongs to T(X). It is clear that τ ≥ E . Furthermore, if τ0 ≥ E and τ0 ∈ T(X) then τ0 ≥ τ and so τ = supT(X) E . It remains to refer to 1.2.14. B 9.1.14. Remark. The explicit formula for the greatest lower bound of E is more involved: inf T(X) E = t(inf T (X) E ). However, the matter becomes simpler when the topologies are given by means of their collections of open sets in accordance with 9.1.12. Namely, U ∈ Op(inf T(X) E ) ⇔ (∀ τ ∈ E ) U ∈ Op(τ ). In other words, Op(inf T(X) E ) =

\

Op(τ ).

τ ∈E

In this connection it is in common parlance to speak of the intersection of the set E of topologies (rather than of the greatest lower bound of E ). CB 9.2. Continuity 9.2.1. Remark. The presence of a topology on a set obviously makes it possible to deal with such things as the interior and closure of a subset, convergence of filters and nets, etc. We have already made use of this circumstance while introducing multinormed spaces. Observe for the sake of completeness that in every topological space the following analogs of 4.1.19 and 4.2.1 are valid:

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Chapter 9

9.2.2. Birkhoff Theorem. For a nonempty subset U and a point x of a topological space the following statements are equivalent: (1) x is an adherent point of U ; (2) there is some filter containing U and converging to x; (3) there is a net of elements of U which converges to x. CB 9.2.3. For a mapping f between topological spaces the following conditions are equivalent: (1) the inverse image under f of an open set is open; (2) the inverse image under f of a closed set is closed; (3) the image under f of the neighborhood filter of an arbitrary point x is coarser than the neighborhood filter of f (x); (4) for all x, the mapping f transforms each filter convergent to x into a filter convergent to f (x); (5) for all x, the mapping f sends a net that converges to x to a net that converges to f (x). CB 9.2.4. Definition. A mapping acting between topological spaces X and Y and satisfying one (and hence all) of the equivalent conditions 9.2.3 (1)–9.2.3 (5) is called continuous. A continuous one-to-one mapping f from X onto Y whose inverse f −1 acts continuously from Y to X is a homeomorphism or a topological mapping or a topological isomorphism between X and Y . 9.2.5. Remark. If f : (X, τX ) → (Y, τY ) meets 9.2.3 (5) at some point x in X then it is customary to say that f is continuous at x (cf. 4.2.2). Observe that the difference is immaterial between the definitions of the continuity property at a point of X and the general continuity property (on X). Indeed, if we let τx (x) := τX (x) and τx (x) := fil {x} for x ∈ X, x 6= x, then the continuity property of f at x (with respect to the topology τX in X) amounts to that of f : (X, τx ) → (Y, τY ) (at every point of the space X with topology τx ). 9.2.6. Let τ1 , τ2 ∈ T(X). Then τ1 ≥ τ2 if and only if IX : (X, τ1 ) → (X, τ2 ) is continuous. CB 9.2.7. Let f : (X, τ ) → (Y, ω) be a continuous mapping and let τ1 ∈ T(X) and ω1 ∈ T(Y ) be such that τ1 ≥ τ and ω ≥ ω1 . Then f : (X, τ1 ) → (Y, ω1 ) is continuous. C By hypothesis the following diagram commutes: (X, τ ) IX ↑ (X, τ1 )

f

−→ f

−→

(Y, ω) ↓ IY (Y, ω1 )

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It suffices to observe that every composition of continuous mappings is continuous. B 9.2.8. Inverse Image Topology Theorem. Let f : X → (Y, ω). Put T0 := {τ ∈ T(X) : f : (X, τ ) → (Y, ω) is continuous}. Then the topology f −1 (ω) := inf T0 belongs to T0 . C From 9.2.3 (1) it follows that τ ∈ T0 ⇔ (x ∈ X ⇒ f −1 (ω(f (x))) ⊂ τ (x)). Let τ (x) := f −1 (ω(f (x))). Undoubtedly, t(τ ) = τ . Furthermore, f (τ (x)) = f (f −1 (ω(f (x)))) ⊃ ω(f (x)); i.e., τ ∈ T0 by 9.2.3 (3). Thus, f −1 (ω) = τ . B 9.2.9. Definition. The topology f −1 (ω) is the inverse image of ω under a mapping f or simply the inverse image topology under f . 9.2.10. Remark. Theorem 9.2.8 is often verbalized as follows: “The inverse image topology under a mapping is the weakest topology on the set of departure in which the mapping is continuous.” Moreover, it is easy for instance from 9.1.14 that the open sets of the inverse image topology are precisely the inverse images of open sets. In particular, (xξ → x in f −1 (ω)) ⇔ (f (xξ ) → f (x) in ω); likewise, (F → x in f −1 (ω)) ⇔ (f (F ) → f (x) in ω) for a filter F . CB 9.2.11. Image Topology Let f : (X, τ ) → Y . Put Š0 := {ω ∈ T(Y ) : f : (X, τ ) → (Y, ω) is continuous}. Then the topology f (τ ) := sup Š0 belongs to Š0 . C Appealing to 9.1.13, observe that f (τ )(y) = (supT(Y ) Š0 )(y) = (supT (Y ) Š0 )(y) = sup{ω(y) : ω ∈ Š0 } for y ∈ Y . By virtue of 9.2.3 (3), ω ∈ Š0 ⇔ (x ∈ X ⇒ f (τ (x)) ⊃ ω(f (x))). Comparing the formulas, infer that f (τ ) ∈ Š0 . B 9.2.12. Definition. The topology f (τ ) is the image of τ under a mapping f or simply the image topology under f .

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Chapter 9

9.2.13. Remark. Theorem 9.2.11 is often verbalized as follows: “The image of a topology under a mapping is the strongest topology on the set of arrival in which the mapping is continuous.” 9.2.14. Theorem. Let (fξ : X → (Yξ , ωξ ))ξ∈„ be a family of mappings. Further, put τ := supξ∈„ fξ−1 (ωξ ). Then τ is the weakest (= least) topology on X making all the mappings fξ (ξ ∈ „) continuous. C Using 9.2.8, note that (fξ : (X, τ ) → (Yξ , ωξ ) is continuous) ⇔ τ ≥ fξ−1 (ωξ ). . 9.2.15. Theorem. Let (fξ : (Xξ , τξ ) → Y )ξ∈„ be a family of mappings. Further, assign ω := inf ξ∈„ fξ (τξ ). Then ω is the strongest (= greatest) topology on Y making all the mappings fξ (ξ ∈ „) continuous. C Appealing to 9.2.11, conclude that (fξ : (Xξ , τξ ) → (Y, ω) is continuous) ⇔ ω ≤ fξ (τξ ). . 9.2.16. Remark. The messages of 9.2.14 and 9.2.15 are often referred to as the theorems on topologizing by a family of mappings. 9.2.17. Examples. (1) Let (X, τ ) be a topological space and let X0 be a subset of X. Denote the identical embedding of X0 into X by ι : X0 → X. Put τ0 := ι−1 (τ ). The topology τ0 is the induced topology (by τ in X0 ), or the relative or subspace topology; and the space (X0 , τ0 ) is a subspace of (X, τ ). (2) Let (Xξ , τξ )ξ∈„ be a family of topological spaces and let X := Q X be the product of (Xξ )ξ∈„ . Put τ := supξ∈„ Pr−1 ξ (τξ ), where Prξ : X → Xξ ξ∈„ ξ is the coordinate projection (onto Xξ ); i.e., Prξ x = xξ (ξ ∈ „). The topology τ is the product topology or the product of the topologies (τξ )ξ∈„ , or the Tychonoff topology of X. The space (X, τ ) is the Tychonoff product of the topological spaces under study. In particular, if Xξ := [0, 1] for all ξ ∈ „ then X := [0, 1]„ (with the Tychonoff topology) is a Tychonoff cube. When „ := N, the term “Hilbert cube” is applied. 9.3. Types of Topological Spaces 9.3.1. For a topological space the following conditions are equivalent: (1) every singleton of the space is closed; (2) the intersection of all neighborhoods of each point in the space consists solely of the point;

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(3) each one of any two points in the space has a neighborhood disjoint from the other. C To prove, it suffices to observe that y ∈ cl{x} ⇔ (∀ V ∈ τ (y)) x ∈ V ⇔ x ∈ ∩{V : V ∈ τ (y)}, where x and y are points of a space with topology τ . B 9.3.2. Definition. A topological space satisfying one (and hence all) of the equivalent conditions 9.3.1 (1)–9.3.1 (3), is called a separated space or a T1 -space. The topology of a T1 -space is called a separated topology or (rarely) a T1 -topology. 9.3.3. Remark. By way of expressiveness, one often says: “A T1 -space is a space with closed points.” 9.3.4. For a topological space the following conditions are equivalent: (1) each filter has at most one limit; (2) the intersection of all closed neighborhoods of a point in the space consists of the sole point; (3) each one of any two points of the space has a neighborhood disjoint from some neighborhood of the other point. C (1) ⇒ (2): If y ∈ ∩U ∈τ (x) cl U then U ∩ V 6= ∅ for all V ∈ τ (y), provided that U ∈ τ (x). Therefore, the join F := τ (x) ∨ τ (y) is available. Clearly, F → x and F → y. By hypothesis, x = y. (2) ⇒ (3): Let x, y ∈ X, x 6= y (if such points are absent then either X = ∅ or X is a singleton and nothing is left unproven). There is an neighborhood U in τ (x) such that U = cl U and y ∈ / U . Consequently, the complement V of U to X is open. Furthermore, U ∩ V = ∅. (3) ⇒ (1): Let F be a filter on X. If F → x and F → y then F ⊃ τ (x) and F ⊃ τ (y). Thus, U ∩ V 6= ∅ for U ∈ τ (x) and V ∈ τ (y), which means that x = y. B 9.3.5. Definition. A topological space satisfying one (and hence all) of the equivalent conditions 9.3.4 (1)–9.3.4 (3) is a Hausdorff space or a T2 -space. A natural meaning is ascribed to the term “Hausdorff topology.” 9.3.6. Remark. By way of expressiveness, one often says: “A T2 -space is a space with unique limits.” 9.3.7. Definition. Let U and V be subsets of a topological space. It is said that V is a neighborhood of U or about U , provided int V ⊃ U . If U is nonempty then all neighborhoods of U constitute some filter that is the neighborhood filter of U .

160

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9.3.8. For a topological space the following conditions are equivalent: (1) the intersection of all closed neighborhoods of an arbitrary closed set consists only of the members of the set; (2) the neighborhood filter of each point has a base of closed sets; (3) if F is a closed set and x is a point not in F then there are disjoint neighborhoods of F and x, respectively. C (1) ⇒ (2): If x ∈ X and U ∈ τ (x) then V := X \ int U is closed and x ∈ / V. By hypothesis there is a set F in Cl(τ ) such that x ∈ / F and int F ⊃ V . Put G := X \ F . Clearly, G ∈ τ (x). Moreover, G ⊂ X \ int F = cl(X \ int F ) ⊂ X \ V ⊂ int U ⊂ U . Consequently, cl G ⊂ U . (2) ⇒ (3): If x ∈ X and F ∈ Cl(τ ) with x ∈ / F then X \ F ∈ τ (x). Thus, there is a closed neighborhood U in τ (x) lying in X \ F . Thus, X \ U is a neighborhood of F disjoint from U . (3) ⇒ (1): If F ∈ Cl(τ ) and int G ⊃ F ⇒ y ∈ cl G, then U ∩ G 6= ∅ for every U in τ (y) and every neighborhood G of F . This means that y ∈ F . B 9.3.9. Definition. A T3 -space is a topological space satisfying one (and hence all) of the equivalent conditions 9.3.8 (1)–9.3.8 (3). A separated T3 -space is called regular. 9.3.10. Urysohn Little Lemma. For a topological space the following conditions are equivalent: (1) the neighborhood filter of each nonempty closed set has a base of closed sets; (2) if two closed sets are disjoint then they have disjoint neighborhoods. C (1) ⇒ (2): Let F1 and F2 be closed sets in some space X with F1 ∩ F2 = ∅. Put G := X \ F1 . Obviously, G is open and G ⊃ F2 . If F2 = ∅, then there is nothing to be proven. It may be assumed consequently that F2 6= ∅. Then there is a closed set V2 such that G ⊃ V2 ⊃ int V2 ⊃ F2 . Put V1 := X \ V2 . It is clear that V1 is open, and V1 ∩ V2 = ∅. Moreover, V1 ⊃ X \ G = X \ (X \ F1 ) = F1 . (2) ⇒ (1): Let F = cl F , G = int G and G ⊃ F . Put F1 := X \ G. Then F1 = cl F1 and so there are open sets U and U1 satisfying U ∩ U1 = ∅, with F ⊂ U and F1 ⊂ U1 . Finally, cl U ⊂ X \ U1 ⊂ X \ F1 = G. B 9.3.11. Definition. A T4 -space is a topological space meeting one (and hence both) of the equivalent conditions 9.3.10 (1) and 9.3.10 (2). A separated T4 -space is called normal. 9.3.12. Continuous Function Recovery Lemma. Let a subset T be dense in R and let t 7→ Ut (t ∈ T ) be a family of subsets of a topological space X. There is a unique continuous function f : X → R such that {f < t} ⊂ Ut ⊂ {f ≤ t} (t ∈ T )

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if and only if (t, s ∈ T & t < s) ⇒ cl Ut ⊂ int Us . C ⇒: Take t < s. Since {f ≤ t} is closed and {f < s} is open, the inclusions hold: cl Ut ⊂ {f ≤ t} ⊂ {f < s} ⊂ int Us . ⇐: Since Ut ⊂ cl Ut ⊂ int Us ⊂ Us for t < s, the family t 7→ Ut (t ∈ T ) increases by inclusion. Therefore, f exists by 3.8.2 and is unique by 3.8.4. Consider the families t 7→ Vt := cl Ut and t 7→ Wt := int Ut . These families increase by inclusion. Consequently, on applying 3.8.2 once again, find functions g, h : X → R satisfying {g < t} ⊂ Vt ⊂ {g ≤ t},

{h < t} ⊂ Wt ⊂ {h ≤ t}

for all t ∈ T . If t, s ∈ T and t < s, then in view of 3.8.3 Wt = int Ut ⊂ Ut ⊂ Us ⇒ f ≤ h; Vt = cl Ut ⊂ int Us = Ws ⇒ h ≤ g; Ut ⊂ Us ⊂ cl Us = Vs ⇒ g ≤ f. So, f = g = h. Taking account of 3.8.4 and 9.1.5 and given t ∈ R, find {f < t} = {h < t} = ∪{Ws : s < t, s ∈ T } ∈ Op(τX ); {f ≤ t} = {g ≤ t} = ∩{Vs : t < s, s ∈ T } ∈ Cl(τX ). These inclusions readily provide continuity for f . B 9.3.13. Urysohn Great Lemma. Let X be a T4 -space. Assume further that F is a closed set in X and G is a neighborhood of F . Then there is a continuous function f : X → [0, 1] such that f (x) = 0 for x ∈ F and f (x) = 1 for x ∈ / G. C Put Ut := ∅ for t < 0 and Ut := X for t > 1. Consider the set T of the dyadic-rational points of the interval [0, 1]; i.e., T := ∪n∈N Tn with Tn := {k2−n+1 : k := 0, 1, . . . , 2n−1 }. It suffices to define Ut for all t in T so that the family t 7→ Ut (t ∈ T := T ∪ (R \ [0, 1])) satisfy the criterion of 9.3.12. This is done by way of induction. If t ∈ T1 , i.e., t ∈ {0, 1}; then put U0 := F and U1 := G. Assume now that, for t ∈ Tn and n ≥ 1, some set Ut has already been constructed, satisfying cl Ut ⊂ int Us whenever t, s ∈ Tn and t < s. Take t ∈ Tn+1 and find the two points tl and tr in Tn nearest to t: tl := sup{s ∈ Tn : s ≤ t}; tr := inf{s ∈ Tn : t ≤ s}.

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If t = tl or t = tr , then Ut exists by the induction hypothesis. If t 6= tl and t 6= tr , then tl < t < tr and again by the induction hypothesis cl Utl ⊂ int Utr . By virtue of 9.3.11 there is a closed set Ut such that cl Utl ⊂ int Ut ⊂ Ut = cl Ut ⊂ int Utr . It remains to show that the resulting family satisfies the criterion of 9.3.12. To this end, take t, s ∈ Tn+1 with t < s. If tr = sl , then for s > sl by construction cl Ut ⊂ cl Utr = cl Usl ⊂ int Us . For t < tr = sl similarly deduce the following: cl Ut ⊂ int Utr = inf Usl ⊂ int Us . If tr < sl then, on using the induction hypothesis, infer that cl Ut ⊂ cl Utr ⊂ int Usl ⊂ int Us , what was required. B 9.3.14. Urysohn Theorem. A topological space X is a T4 -space if and only if to every pair of disjoint closed sets F1 and F2 in X there corresponds a continuous function f : X → [0, 1] such that f (x) = 0 for x ∈ F1 and f (x) = 1 for x ∈ F2 . C ⇒: It suffices to apply 9.3.13 with F := F1 and G := X \ F2 . ⇐: If F1 ∩ F2 = ∅ and F1 and F2 are closed sets, then for a corresponding function f the sets G1 := {f < 1 /2 } and G2 := {f > 1 /2 } are open and disjoint. Moreover, G1 ⊃ F1 and G2 ⊃ F2 . B 9.3.15. Definition. A topological space X is a T31 /2 -space, if to a closed set F in X and a point x not in F there corresponds a continuous function f : X → [0, 1] such that f (x) = 1 and y ∈ F ⇒ f (y) = 0. A separated T31 /2 -space is a Tychonoff space or a completely regular space. 9.3.16. Every normal space is a Tychonoff space. C Straightforward from 9.3.1 and 9.3.14. B 9.4. Compactness 9.4.1. Let B be a filterbase on a topological space and let cl B := ∩{cl B : B ∈ B} be the set of adherent points of B (also called the adherence of B). Then

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(1) cl B = cl fil B; (2) B → x ⇒ x ∈ cl B; (3) (B is an ultrafilter and x ∈ cl B) ⇒ B → x. C Only (3) needs demonstrating, since (1) and (2) are evident. Given U ∈ τ (x) and B ∈ B, observe that U ∩ B 6= ∅. In other words, the join F := τ (x) ∨ B is available. It is clear that F → x. Furthermore, F = B, because B is an ultrafilter. B 9.4.2. Definition. A subset C of a topological space X is a compact set in X if each open cover of C has a finite subcover (cf. 4.4.1). 9.4.3. Theorem. Let X be a topological space and let C be a subset of X. The following statements are equivalent: (1) C is compact; (2) if a filterbase B lacks adherent points in C then there is a member B of B such that B ∩ C = ∅; (3) each filterbase containing C has an adherent point in C; (4) each ultrafilter containing C has a limit in C. C (1) ⇒ (2): Since cl B ∩ C = ∅; therefore, C ⊂ X \ cl B. Thus, C ⊂ X \ ∩{cl B : B ∈ B} = ∪{X \ cl B : B ∈ B}. Consequently, there is a finite subset B0 of B such that C ⊂ ∪{X \ cl B0 : B0 ∈ B0 } = X \ ∩{cl B0 : B0 ∈ B0 }. Let B ∈ B satisfy B ⊂ ∩{B0 : B0 ∈ B0 } ⊂ ∩{cl B0 : B0 ∈ B0 }. Then C ∩ B ⊂ C ∩ (∩{cl B0 : B0 ∈ B0 }) = ∅. (2) ⇒ (3): In case C = ∅, there is nothing to be proven. If C 6= ∅, then for B ∈ B by hypothesis B ∩ C 6= ∅ because C ∈ B. Thus, cl B ∩ C 6= ∅. (3) ⇒ (4): It suffices to appeal to 9.4.1. (4) ⇒ (1): Assume that C 6= ∅ (otherwise, nothing is left to proof). Suppose that C is not compact. Then there is a set E of open sets such that C ⊂ ∪{G : G ∈ E } and at the same time, for every finite subset E0 of E , the inclusion C ⊂ ∪{G : G ∈ E0 } fails. Put ( ) \ B := X \ G : E0 is a finite subset of E . G∈E0

It is clear that B is a filterbase. Furthermore, cl B = ∩{cl B : B ∈ B} = ∩{X \ G : G ∈ E } = X \ ∪{G : G ∈ E } ⊂ X \ C. Now choose an ultrafilter F that is coarser than B, which is guaranteed by 1.3.10. By supposition each member of B meets C. We may thus assume that C ∈ F (adjusting the choice of F , if need be). Then F → x for some x in C and so, by 9.4.1 (2), cl F ∩ C 6= ∅. At the same time cl F ⊂ cl B. We arrive at a contradiction. B

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9.4.4. Remark. The equivalence (1) ⇔ (4) in Theorem 9.4.3 is called the Bourbaki Criterion and verbalized for X = C as follows: “A space is compact if and only if every ultrafilter on it converges” (cf. 4.4.7). An ultranet is a net whose tail filter is an ultrafilter. The Bourbaki Criterion can be expressed as follows: “Compactness amounts to convergence of ultranets.” Many convenient tests for compactness are formulated in the language of nets. For instance: “A space X is compact if and only if each net in X has a convergent subnet.” 9.4.5. Weierstrass Theorem. The image of a compact set under a continuous mapping is compact (cf. 4.4.5). CB 9.4.6. Let X0 be a subspace of a topological space X and let C be a subset of X0 . Then C is compact in X0 if and only if C is compact in X. C ⇒: Immediate from 9.4.5 and 9.2.17 (1). ⇐: Let B be a filterbase on X0 . Further, let V := clX0 B stand for the adherence of B relative to X0 . Suppose that V ∩C = ∅. Since B is also a filterbase on X, it makes sense to speak of the adherence W := clX B of B relative to X. It is clear that V = W ∩ X0 and, consequently, W ∩ C = ∅. Since C is compact in X, by 9.4.3 there is some B in B such that B ∩ C = ∅. Using 9.4.3 once again, infer that C is compact in X0 . B 9.4.7. Remark. The claim of 9.4.6 is often expressed as follows: “Compactness is an absolute concept.” It means that for C to be or not to be compact depends on the topology induced in C rather than on the ambient space inducing the topology. For that reason, it is customary to confine study to compact spaces, i.e. to sets “compact in themselves.” A topology τ on a set C, making C into a compact space, is usually called a compact topology on C. Also, such C is referred to as “compact with respect to τ .” 9.4.8. Tychonoff Theorem. The Tychonoff product of compact spaces is compact. Q C Let X := ξ∈„ Xξ be the product of such spaces. If at least one of the spaces Xξ is nonempty then X = ∅ and nothing is left to proof. Let X 6= ∅ and let F be an ultrafilter on X. By 1.3.12, given ξ ∈ „ and considering the coordinate projection Prξ : X → Xξ , observe that Prξ (F ) is an ultrafilter on Xξ . Consequently, in virtue of 9.4.3 there is some xξ in Xξ such that Prξ (F ) → xξ . Let x : ξ 7→ xξ . It is clear that F → x (cf. 9.2.10). Appealing to 9.4.3 once more, infer that X is compact. B 9.4.9. Every closed subset of a compact space is compact. C Let X be compact and C ∈ Cl(X). Assume further that F is an ultrafilter on X and C ∈ F . By Theorem 9.4.3, F has a limit x in X: that is, F → x. By the Birkhoff Theorem, x ∈ cl C = C. Using 9.4.3 again, conclude that C is compact. B

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9.4.10. Every compact subset of a Hausdorff space is closed. C Let C be compact in a Hausdorff space X. If C = ∅ then there is nothing to prove. Let C 6= ∅ and x ∈ cl C. By virtue of 9.2.2 there is a filter F0 on X such that C ∈ F0 and F0 → x. Let F be an ultrafilter finer than F0 . Then F → x and C ∈ F . By 9.4.3, F has a limit in C. By 9.3.4 every limit in X is unique. Consequently, x ∈ C. B 9.4.11. Let f : (X, τ ) → (Y, ω) be a continuous one-to-one mapping with f (X) = Y . If τ is a compact topology and ω is a Hausdorff topology, then f is a homomorphism. C It suffices to establish that f −1 is continuous. To this end, we are to demonstrate that F ∈ Cl(τ ) ⇒ f (F ) ∈ Cl(ω). Take F ∈ Cl(τ ). Then F is compact by 9.4.9. Successively applying 9.4.5 and 9.4.10, infer that f (F ) is closed. B 9.4.12. Let τ1 and τ2 be two topologies on a set X. If (X, τ1 ) is a compact space and (X, τ2 ) is a Hausdorff space with τ1 ≥ τ2 , then τ1 = τ2 . CB 9.4.13. Remark. The message of 9.4.12 is customarily verbalized as follows: “A compact topology is minimal among Hausdorff topologies.” 9.4.14. Theorem. Every Hausdorff compact space is normal. C Let X be the space under study and let B be some filterbase on X. Assume further that U is a neighborhood of cl B. It is clear that X \ int U is compact (cf. 9.4.9) and cl B ∩ (X \ int U ) = ∅. By Theorem 9.4.3 there is a member B of B such that B∩(X\int U ) = ∅; i.e., B ⊂ U . Putting, if need be, B := {cl B : B ∈ B}, we may assert that cl B ⊂ U . To begin with, take x ∈ X and put B := τ (x). By virtue of 9.3.4, cl B = {x} and, consequently, the filter τ (x) has a base of closed sets. Thus, X is regular. Now take nonempty closed subset F of X. Take as B the neighborhood filter of F . By 9.3.8, cl B = F , and, as is already established, B has a base of closed sets. In accordance with 9.3.9, X is a normal space. B 9.4.15. Corollary. Each Hausdorff compact space is (to within a homeomorphism) a closed subset of a Tychonoff cube. C The compactness property of a closed subset of a Tychonoff cube follows from 9.4.8 and 9.4.9. Moreover, every cube is a Hausdorff space and so such is each of its subspaces. Now take some Hausdorff compact space X. Let Q be the collection of all continuous function from X to [0, 1]. Define the mapping ‰ : X → [0, 1]Q as ‰(x)(f ) := f (x) where x ∈ X and f ∈ Q. From 9.4.14 and 9.3.14 infer that ‰ carries X onto ‰(X) in a one-to-one fashion. Furthermore, ‰ is continuous. Application to 9.4.11 completes the proof. B

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9.4.16. Remark. Corollary 9.4.15 presents a part of a more general assertion. Namely, a Tychonoff space is (to within a homeomorphism) a subspace of a Tychonoff cube. CB 9.4.17. Remark. Sometimes a Hausdorff compact space is also called a compactum (cf. 4.5 and 4.6). 9.4.18. Dieudonn` e Lemma. Let F be a closed set and let G1 , . . . , Gn be open sets in a normal topological space, with F ⊂ G1 ∪ . . . ∪ Gn . There are closed sets F1 , . . . , Fn such that F = F1 ∪ . . . ∪ Fn and Fk ⊂ Gk (k := 1, . . . , n). C It suffices to settle the case n := 2. For k := 1, 2 the set Uk := F \ Gk is closed. Moreover, U1 ∩ U2 = ∅. By 9.3.10 there are open V1 and V2 such that U1 ⊂ V1 , U2 ⊂ V2 and V1 ∩V2 = ∅. Put Fk := F \Vk . It is clear that Fk is closed and Fk ⊂ F \Uk = F \(F \Gk ) ⊂ Gk for k := 1, 2. Finally, F1 ∪F2 = F \(V1 ∩V2 ) = F . B 9.4.19. Remark. From 9.3.14 we deduce that under the hypotheses of 9.4.18 there are continuous functions h1 , . . . , hn : X → [0, 1] such that hk |G0k = 0 and Pn k=1 hk (x) = 1 for every point x in some neighborhood about F . (As usual, 0 Gk := X \ Gk .) 9.4.20. Definition. A topology is called locally compact if each point possesses a compact neighborhood. A locally compact space is a set furnished with a Hausdorff locally compact topology. 9.4.21. A topological space is locally compact if and only if it is homeomorphic with a punctured compactum (= a compactum with a deleted point), i.e. the complement of a singleton to a compactum. C ⇐: In virtue of the Weierstrass Theorem it suffices to observe that each point of a punctured compactum possesses a closed neighborhood (since every compactum is regular). It remains to make use of 9.4.9 and 9.4.6. ⇒: Put the initial space X in X · := X ∪ {∞}, adjoining to X a point ∞ taken elsewhere. Take the complements to X · of compact subsets of X as a base for the neighborhood filter about ∞. A neighborhood of a point x of X in X · is declared to be a superset of a neighborhood of x in X. If A is an ultrafilter in X · and K is a compactum in X then A converges to a point in K provided that K ∈ A. If A contains the complement of each compactum K in X to X, then A converges to ∞. B 9.4.22. Remark. If a locally compact space X is not compact in its own right then X · of 9.4.20 is the one-point or Alexandroff compactification of X. 9.5. Uniform and Multimetric Spaces 9.5.1. Definition. Let X be a nonempty set and let UX be a filter on X 2 . The filter UX is a uniformity on X if

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(1) UX ⊂ fil {IX }; (2) U ∈ UX ⇒ U −1 ∈ UX ; (3) (∀ U ∈ UX )(∃ V ∈ UX ) V ◦ V ⊂ U . The uniformity of the empty set is by definition {∅}. The pair (X, UX ), as well as the underlying set X, is called a uniform space. 9.5.2. Given a uniform space (X, UX ), put x ∈ X ⇒ τ (x) := {U (x) : U ∈ UX }. The mapping τ : x 7→ τ (x) is a topology on X. C Clearly, τ is a pretopology. If W ∈ τ (x) then W = U (x) for some U in UX . Choose a member V of UX so that V ◦ V ⊂ U . If y ∈ V (x) then V (y) ⊂ V (V (x)) = V ◦ V (x) ⊂ U (x) ⊂ W . In other words, the set W is a neighborhood about y for every y in V (x). Therefore, V (x) lies in int W . Consequently, int W is a neighborhood about x. It remains to refer to 9.1.6. B 9.5.3. Definition. The topology τ appearing in 9.5.2 is the topology of the uniform space (X, UX ) under consideration or the uniform topology on X. It is also denoted by τ (UX ), τX , etc. 9.5.4. Definition. A topological space (X, τ ) is called uniformizable provided that there is a uniformity U on X such that τ coincides with the uniform topology τ (U ). 9.5.5. Examples. (1) A metric space (with its metric topology) is uniformizable (with its metric uniformity). (2) A multinormed space (with its topology) is uniformizable (with its uniformity). (3) Let f : X → (Y, UY ) and f −1 (UY ) := f ×−1 (UY ), where as usual f × (x1 , x2 ) := (f (x1 ), f (x2 )) for (x1 , x2 ) ∈ X 2 . Evidently, f −1 (UY ) is a uniformity on X. Moreover, τ (f −1 (UY )) = f −1 (τ (UY )). The uniformity f −1 (UY ) is the inverse image of UY under f . Therefore, the inverse image of a uniform topology is uniformizable. Q(4) Let (Xξ , Uξ )ξ∈„ be a family of uniform spaces. Assume further that X := ξ∈„ Xξ is the product of the family. Put UX := supξ∈„ Pr−1 ξ (Uξ ). The uniformity UX is the Tychonoff uniformity. It is beyond a doubt that the uniform topology τ (UX ) is the Tychonoff topology of the product of (Xξ , τ (Uξ ))ξ∈„ . CB

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(5) Each Hausdorff compact space is uniformizable in a unique fashion. C By virtue of 9.4.15 such space X may be treated as a subspace of a Tychonoff cube. From 9.5.5 (3) and 9.5.5 (4) it follows that X is uniformizable. Since each entourage of a uniform space includes a closed entourage; therefore, the compactness property of the diagonal IX of X 2 implies that every neighborhood of IX belongs to UX . On the other hand, each entourage is always a neighborhood of the diagonal. B (6) Assume that X and Y are nonempty sets, UY is a uniformity on Y and B is an upward-filtered subset of P(X). Given B ∈ B and θ ∈ UY , put UB,θ := {(f, g) ∈ Y X × Y X : g ◦ IB ◦ f −1 ⊂ θ}. Then U := fil {UB,θ : B ∈ B, θ ∈ UY } is a uniformity on Y X . This uniformity has a cumbersome (but exact) title, the “uniformity of uniform convergence on the members of B.” Such is, for instance, the uniformity of the Arens multinorm (cf. 8.3.8). If B is the collection of all finite subsets of X, then U coincides with the Tychonoff uniformity on Y X . This uniformity is called weak, and the corresponding uniform topology is called the topology of pointwise convergence or, rarely, that of simple convergence. If B is a singleton {X}, then the uniformity U is called strong and the corresponding topology τ (U ) in Y X is the topology of uniform convergence on X. 9.5.6. Remark. It is clear that, in a uniform (or uniformizable) space, the concepts make sense such as uniform continuity, total boundedness, completeness, etc. It is beyond a doubt that in such space the analogs of 4.2.4–4.2.9, 4.5.8, 4.5.9, and 4.6.1–4.6.7 are preserved. It is a rewarding practice to ponder over a possibility of completing a uniform space, to validate a uniform version of the Hausdorff Criterion, to inspect the proof of the Ascoli–Arzel´a Theorem in an abstract uniform setting, etc. 9.5.7. Definition. Let X be a set and put R·+ := {x ∈ R· : x ≥ 0}. A mapping d : X 2 → R·+ is called a semimetric or a pseudometric on X, provided that (1) d(x, x) = 0 (x ∈ X); (2) d(x, y) = d(y, x) (x, y ∈ X); (3) d(x, y) ≤ d(x, z) + d(z, y) (x, y, z ∈ X). A pair (X, d) is a semimetric space. 9.5.8. Given a semimetric space (X, d), let Ud := fil {{d ≤ ε} : ε > 0}. Then Ud is a uniformity. CB 9.5.9. Definition. Let M be a (nonempty) set of semimetrics on X. Then the pair (X, M) is a multimetric space with multimetric M. The multimetric uniformity on X is defined as UM := sup{Ud : d ∈ M}.

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9.5.10. Definition. A uniform space is called multimetrizable, if its uniformity coincides with some multimetric uniformity. A multimetrizable topological space is defined by analogy. 9.5.11. Assume that X, Y , and Z are sets, T is a dense subset of R, and (Ut )t∈T and (Vt )t∈T are increasing families of subsets of X × Z and Z × Y , respectively. Then there are unique functions f : X × Z → R, g : Z × Y → R and h : X × Y → R such that {f < t} ⊂ Ut ⊂ {f ≤ t},

{g < t} ⊂ Vt ⊂ {g ≤ t},

{h < t} ⊂ Vt ◦ Ut ⊂ {h ≤ t} (t ∈ T ). Moreover, the presentation holds: h(x, y) = inf{f (x, z) ∨ g(z, y) : z ∈ Z}. C The sought functions exist by 3.8.2. The claim of uniqueness is straightforward from 3.8.4. The presentation of h via f and g raises no doubts. B 9.5.12. Definition. Let f : X × Z → R and g : Z × Y → R. The function h, given by 9.5.11, is called the ∨-convolution (read: vel-convolution) of f and g and is denoted by f 9.5.13. Definition. A mapping f : X 2 → R·+ is a K-ultrametric with K ∈ R, K ≥ 1, if (1) f (x, x) = 0 (x ∈ X); (2) f (x, y) = f (y, x) (x, y ∈ X); (3) 1 /K f (x, u) ≤ f (x, y) ∨ f (y, z) ∨ f (z, u) (x, y, z, u ∈ X). 9.5.14. Remark. Condition 9.5.13 (3) is often referred to as the (strong) ultrametric inequality. In virtue of 9.5.12 this inequality may be rewritten as K −1 f ≤ f 9.5.15. 2-Ultrametric Lemma. To every 2-ultrametric f : X 2 → R·+ there corresponds a semimetric d such that 1 /2 f ≤ d ≤ f . C Let f1 := f and fn+1 := fn We are left with proving that 1 /2 f ≤ d. To this end, show that fn ≥ 1 /2 f for n ∈ N. Proceed by way of induction. The desired inequalities are obvious when n := 1, 2. Assume now that f ≥ f1 ≥ . . . ≥ fn ≥ 1 /2 f and at the same time fn+1 (x, y) < 1 /2 f (x, y) for some (x, y) in X 2 and n ≥ 2.

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Chapter 9 For suitable z1 , . . . , zn in X by construction t := f (x, z1 ) + f (z1 , z2 ) + . . . + f (zn−1 , zn ) + f (zn , y) < 1 /2 f (x, y).

If f (x, z1 ) ≥ t /2 then t /2 ≥ f (z1 , z2 ) + . . . + f (zn , y) ≥ 1 /2 f (z1 , y). It follows that t ≥ f (x, z1 ) and t ≥ f (z1 , y). On account of 9.5.13 (3), 1 /2 f (x, y) ≤ f (x, z1 ) ∨ f (z1 , y) ≤ t. Whence we come to 1 /2 f (x, y) > t ≥ 1 /2 f (x, y), which is false. Thus, f (x, z1 ) < t /2 . Find m ∈ N, m < n, satisfying f (x, z1 ) + . . . + f (zm−1 , zm ) < t /2 ; f (x, z1 ) + . . . + f (zm , zm+1 ) ≥ t /2 . This is possible, since assuming m = n would entail the false inequality f (zn , y) ≥ /2 . (We would have t /2 ≥ f (x, z1 ) + . . . + f (zn−1 , zn ) ≥ 1 /2 f (x, zn ) and so 1 /2 f (x, y) > t ≥ f (x, z n ) ∨ f (zn , y) ≥ 1 /2 f (x, y).) We obtain the inequality t

f (zm+1 , zm+2 ) + . . . + f (zn−1 , zn ) + f (zn , y) < t /2 . Using the induction hypothesis, conclude that f (x, zm ) ≤ 2(f (x, z1 ) + . . . + f (zm−1 , zm )) ≤ t; f (zm , zm+1 ) ≤ t; f (zm+1 , y) ≤ 2(f (zm+1 , zm+2 ) + . . . + f (zn , y)) ≤ t. Consequently, by the definition of 2-ultrametric 1

/2 f (x, y) ≤ f (x, zm ) ∨ f (zm , zm+1 ) ∨ f (zm+1 , y) ≤ t < 1 /2 f (x, y).

We arrive at a contradiction, completing the proof. B 9.5.16. Theorem. Every uniform space is multimetrizable. C Let (X, UX ) be a uniform space. Take V ∈ UX . Put V1 := V ∩ V −1 . If now −1 Vn ∈ UX then find a symmetric entourage V = V , a member of UX , satisfying V ◦ V ◦ V ⊂ Vn . Define Vn+1 := V . Since by construction Vn ⊃ Vn+1 ◦ Vn+1 ◦ Vn+1 ⊃ Vn+1 ◦ IX ◦ IX ⊃ Vn+1 ;

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therefore, (Vn )n∈N is a decreasing family. Given t ∈ R, define a set Ut by the rule  ∅,       IX , Ut := Vinf{n∈N : t≥2−n } ,    V1 ,    2 X ,

t<0 t=0 0 1.

By definition, the family t 7→ Ut (t ∈ R) increases. Consider a unique function f : X 2 → R satisfying the next conditions (cf. 3.8.2 and 3.8.4) {f < t} ⊂ Ut ⊂ {f ≤ t} (t ∈ R). If Wt := U2t for t ∈ R then Us ◦ Us ◦ Us ⊂ Wt for s < t. Consequently, in virtue of 3.8.3 and 9.2.1 the mapping f is a 2-ultrametric. Using 9.5.15, find a semimetric dV such that 1 /2 f ≤ dV ≤ f . Clearly, UdV = fil {Vn : n ∈ N}. It is also beyond a doubt that UM = UX for the multimetric M := {dV : V ∈ UX }. B 9.5.17. Corollary. A topological space is uniformizable if and only if it is a T31 /2 -space. CB 9.5.18. Corollary. A Tychonoff space is the same as a separated multimetric space. CB 9.6. Covers, and Partitions of Unity 9.6.1. Definition. Let E and F be covers of a subset of U in X; i.e., E , F ⊂ P(X) and U ⊂ (∪E ) ∩ (∪F ). It is said that F coarsens E or E refines F , if each member of E is included in some member of F ; i.e., (∀ E ∈ E ) (∃ F ∈ F ) E ⊂ F . It is also said that E is a refinement of F . Observe that if F is a subcover of E (i.e., F ⊂ E ) then E refines F . 9.6.2. Definition. A cover E of a set X is called locally finite (with respect to a topology τ on X), if each point in X possesses a neighborhood (in the sense of τ ) meeting only finite many members of E . In the case of the discrete topology on X, such cover is called point finite. If X is regarded as furnished with a prescribed topology τ then, speaking of a locally finite cover of X, we imply the topology τ .

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9.6.3. Lefschetz Lemma. Let E be a point finite open cover of a normal space X. Then there is an open cover {GE : E ∈ E } such that cl GE ⊂ E for all E ∈ E. C Let the set S comprise the mappings s : E → Op(X) such that ∪s(E ) = X and for E ∈ E either s(E) = E or cl s(E) ⊂ E. Given functions s1 and s2 , put s1 ≤ s2 := (∀ E ∈ E ) (s1 (E) 6= E ⇒ s2 (E) = s1 (E)). It is evident that (S, ≤) is an ordered set and IE ∈ S. Show that S is inductive. Given a chain S0 in S, for all E ∈ E put s0 (E) := ∩{s(E) : s ∈ S0 }. If s0 (E) = E then s(E) = E for all s ∈ S0 . If the case s0 (E) 6= E observe that s0 (E) = ∩{s(E) : s(E) 6= E, s ∈ S0 }. Since the order of S0 is linear, infer that s0 (E) = s(E) for s ∈ S0 with s(E) 6= E. Hence, s0 (E ) ⊂ Op(X) and s0 ≥ S0 . It remains to verify that s0 is a cover of X (and so s0 ∈ S). By the hypothesis of point finiteness, given x ∈ X, there are some E1 , . . . , En in E such that x ∈ E1 ∩ . . . ∩ En and x ∈ / E for the other members E of E . If s0 (Ek ) = Ek for some k, then there is nothing to prove, for x ∈ ∪s0 (E ). In the case when s0 (Ek ) 6= Ek for every k, there are s1 , . . . , sn ∈ S0 meeting the conditions sk (Ek ) 6= Ek (k := 1, 2 . . . , n). Since S0 is a chain, it may be assumed that sn ≥ {s1 , . . . , sn−1 }. Moreover, x ∈ sn (E) ⊂ E for an appropriate E in E . It is clear that E ∈ {E1 , . . . , En } (because x ∈ / E for the other members E of E ). Since s0 (E) = sn (E), it follows that x ∈ s0 (E). By the Kuratowski–Zorn Lemma there is a maximal element s in S. Take E ∈ E . If F := X \ ∪ s(E \ {E}), then F is closed and s(E) is a neighborhood of F . For a suitable G in Op(X) by 9.3.10 F ⊂ G ⊂ cl G ⊂ s(E). Put s(E) := G and s(E) := s(E) for E 6= E (E ∈ E ). It is clear that s ∈ S. If s(E) = E, then s ≥ s and so s = s. Moreover, s(E) ⊂ cl G ⊂ s(E) = E; i.e., cl s(E) ⊂ E. If s(E) 6= E, then cl s(E) ⊂ E by definition. Thus, s is a sought cover. B 9.6.4. Definition. Let f be a numeric or scalar-valued function on a topological space X, i.e. f : X → F. The set supp(f ) := cl{x ∈ X : f (x) 6= 0} is the support of f . If supp(f ) is a compact set then f is a compactly-supported function or a function of compact support. The designation spt (f ) := supp(f ) is used sometimes. 9.6.5. Let (fe )e∈E be a family of numeric functions on X and let E := {supp(fe ) : e ∈ E } be the family of their supports. If E is a point finite cover of X then the family (fe )e∈E is summable pointwise. If in addition E is locally finite and every P member of (fe )e∈E is continuous, then the sum e∈E fe is also continuous. C It suffices to observe that in a suitable neighborhood about a point in X only finitely many members of the family (fe )e∈E are distinct from zero. B 9.6.6. Definition. It is said that a family of functions (f : X → [0, 1])f ∈F is a partition of unity on a subset U of X, if the supports of the members of the family

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P composes a point finite cover of X, and f ∈F f (x) = 1 for all x ∈ U . The empty family of functions in this context is treated as summable to unity at each point. The term “continuous partition of unity” and the like are understood naturally. 9.6.7. Definition. Let E be a cover of a subset U of a topological space and let F be a continuous partition of unity on U . If the family of supports {supp(f ) : f ∈ F } refines E then F is a partition of unity subordinate to E . A possibility of finding such an F for E is also verbalized as follows: “E admits a partition of unity.” 9.6.8. Each locally finite open cover of a normal space admits a partition of unity. C By the Lefschetz Lemma, such cover {Uξ : ξ ∈ „} has an open refinement {Vξ : ξ ∈ „} satisfying the condition cl Vξ ⊂ Uξ for all ξ ∈ „. By the Urysohn Theorem, there is a continuous function gξ : X → [0, 1] such that gξ (x) = 1 for x ∈ Vξ and gξ (x) = 0 for x ∈ X \Uξ . Consequently, supp(gξ ) ⊂ Uξ . In virtue of 9.6.5 the family (gξ )ξ∈„ is summable pointwise to a continuous function g. Moreover, g(x) > 0 for all x ∈ X by construction. Put fξ := gξ /g (ξ ∈ „). The family (fξ )ξ∈„ is what we need. B 9.6.9. Definition. A topological space X is called paracompact, if each cover of X has a locally finite open refinement. 9.6.10. Remark. The theory of paracompactness contains deep and surprising facts. 9.6.11. Theorem. Every metric space is paracompact. 9.6.12. Theorem. A Hausdorff topological space is paracompact if and only if its every open cover admits a partition of unity. 9.6.13. Remark. The metric space RN possesses a number of additional structures providing a stock of well-behaved, smooth (= infinitely differentiable) functions (cf. 4.8.1). 9.6.14. Definition. A mollifier or a mollifying kernel on RN is a real-valued smooth function a having unit (Lebesgue) integral and such that a(x) > 0 for |x| < 1 and a(x) = 0 for |x| ≥ 1. In this event, supp(a) = {x ∈ RN : |x| ≤ 1} is the (unit Euclidean) ball B := BR N . 9.6.15. Definition. A delta-like sequence is a family of real-valued (smooth) functions (b R ε )ε>0 such that, first, limε→0 (sup | supp(bε )|) = 0 and, second, the equality holds R N bε (x) dx = 1 for all ε > 0. The terms “δ-sequence” and “δ-like sequence” are also in current usage. Such a sequence is often assumed countable without further specification.

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9.6.16. Example. The function a(x) := t exp(−(|x|2 −1)−1 ) is taken as a most popular mollifier when extended by zeroR beyond the open ball int B, with the constant t determined from the condition R N a(x) dx = 1. Each mollifier generates the delta-like sequence aε (x) := ε−N a(x/ε) (x ∈ RN ). 9.6.17. Definition. Let f ∈ L1,loc (RN ); i.e., let f be a locally integrable function, that is, a function whose restriction to each compact subset of RN is integrable. For a compactly-supported integrable function g the convolution f ∗ g is defined as Z f ∗ g(x) := f (x − y)g(y) dy (x ∈ RN ). RN

9.6.18. Remark. The role of a mollifying kernel and the corresponding deltalike sequence (aε )ε>0 becomes clear from inspecting the aftermath of applying the smoothing process f 7→ (f ∗aε )ε>0 to a function f belonging to L1,loc (RN ) (cf. 10.10.7 (5)). 9.6.19. The following statements are valid: (1) to every compact set K in the space RN and every neighborhood U of K there corresponds a truncator (= a bump function) ψ := ψK,U , i.e. a smooth mapping ψ : RN → [0, 1] such that K ⊂ int{ψ = 1} and supp(ψ) ⊂ U ; (2) assume that U1 , . . . , Un ∈ Op(RN ) and U1 ∪ . . . ∪ Un is a neighN borhood of a compact set K; there Pn are smooth functions ψ1 , . . . , ψn : R → [0, 1] such that supp(ψk ) ⊂ Uk and k=1 ψk (x) = 1 for x in some neighborhood of K. C (1) Put ε := d(K, RN \ U ) := inf{|x − y| : x ∈ K, y ∈ / U }. It is clear that ε > 0. Given β > 0, denote the characteristic function of K + εB by χβ . Take a delta-like sequence (bγ )γ>0 of positive functions and put ψ := χβ ∗ bγ . When γ ≤ β and β + γ ≤ ε with γ := sup | supp(bγ )|, observe that ψ is a sought function. (2) By the Diedonn`e Lemma there are closed sets Fk , with Fk ⊂ Uk , composing a cover of K. P Put Kk := Fk ∩ K and choose some P truncators ψk := ψKk ,Uk . The n n . . . , n), defined on { k=1 ψk > 0}, meet the claim functions ψk / k=1 ψk (k := 1, P n after extension by zero onto { k=1 ψk = 0} and multiplication by a truncator corresponding to an appropriate neighborhood of K. B 9.6.20. Countable Partition Theorem. Let E be a family of open sets in RN and Š := ∪E . There is a countable partition of unity which is composed of smooth compactly-supported functions on RN and subordinate to the cover E of Š. C Refine from E a countable locally finite cover A of Š with compact sets so that the family (α := int α)α∈A be also an open cover of Š. Choose an open cover (Vα )α∈A of Š from the condition cl Vα ⊂ α for α ∈ A. In Pvirtue of 9.6.19 (1) there are truncators ψ α := ψcl Vα ,α . Putting ψα (x) := ψ α (x)/ α∈A ψ α (x) for x ∈ Š and ψα (x) := 0 for x ∈ RN \ Š, arrive to a sought partition. B

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9.6.21. Remark. It is worth observing that the so-constructed partition of unity (ψα )α∈A possesses the property that to each compact subset K ofPŠ there correspond a finite subset A0 of A and a neighborhood U of K such that α∈A0 ψα (x) = 1 for all x ∈ U (cf. 9.3.17 and 9.6.19 (2)). Exercises 9.1. 9.2. 9.3. 9.4. What are

Give examples of pretopological and topological spaces and constructions leading to them. Is it possible to introduce a topology by indicating convergent filters or sequences? Establish relations between topologies and preorders on a finite set. Describe topological spaces in which the union of every family of closed sets is closed. the continuous mappings between such spaces?

9.5. Let (fξ : X0 = 0 O Yξ , τξ ))ξ∈„ be a family of mappings. A topology σ on X is called admissible (in the case under study), if for every topological space (Z, ω) and every mapping g : Z0 = 0 O the following statement holds: g : (Z, ω)0 = 0 O X, σ) is continuous if and only if so is each mapping fξ ◦ g (ξ ∈ „). Demonstrate that the weakest topology on X making every fξ (ξ ∈ „) continuous is the strongest admissible topology (in the case under study).

9.6. Let (fξ : (Xξ , σξ )0 = 0 O )ξ∈„ be a family of mappings. A topology τ on Y is called admissible (in the case under study), if for every topological space (Z, ω) and every mapping g : Y 0 = 0 O the following statement is true: g : (Y, τ )0 = 0 O Z, ω) is continuous if and only if each mapping g ◦ fξ (ξ ∈ „) is continuous. Demonstrate that the strongest topology on Y making every fξ (ξ ∈ „) continuous is the weakest admissible topology (in the case under study). 9.7. Prove that in the Tychonoff product of topological spaces, the closure of the product of subsets of the factors is the product of closures:

! cl

Y ξ∈„

Aξ

=

Y

cl Aξ .

ξ∈„

9.8. Show that a Tychonoff product is a Hausdorff space if and only if so is every factor. 9.9. Establish compactness criteria for subsets of classical Banach spaces. 9.10. A Hausdorff space X is called H-closed, if X is closed in every ambient Hausdorff space. Prove that a regular H-closed space is compact. 9.11. Study possibilities of compactifying a topological space. 9.12. Prove that the Tychonoff product of uncountably many real axes fails to be a normal space. 9.13. Show that each continuous function on the product of compact spaces depends on at most countably many coordinates in an evident sense (specify it!). 9.14. Let A be a compact subset and let B be a closed subset of a uniform space, with A ∩ B = ∅. Prove that V (A) ∩ V (B) = ∅ for some entourage V . 9.15. Prove that a completion (in an appropriate sense) of the product of uniform spaces is uniformly homeomorphic (specify!) to the product of completions of the factors. 9.16. A subset A of a separated uniform space is called precompact if a completion of A is compact. Prove that a set is precompact if and only if it is totally bounded. 9.17. Which topological spaces are metrizable? 9.18. Given a uniformizable space, describe the strongest uniformity among those inducing the initial topology. 9.19. Verify that the product of a paracompact space and a compact space is paracompact. Is paracompactness preserved under general products?

Chapter 10 Duality and Its Applications

10.1. Vector Topologies 10.1.1. Definition. Let (X, F, +, · ) be a vector space over a basic field F. A topology τ on X is a topology compatible with vector structure or, briefly, a vector topology, if the following mappings are continuous: + : (X × X, τ × τ ) → (X, τ ), · : (F × X, τF × τ ) → (X, τ ). The space (X, τ ) is then referred to as a topological vector space. 10.1.2. Let τX be a vector topology. The mappings x 7→ x + x0 ,

x 7→ αx (x0 ∈ X, α ∈ F \ 0)

are topological isomorphisms in (X, τX ). CB 10.1.3. Remark. It is beyond a doubt that a vector topology τ on a space X possesses the next “linearity” property: τ (αx + βy) = ατ (x) + βτ (y)

(α, β ∈ F \ 0; x, y ∈ X),

where in accordance with the general agreements (cf. 1.3.5 (1)) Uαx+βy ∈ ατ (x) + βτ (y) ⇔ (∃ Ux ∈ τ (x) & Uy ∈ τ (y)) αUx + βUy ⊂ Uαx+βy . In this regard a vector topology is often called a linear topology and a topological vector space, a linear topological space. This terminology should be used on the understanding that a topology may possess the “linearity” property while failing to be linear. For instance, such is the discrete topology of a nonzero vector space.

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10.1.4. Theorem. Let X be a vector space and let N be a filter on X. There is a vector topology τ on X such that N = τ (0) if and only if the following conditions are fulfilled: (1) N + N = N ; (2) N consists of absorbing sets; (3) N has a base of balanced sets. Moreover, τ (x) = x + N for all x ∈ X. C ⇒: Let τ be a vector topology and N = τ (0). From 10.1.2 infer that τ (x) = x + N for x ∈ X. It is also clear that (1) reformulates the continuity property of addition at zero (of the space X 2 ). Condition (2) may be rewritten as τF (0)x ⊃ N for every x in X, which is the continuity property of the mapping α 7→ αx at zero (of the space R) for every x in X. Condition (3) with account taken of (2) may in turn be rendered in the form τF (0)N = N , which is the continuity property of scalar multiplication at zero (of the space F × X). ⇐: Let N be a filter satisfying (1)–(3). It is evident that N ⊂ fil {0}. Put τ (x) := x + N . Then τ is a pretopology. From the definition of τ and (1) it follows that τ is a topology, with every translation continuous and addition continuous at zero in X 2 . Thus, addition is continuous at every point of X 2 . The validity of (2) and (3) means that the mapping (λ, x) 7→ λx is jointly continuous at zero and continuous at zero in the first argument given the second argument. By virtue of the identity λx − λ0 x0 = λ0 (x − x0 ) + (λ − λ0 )x0 + (λ − λ0 )(x − x0 ), we are left with examining the continuity property of scalar multiplication at zero in the second argument given the first argument. In other words, it is necessary to show that λN ⊃ N for λ ∈ F. With this in mind, find n ∈ N such that |λ| ≤ n. Let V in N and W in N be such that
W is balanced and W1 + . . . + Wn ⊂ V , λ where Wk := W . Then λW = n /n W ⊂ nW ⊂ W1 + . . . + Wn ⊂ V . B 10.1.5. Theorem. The set VT(X) of all vector topologies on X presents a complete lattice. Moreover, supVT(X) E = supT(X) E for every subset E of VT(X). C Let τ := supT(X) E . Since for τ ∈ E each translation by a vector is a topological isomorphism in (X, τ ); therefore, this mapping is a topological isomorphism in (X, τ ). Using 9.1.13, observe that the filter τ (0) meets conditions 10.1.4 (1)–10.1.4 (3), since these conditions are fulfilled for every filter τ (0) with τ ∈ E . It remains to refer to 1.2.14. B

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10.1.6. Theorem. The inverse image of a vector topology under a linear operator is a vector topology. C Take T ∈ L (X, Y ) and ω ∈ VT(Y ). Put τ := T −1 (ω). If xγ → x and yγ → y in (X, τ ) then by 9.2.8 T xγ → T x and T yγ → T y. So T (xγ +yγ ) → T (x+y). This means in virtue of 9.2.10 that xγ +yγ → x+y in (X, τ ). Thus, τ (x) = x+τ (0) for all x ∈ X and, moreover, τ (0) + τ (0) = τ (0). Successively applying 3.4.10 and 3.1.8 to the linear correspondence T −1 , observe that the filter τ (0) = T −1 (ω(0)) consists of absorbing sets and has a base of balanced sets. The reason is as follows: by 10.1.4 the filter ω(0) possesses these two properties. Once again using 10.1.4, conclude that τ ∈ VT(X). B 10.1.7. The product of vector topologies is a vector topology. C Immediate from 10.1.5 and 10.1.6. B 10.1.8. Definition. Let A and B be subsets of a vector space. It is said that A is B-stable if A + B ⊂ A. 10.1.9. To every vector topology τ on X there corresponds a unique uniformity Uτ having a base of IX -stable sets and such that τ = τ (Uτ ). C Given U ∈ τ (0), put VU := {(x, y) ∈ X 2 : y − x ∈ U }. Observe the obvious properties: IX ⊂ VU ;

VU + IX = VU ;

VU1 ∩U2 ⊂ VU1 ∩ VU2 ;

(VU )−1 = V−U ;

VU1 ◦ VU2 ⊂ VU1 +U2

for all U , U1 , U2 ∈ τ (0). Using 10.1.4, infer that Uτ := fil {VU : U ∈ τ (0)} is a uniformity and τ = τ (Uτ ). It is also beyond a doubt that Uτ has a base of IX -stable sets. If now U is another uniformity such that τ (U ) = τ , and W is some IX -stable entourage in U ; then W = VW (0) . Whence the sought uniqueness follows. B 10.1.10. Definition. Let (X, τ ) be a topological vector space. The uniformity Uτ , constructed in 10.1.9, is the uniformity of X. 10.1.11. Remark. Considering a topological vector space, we assume it to be furnished with the corresponding uniformity without further specification. 10.2. Locally Convex Topologies 10.2.1. Definition. A vector topology is locally convex if the neighborhood filter of each point has a base of convex sets.

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10.2.2. Theorem. Let X be a vector space and let N be a filter on X. There is a locally convex topology τ on X such that N = τ (0) if and only if (1) 1 /2 N = N ; (2) N has a base of absorbing absolutely convex sets. C ⇒: By virtue of 10.1.2 the mapping x 7→ 2x is a topological isomorphism. This means that 1 /2 N = N . Now take U ∈ N . By hypothesis there is a convex set V in N such that V ⊂ U . Applying 10.1.4, find a balanced set W satisfying W ⊂ V . Using the Motzkin formula and 3.1.14, show that the convex hull co(W ) is absolutely convex. Moreover, W ⊂ co(W ) ⊂ V ⊂ U . ⇐: An absolutely convex set is balanced. Consequently, N satisfies 10.1.4 (2) and 10.1.4 (3). If V ∈ N and W is a convex set, W ∈ N and W ⊂ V ; then 1 /2 W ∈ N . Furthermore, 1 /2 W + 1 /2 W ⊂ W ⊂ V because of the convexity property of W . This means that N + N = N . It remains to refer to 10.1.4. B 10.2.3. Corollary. The set LCT (X) of all locally convex topologies on X is a complete lattice. Moreover, supLCT (X) E = supT(X) E for every subset E of LCT (X). /. 10.2.4. Corollary. The inverse image of a locally convex topology under a linear operator is a locally convex topology. CB 10.2.5. Corollary. The product of locally convex topologies is a locally convex topology. CB 10.2.6. The topology of a multinormed space is locally convex. CB 10.2.7. Definition. Let τ be a locally convex topology on X. The set of all everywhere-defined continuous seminorms on X is called the mirror (rarely, the spectrum) of τ and is denoted by Mτ . The multinormed space (X, Mτ ) is called associated with (X, τ ). 10.2.8. Theorem. Each locally convex topology coincides with the topology of the associated multinormed space. C Let τ be a locally convex topology on X and let ω := τ (Mτ ) be the topology of the associated space (X, Mτ ). Take V ∈ τ (0). By 10.2.2 there is an absolutely convex neighborhood B of zero, B ∈ τ (0), such that B ⊂ V . In virtue of 3.8.7 {pB < 1} ⊂ B ⊂ {pB ≤ 1}. It is obvious that pB is continuous (cf. 7.5.1); i.e., pB ∈ Mτ , and so {pB < 1} ∈ ω(0). Consequently, V ∈ ω(0). Using 5.2.10, infer that ω(x) = x+ω(0) ⊃ x+τ (0) = τ (x); i.e., ω ≥ τ . Furthermore, τ ≥ ω by definition. B

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10.2.9. Definition. A vector space, endowed with a separated locally convex topology, is a locally convex space. 10.2.10. Remark. Theorem 10.2.8 in slightly trimmed form is often verbalized as follows: “The concept of locally convex space and the concept of separated multinormed space have the same scope.” For that reason, the terminology connected with the associated multinormed space is lavishly applied to studying a locally convex space (cf. 5.2.13). 10.2.11. Definition. Let τ be a locally convex topology on X. The symbol (X, τ )0 (or, in short, X 0 ) denotes the subspace of X # that comprises all continuous linear functionals. The space (X, τ )0 is the dual (or τ -dual) of (X, τ ). 10.2.12. (X, τ )0 = ∪ {|∂|(p) : p ∈ Mτ }. CB 10.2.13. Prime Theorem. The prime mapping τ 7→ (X, τ )0 from LCT (X) to Lat(X # ) preserves suprema; i.e., (X, sup E )0 = sup{(X, τ )0 : τ ∈ E } for every subset E of LCT (X). C If E = ∅ then sup E is the trivial topology τ◦ of X and, consequently, (X, τ◦ )0 = 0 = inf Lat(X # ) = supLat(X # ) ∅. By virtue of 9.2.7 the prime mapping increases. Given a nonempty E , from 2.1.5 infer that (X, sup E )0 ≥ sup{(X, τ )0 : τ ∈ E }. If f ∈ (X, sup E )0 , then in view of 10.2.12 and 9.1.13 there are topologies τ1 , . . . , τn ∈ E such that f ∈ (X, τ1 ∨ . . . ∨ τn )0 . Using 10.2.12 and 5.3.7, find p1 ∈ Mτ1 , . . . , pn ∈ Mτn satisfying f ∈ |∂|(p1 ∨ . . . ∨ pn ). Recalling 3.5.7 and 3.7.9, observe that |∂|(p1 + . . . + pn ) = |∂|(p1 ) + . . . + |∂|(pn ). Finally, f ∈ (X, τ1 )0 + . . . + (X, τn )0 = (X, τ1 )0 ∨ . . . ∨ (X, τn )0 . . 10.3. Duality Between Vector Spaces 10.3.1. Definition. Let X and Y be vector spaces over the same ground field F. Also, Assume given a bilinear form (or, as it is called sometimes, a bracketing) h· | ·i acting from X × Y to F, i.e. a mapping linear in each of its arguments. Given x ∈ X and y ∈ Y , put hx | : y 7→ hx | yi, | yi : x 7→ hx | yi,

h· | : X → F Y , | ·i : Y → F X,

hX | ⊂ Y # ; | Y i ⊂ X #.

The mappings h· | and | ·i are the bra-mapping and the ket-mapping of the initial bilinear form. By analogy, a member of hX | is a bra-functional on X and a member of | Y i is a ket-functional on Y .

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10.3.2. The bra-mapping and the ket-mapping are linear operators. CB 10.3.3. Definition. A bracketing of some vector spaces X and Y is a pairing, if its bra-mapping and ket-mapping are monomorphisms. In this case we say that X and Y are (set) in duality, or present a duality pair, or that Y is the pair-dual of X, etc. This is written down as X ↔ Y . Each of the bra-mapping and the ketmapping is then referred to as dualization. For suggestiveness, the corresponding pairing of some spaces in duality is also called their duality bracket. 10.3.4. Examples. (1) Let X ↔ Y with duality bracket h· | ·i. Given (y, x) ∈ Y × X, put hy | xi := hx | yi. It is immediate that the new bracketing is a pairing of Y and X. Moreover, the pairs of dualizations with respect to the old (direct) and new (reverse) duality brackets are the same. It thus stands to reason to draw no distinction between the two duality brackets unless in case of an emergency (cf. 10.3.3). For instance, Y is the pair-dual of X in the direct duality bracket if and only if X is the pair-dual of Y in the reverse duality bracket. Therefore, the hair-splitting is neglected and a unified term “pair-dual” is applied to each of the spaces in duality, with duality treated as a whole abstract phenomenon. Observe immediately that the mapping hx | yiR := Rehx | yi sets in duality the real carriers XR and YR . By way of taking liberties, the previous notation is sometimes reserved for the arising duality XR ↔ YR ; i.e., it is assumed that hx | yi := hx | yiR , on considering x and y as members of the real carriers. (2) Let H be a Hilbert space. The inner product on H sets H and H∗ in duality. The prime mapping is then coincident with the ket-mapping. (3) Let (X, τ ) be a locally convex space and let X 0 be the dual of X. The natural evaluation mapping (x, x0 ) 7→ x0 (x) sets X and X 0 in duality. (4) Let X be a vector space and let X # := L (X, F) be the (algebraic) dual of X. It is clear that the evaluation mapping (x, x# ) 7→ x# (x) sets the spaces in duality. 10.3.5. Definition. Let X ↔ Y . The inverse image in X of the Tychonoff topology on F Y under the bra-mapping, further denoted by σ(X, Y ), is the bratopology or the weak topology on X induced by Y . The bra-topology σ(Y, X) of Y ↔ X is the ket-topology of X ↔ Y or the weak topology on Y induced by X. 10.3.6. The bra-topology is the weakest topology making every ket-functional continuous. The ket-topology is the weakest topology making every bra-functional continuous. C xγ → x (in σ(X, Y )) ⇔ hxγ | → hx | (in F Y ) ⇔ (∀ y ∈ Y ) hxγ | (y) → hx | (y) ⇔ (∀ y ∈ Y ) hxγ | yi → hx | yi ⇔ (∀ y ∈ Y ) | yi(xγ ) → | yi(x) ⇔ (∀ y ∈ Y ) xγ → x (in | yi−1 (τF )) B

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10.3.7. Remark. The notation σ(X, Y ) agrees perfectly with that of the weak multinorm in 5.1.10 (4). Namely, σ(X, Y ) is the topology of the multinorm {|h· | yi| : y ∈ Y }. Likewise, σ(Y, X) is the topology of the multinorm {|hx | ·i| : x ∈ X}. CB 10.3.8. The spaces (X, σ(X, Y )) and (Y, σ(Y, X)) are locally convex. C Immediate from 10.2.4 and 10.2.5. B 10.3.9. Dualization Theorem. Each dualization is an isomorphism between the pair-dual and the weak dual of the pertinent member of a duality pair. C Consider a duality pair X ↔ Y . We are to prove exactness for the sequences h·|

0 → X −→(Y, σ(Y, X))0 → 0;

|·i

0 → Y −→(X, σ(X, Y ))0 → 0.

Since the ket-mapping of X ↔ Y is the bra-mapping of Y ↔ X, it suffices to show that the first sequence is exact. The bra-mapping is a monomorphism by definition. Furthermore, from 10.2.13 and 10.3.6 it follows that (Y, σ(Y, X))0 = (Y, sup{hx |−1 (τF ) : x ∈ X})0 = sup{(Y, hx |−1 (τF ))0 : x ∈ X} = lin({(Y, f −1 (τF ))0 : f ∈ hX |}) = hX |, since in view of 5.3.7 and 2.3.12 (Y, f −1 (τF ))0 = {λf : λ ∈ F} (f ∈ Y # ). . 10.3.10. Remark. Theorem 10.3.9 is often referred to as the theorem on the general form of a weakly continuous functional. Here a useful convention reveals itself: apply the base form “weak” when using objects and properties that are related to weak topologies. Observe immediately that, in virtue of 10.3.9, Example 10.3.4 (3) actually lists all possible duality brackets. That is why in what follows we act in accordance with 5.1.11, continuing the habitual use of the designation (x, y) := hx | yi, since it leads to no misunderstanding. For the same reason, given a vector space X, we draw no distinction between the pair-dual of a space X and the weak dual of X. In other words, considering a duality pair X ↔ Y , we sometimes identify X with (Y, σ(Y, X))0 and Y with (X, σ(X, Y ))0 , which justifies writing X 0 = Y and Y 0 = X. 10.3.11. Remark. A somewhat obsolete convention relates to X ↔ X 0 with X a normed space. The ket-topology σ(X 0 , X) is customarily called the weak∗ topology (read: weak-star topology) in X 0 , which reflects the concurrent notation X ∗ for X 0 . The term “weak∗ ” proliferates in a routine fashion elsewhere.

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10.4. Topologies Compatible with Duality 10.4.1. Definition. Take a duality pair X ↔ Y and let τ be a locally convex topology on X. It is said that τ is compatible with duality (between X and Y by pairing X ↔ Y ), provided that (X, τ )0 = | Y i. A locally convex topology ω on Y is compatible with duality (by pairing X ↔ Y ), if ω is compatible with duality (by pairing Y ↔ X); i.e., if the equality holds: (Y, ω)0 = hX |. A unified concise term “compatible topology” is also current in each of the above cases. 10.4.2. Weak topologies are compatible. C Follows from 10.3.9. B 10.4.3. Let τ (X, Y ) stand for the least upper bound of the set of all locally convex topologies on X compatible with duality (between X and Y ). Then the topology τ (X, Y ) is also compatible. C Denote the set of all compatible topologies on X by E . Theorem 10.2.13 readily yields the equalities (X, τ (X, Y ))0 = sup{(X, τ )0 : τ ∈ E } = sup{| Y i : τ ∈ E } = | Y i, because E is nonempty by 10.4.2. B 10.4.4. Definition. The topology τ (X, Y ), constructed in 10.4.3 (i.e., the finest locally convex topology on X compatible with duality by pairing X ↔ Y ), is the Mackey topology (on X induced by X ↔ Y ). 10.4.5. Mackey–Arens Theorem. A locally convex topology τ on X is compatible with duality between X and Y if and only if σ(X, Y ) ≤ τ ≤ τ (X, Y ). C By 10.2.13 the prime mapping τ 7→ (X, τ )0 preserves suprema and, in particular, increases. Therefore, given τ in the interval of topologies, from 10.4.2 and 10.4.3 obtain | Y i = (X, σ(X, Y )) ⊂ (X, τ )0 ⊂ (X, τ (X, Y ))0 = | Y i. The remaining claim is obvious. B 10.4.6. Mackey Theorem. bounded sets in stock.

All

compatible

topologies have the same

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C A stronger topology has fewer bounded sets. So, to prove the theorem it suffices to show that if some set U is weakly bounded in X (= is bounded in the bra-topology) then U is bounded in the Mackey topology. Take a seminorm p from the mirror of the Mackey topology and demonstrate that p(U ) is bounded in R. Put X0 := X/ ker p and p0 := pX/ ker p . By 5.2.14, p0 is clearly a norm. Let ϕ : X → X0 be the coset mapping. It is beyond a doubt that ϕ(U ) is weakly bounded in (X0 , p0 ). From 7.2.7 it follows that ϕ(U ) is bounded in the norm p0 . Since p0 ◦ ϕ = p; therefore, U is bounded in (X, p). B 10.4.7. Corollary. Let X be a normed space. Then the Mackey topology τ (X, X 0 ) coincides with the initial norm topology on X. C It suffices to refer to the Kolmogorov Normability Criterion implying that the space X with the topology τ (X, X 0 ) finer than the original topology is normable. Appealing to 5.3.4 completes the proof. B 10.4.8. Strict Separation Theorem. Let (X, τ ) be a locally convex space. Assume further that K and V are nonempty convex subsets of X with K compact, V closed and K ∩ V = ∅. Then there is a functional f , a member of (X, τ )0 , such that sup Re f (K) < inf Re f (V ). C A locally convex space is obviously a regular space. Since K is compact, it thus follows that, for an appropriate convex neighborhood of zero, say W , the set U := K + W does not meet V (it suffices to consider the filterbases comprising all subsets of the form K + W and V + W , with W a closed neighborhood of zero). By 3.1.10, U is convex. Furthermore, K ⊂ int U = core U . By the Eidelheit Separation Theorem, there is a functional l, a member of (XR )# , such that the hyperplane {l = 1} in XR separates V from U and does not meet the core of U . Obviously, l is bounded above on W and so l ∈ (XR , τ )0 by 7.5.1. If f := Re−1 l then, of view of 3.7.5, f ∈ (X, τ )0 . It is clear that f is a sought functional. B 10.4.9. Mazur Theorem. All compatible topologies have the same closed convex sets in stock. C A stronger topology has more closed sets. So, to prove the theorem it suffices in view of 10.4.5 to show that if U is a convex set closed in the Mackey topology then U is weakly closed. The last claim is beyond a doubt since by Theorem 10.4.8 U is the intersection of weakly closed sets of the form {Re f ≤ t}, with f a (weakly) continuous linear functional and t ∈ R. B 10.5. Polars 10.5.1. Definition. Let X and Y be sets and let F ⊂ X × Y be a correspon-

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dence. Given a subset U of X and a subset V of Y, put π(U ) := πF (U ) := {y ∈ Y : F −1 (y) ⊃ U }; π −1 (V ) := πF−1 (V ) := {x ∈ X : F (u) ⊃ V }. The set π(U ) is the (direct) polar of U (under F ), and the set π −1 (V ) is the (reverse) polar of V (under F ). 10.5.2. The following statements are valid: (1) π(u) := π({u}) = F (u) and π(U ) = ∩u∈U π(u); (2) π(∪ξ∈„ Uξ ) = ∩ξ∈„ π(Uξ ); (3) πF−1 (V ) = πF −1 (V ); (4) U1 ⊂ U2 ⇒ π(U1 ) ⊃ π(U2 ); (5) U × V ⊂ F ⇒ (V ⊂ π(U ) & U ⊂ π −1 (V ); (6) U ⊂ π −1 (π(U )). CB 10.5.3. Akilov Criterion. A subset U of X is the polar of some subset of Y if and only if given x ∈ X \ U there is an element y in Y such that U ⊂ π −1 (y),

x∈ / π −1 (y).

C ⇒: If U = π −1 (V ) then U = ∩v∈V π −1 (v) by 10.5.2 (1). ⇐: The inclusion U ⊂ π −1 (y) means that y ∈ π(U ). Thus, by hypothesis U = ∩y∈π(U ) π −1 (y) = π −1 (π(U )). B 10.5.4. Corollary. The set π −1 (π(U )) is the (inclusion) least polar greater than U . CB 10.5.5. Definition. The set πF−1 (πF (U )) is the bipolar of a subset U (under the correspondence F ). 10.5.6. Examples. (1) Let (X, σ) be an ordered set and let U be a subset of X. Then πσ (U ) is the collection of all upper bounds of U (cf. 1.2.7). (2) Let (H, ( · , · )H ) be a Hilbert space and F := {(x, y) ∈ H 2 : (x, y)H = 0}. Then π(U ) = π −1 (U ) = U ⊥ for every subset U of H. The bipolar of U in this case coincides with the closed linear span of U , that is, the closure of the linear span of U . (3) Let X be a normed space and let X 0 be the dual of X. Consider F := {(x, x0 ) : x0 (x) = 0}. Then π(X0 ) = X0⊥ and π −1 (X0 ) = ⊥X0 for a subspace X0 of X and a subspace X0 of X 0 (cf. 7.6.8). Moreover, π −1 (π(X0 )) = cl X0 by 7.5.14.

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Chapter 10 10.5.7. Definition. Let X ↔ Y . Put pol := {(x, y) ∈ X × Y : Rehx | yi ≤ 1}; abs pol := {(x, y) ∈ X × Y : |hx | yi| ≤ 1}.

To refer to direct or inverse polars under pol , we use the unified term “polar” (with respect to X ↔ Y ) and the unified designations π(U ) and π(V ). In the case of the correspondence abs pol , we speak of absolute polars (with respect to X ↔ Y ) and write U ◦ and V ◦ (for U ⊂ X and V ⊂ Y ). 10.5.8. Bipolar Theorem. The bipolar π 2 (U ) := π(π(U )) is the (inclusion) least weakly closed conical segment greater than U . C Straightforward from 10.4.8 and the Akilov Criterion. B 10.5.9. Absolute Bipolar Theorem. The absolute bipolar U ◦◦ := (U ◦ )◦ is the (inclusion) least weakly closed absolutely convex set greater than U . C It suffices, first, to observe that the polar of a balanced set U coincides with the absolute polar of U and, second, to apply 10.5.8. B 10.6. Weakly Compact Convex Sets 10.6.1. Let X be a locally convex real vector space and let p : X → R be a continuous sublinear functional on X. Then the (topological) subdifferential ∂(p) is compact in the topology σ(X 0 , X). Q C Put Q := x∈X [−p(−x), p(x)] and endow Q with the Tychonoff topology. Evidently ∂(p) ⊂ Q, with the Tychonoff topology on Q and σ(X 0 , X) inducing the same topology on ∂(p). It is beyond a doubt that the set ∂(p) is closed in Q by the continuity of p. Taking note of the Tychonoff Theorem and 9.4.9, conclude that ∂(p) is a σ(X 0 , X)-compact set. B 10.6.2. The balanced subdifferential of each continuous seminorm is weakly compact. CB 10.6.3. Theorem. Let X be a real vector space. A subset U of X # is the subdifferential of a (unique total) sublinear functional sU : X → R if and only if U is nonempty, convex and σ(X # , X)-compact. C ⇒: Let U = ∂(sU ) for some sU . The uniqueness of sU is ensured by 3.6.6. In view of 10.2.12 it is easy that the mirror of the Mackey topology τ (X, X # ) is the strongest multinorm on X (cf. 5.1.10 (2)). Whence we infer that the functional sU is continuous with respect to τ (X, X # ). In virtue of 10.6.1 the set U is compact in σ(X # , X). The convexity and nonemptiness of U are obvious. ⇐: Put sU (x) := sup{l(x) : l ∈ U }. Undoubtedly, sU is a sublinear functional and dom sU = X. By definition, U ⊂ ∂(sU ). If l ∈ ∂(sU ) and l ∈ / U , then by the Strict Separation Theorem and the Dualization Theorem sU (x) < l(x) for some x in X. This is a contradiction. B

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10.6.4. Definition. The sublinear functional sU , constructed in 10.6.3, is the supporting function of U . The term “support function” is also in current usage. 10.6.5. Kre˘ın–Milman Theorem. Each compact convex set in a locally convex space is the closed convex hull (= the closure of the convex hull) of the set of its extreme points. C Let U be such subset of a space X. It may be assumed that the space X is real and U 6= ∅. By virtue of 9.4.12, U is compact with respect to the # topology σ(X, X 0 ). Since σ(X, X 0 ) is induced in X by the topology σ(X 0 , X 0 ) # on X 0 ; therefore, U = ∂(sU ). Here (cf. 10.6.3) sU : X 0 → R acts by the rule sU (x0 ) := sup x0 (U ). By the Kre˘ın–Milman Theorem in subdifferential form, the set ext U of the extreme points of U is not empty. The closure of the convex hull of ext U is a subdifferential by Theorem 10.6.3. Moreover, this set has sU as its supporting function, thus coinciding with U (cf. 3.6.6). B 10.6.6. Let X ↔ Y and let S be a conical segment in X. Assume further that pS is the Minkowski functional of S. The polar π(S) is the inverse image of the (algebraic) subdifferential ∂(pS ) under the ket-mapping; i.e., π(S) = | ∂(pS ) i−1 R . If S is absolutely convex, then the absolute polar S ◦ is the inverse image of the (algebraic) balanced subdifferential |∂|(pS ) under the ket-mapping; i.e., S ◦ = | |∂|(pS ) i−1 . C If y ∈ YR and y ∈ | ∂(pS ) i−1 R then | yiR belongs to ∂(pS ). Hence, Rehx | yi = hx | yiR = | yiR (x) ≤ pS (x) ≤ 1 for x ∈ S, because S ⊂ {pS ≤ 1} by the Gauge Theorem. Consequently, y ∈ π(S). If, in turn, y ∈ π(S) then | yiR belongs to ∂(pS ). Indeed, 1 > pS (α−1 x) for all x in XR and α > pS (x); i.e., α−1 x ∈ {pS < 1} ⊂ S. Whence hα−1 x | yiR = Rehα−1 x | yi = α−1 Rehx | yi ≤ 1. Finally, observe that | yiR (x) ≤ α. Since α is arbitrary, this inequality means that | yiR (x) ≤ pS (x). In other words, y ∈ −1 | ∂(ps )i−1 R , which implies that π(S) = | ∂(pS )iR . The remaining claim follows from the properties of the complexifier (cf. 3.7.3 and 3.7.9). B 10.6.7. Alaoglu–Bourbaki Theorem. The polar of a neighborhood of zero of each compatible topology is a weakly compact convex set. C Let U be a neighborhood of zero in a space X and let π(U ) be the polar of U (with respect to X ↔ X 0 ). Since U ⊃ {p ≤ 1} for some continuous seminorm p, by 10.5.2 (4), π(U ) ⊂ π({p ≤ 1}) = π(Bp ) = Bp◦ . Using 10.6.6 and recalling that p is the Minkowski functional of Bp , obtain the inclusion π(U ) ⊂ |∂|(p). By virtue of 10.6.2 the topological balanced subdifferential |∂|(p) is σ(X 0 , X)-compact. By definition π(U ) is weakly closed. To infer the σ(X 0 , X)-compactness of π(U ), it remains to appeal to 9.4.9. The convexity property of π(U ) is beyond a doubt. B

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Chapter 10 10.7. Reflexive Spaces

10.7.1. Kakutani Criterion. A normed space is reflexive if and only if its unit ball is weakly compact. C ⇒: Let X be reflexive, i.e. 00 (X) = X 00 . In other words, the image of X under the double prime mapping coincides with X 00 . Since the ball BX 00 is the polar of the ball BX 0 with respect to X 00 ↔ X 0 ; therefore, BX 00 is a σ(X 00 , X 0 )compact set by the Alaoglu–Bourbaki Theorem. It remains to observe that BX 00 is (the image under the double prime mapping of) BX , and σ(X, X 0 ) is (the inverse image under the double prime mapping of) σ(X 00 , X 0 ). ⇐: Consider the duality pair X 00 ↔ X 0 . By definition, the ball BX 00 presents the bipolar of BX (more precisely, the bipolar of (BX )00 ). Using the Absolute Bipolar Theorem and observing that the weak topology σ(X, X 0 ) is induced in X by the topology σ(X 00 , X 0 ), conclude that BX 00 = BX (because of the obvious convexity and closure properties of BX , the latter following from compactness since X is separated). Thus, X is reflexive. B 10.7.2. Corollary. A space X is reflexive if and only if every bounded closed convex set in X is weakly compact. CB 10.7.3. Corollary. Every closed subspace of a reflexive space is reflexive. C By the Mazur Theorem, such subspace and, hence, the unit ball of it are weakly closed. It thus suffices to apply the Kakutani Criterion twice. B 10.7.4. Pettis Theorem. A Banach space and its dual are (or are not) reflexive simultaneously. C If X is reflexive then σ(X 0 , X) coincides with σ(X 0 , X 00 ). Therefore, by the Alaoglu–Bourbaki Theorem, BX 0 is σ(X 0 , X 00 )-compact. Consequently, X 0 is reflexive. If, in turn, X 0 is reflexive, then so is X 00 by what was proven. However, X, as a Banach space, is a closed subspace of X 00 . Thus, X is reflexive by 10.7.3. B 10.7.5. James Theorem. A Banach space is reflexive if and only if each continuous (real) linear functional attains its supremum on the unit ball of the space. 10.8.

The Space C(Q, R)

10.8.1. Remark. Throughout Section 10.8 let Q stand for a nonempty Hausdorff compact space, denoting by C(Q, R) the set of continuous real-valued functions on Q. Unless specified otherwise, C(Q, R) is furnished with the natural pointwise algebraic operations and order and equipped with the sup-norm k · k := k · k∞ related to the Chebysh¨ev metric (cf. 4.6.8). Keeping this in mind, we treat the statements like “C(Q, R) is a vector lattice,” “C(Q, R) is a Banach algebra,” etc. Other structures, if ever introduced in C(Q, R), are specified deliberately.

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10.8.2. Definition. A subset L of C(Q, R) is a sublattice (in C(Q, R)) if f1 ∨ f2 ∈ L and f1 ∧ f2 ∈ L for f1 , f2 ∈ L, where, as usual, f1 ∨ f2 (q) := f1 (q) ∨ f2 (q), f1 ∧ f2 (q) := f1 (q) ∧ f2 (q) (q ∈ Q). 10.8.3. Remark. Observe that to be a sublattice in C(Q, R) means more than to be a lattice with respect to the order induced from C(Q, R). 10.8.4. Examples. (1) ∅, C(Q, R), and the closure of a sublattice. (2) The intersection of each family of sublattices is also a sublattice. (3) Let L be a sublattice and let Q0 be a subset of Q. Put LQ0 := {f ∈ C(Q, R) : (∃ g ∈ L) g(q) = f (q) (q ∈ Q0 )}. Then LQ0 is a sublattice. Moreover, L ⊂ LQ0 . (4) Let Q0 be a compact subset of Q. Given a sublattice L in C(Q, R), put  L Q := f Q : f ∈ L . 0

0

Therefore,  LQ0 = f ∈ C(Q, R) : f Q0 ∈ L Q0 . It is clear that L Q0 is a sublattice of C(Q0 , R). Furthermore, if L is a vector sublattice of C(Q, R) (i.e., a vector subspace and simultaneously a sublattice of C(Q, R)); then L Q is a vector sublattice of C(Q0 , R) (certainly, if Q0 6= ∅). 0

(5) Let Q := {1, 2}. Then C(Q, R) ' R2 . Each nonzero vector sublattice of R is given as {(x1 , x2 ) ∈ R2 : α1 x1 = α2 x2 } 2

for some α1 , α2 ∈ R+ . (6) Let L be a vector sublattice of C(Q, R). For q ∈ Q, the alternative is offered: either L{q} = C(Q, R) or L{q} = {f ∈ C(Q, R) : f (q) = 0}. If q1 and q2 are distinct points of Q and L {q1 ,q2 } 6= 0, then by 10.8.4 (5) there are some α1 , α2 ∈ R+ such that L{q1 ,q2 } = {f ∈ C(Q, R) : α1 f (q1 ) = α2 f (q2 )}. Moreover, if L contains a constant function other than zero (i.e. a nonzero multiple of the constantly-one function 1) then as α1 and α2 in the above presentation of L{q1 ,q2 } the unity, 1, may be taken. CB

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10.8.5. Let L be a sublattice of C(Q, R). A function f in C(Q, R) belongs to the closure of L if and only if for all ε > 0 and (x, y) ∈ Q2 there is a function f := fx,y,ε in L satisfying the conditions f (x) − f (x) < ε,

f (y) − f (y) > −ε.

C ⇒: This is obvious. ⇐: Basing it on 3.2.10 and 3.2.11, assume that f = 0. Take ε > 0. Fix x ∈ Q and consider the function gy := fx,y,ε ∈ L. Let Vy := {q ∈ Q : gy (q) > −ε}. Then Vy is an open set and y ∈ Vy . By a standard compactness argument, there are y1 , . . . , yn ∈ Q such that Q = Vy1 ∪ . . . ∪ Vyn . Put fx := gy1 ∨ . . . ∨ gyn . It is clear that fx ∈ L. Furthermore, fx (x) < ε and fx (y) > −ε for all y ∈ Q. Now let Ux := {q ∈ Q : fx (q) < ε}. Then Ux is open and x ∈ Ux . Using the compactness of Q once again, find x1 , . . . , xm ∈ Q such that Q = Ux1 ∪ . . . ∪ Uxm . Finally, put l := fx1 ∧ . . . ∧ fxm . It is beyond a doubt that l ∈ L and klk < ε. B 10.8.6. Remark. The message of 10.8.5 is often referred to as the Generalized Dini Theorem. (cf. 7.2.10). 10.8.7. Kakutani Lemma. Every sublattice L of C(Q, R) is expressible as \
cl L = cl L{q1 ,q2 } . (q1 ,q2 )∈Q2


C The inclusion of cl L into cl L{q1 ,q2 } for all (q1 , q2 ) ∈ Q2 raises no doubts.
If f ∈ cl L{q1 ,q2 } for all such q1 and q2 , then by 10.8.5, f ∈ cl L. B 10.8.8. Corollary. Every vector sublattice L of C(Q, R) is expressible as \ cl L = L{q1 ,q2 } . (q1 ,q2 )∈Q2

C Observe that every set of the form L{q1 ,q2 } is closed. B 10.8.9. Definition. A subset U of F Q separates the points of Q, if for all q1 , q2 ∈ Q such that q1 6= q2 there is a function u ∈ U assuming different values at q1 and q2 , i.e. u(q1 ) 6= u(q2 ). 10.8.10. Stone Theorem. If a vector sublattice of C(Q, R) contains constant functions and separates the points of Q, then it is dense in C(Q, R). C Given such sublattice L, observe that L{q1 ,q2 } = C(Q, R){q1 ,q2 } for every pair (q1 , q2 ) in Q2 (cf. 10.8.4 (6)). It remains to appeal to 10.8.8. B

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10.8.11. Let µ ∈ C(Q, R)0 . Put N (µ) := {f ∈ C(Q, R) : [0, |f |] ⊂ ker µ}. Then there is a unique closed subset supp(µ) of Q such that f ∈ N (µ) ⇔ f supp(µ) = 0. C By the Interval Addition Lemma [0, |f |] + [0, |g|] = [0, |f | + |g|]. Thus, f, g ∈ N (µ) ⇒ |f | + |g| ∈ N (µ). Since N (µ) is an order ideal; i.e., (f ∈ N (µ) & 0 ≤ |g| ≤ |f | ⇒ g ∈ N (µ)), conclude that N (µ) is a linear set. Moreover, N (µ) is closed. Indeed, assuming fn ≥ 0, fn → f and fn ∈ N (µ), for g ∈ [0, f ] find g ∧ fn → g and g ∧ fn ∈ [0, fn ]. Whence it follows that µ(g) = 0; i.e., f ∈ N (µ). Since N (µ) is an order ideal, from 10.8.8 deduce that \ N (µ) = N (µ){q} . q∈Q

Define the set supp(µ) as q ∈ supp(µ) ⇔ N (µ){q} 6= C(Q, R) ⇔ (f ∈ N (µ) ⇒ f (q) = 0). It is beyond a doubt that supp(µ) is closed. Moreover, the equalities hold: \  N (µ) = N (µ){q} = f ∈ C(Q, R) : f supp(µ) = 0 . q∈supp(µ)

The claim of uniqueness follows from the normality of Q (cf. 9.4.14) and the Urysohn Theorem. B 10.8.12. Definition. The set supp(µ) under discussion in 10.8.11 is the support of µ (cf. 10.9.4 (5)). 10.8.13. Remark. If µ is positive then N (µ) = {f ∈ C(Q, R) : µ(|f |) = 0}. Consequently, when µ(f g) = 0 for all g ∈ C(Q, R), observe that f supp(µ) = 0. By analogy supp(µ) = ∅ ⇔ N (µ) = C(Q, R) ⇔ µ = 0. Therefore, it is quite convenient to work with the support of a positive functional.

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Let F be a closed subset of Q. It is said that F supports or carries µ or that X \ F lacks µ or is void of µ if µ(|f |) = 0 for every continuous function f with supp(f ) ⊂ Q\F . The support supp(µ) of µ carries µ; moreover, supp(µ) is included in every closed subset of Q supporting µ. In other words, the support of µ is the complement of the greatest open set void of µ (cf. 10.10.5 (6)). It stands to reason to observe that in virtue of 3.2.14 and 3.2.15 to every bounded functional µ there correspond some positive (and hence bounded) functionals µ+ , µ− , and |µ| defined as µ+ (f ) = sup µ[0, f ];

µ− (f ) = − inf µ[0, f ];

|µ| = µ+ + µ− ,

given f ∈ C(Q, R)+ . Moreover, C(Q, R)0 is a Kantorovich space (cf. 3.2.16). CB 10.8.14. The supports of µ and |µ| coincide. C By definition N (µ) = N (|µ|). B 10.8.15. Considering a ∈ C(Q, R) with 0 ≤ a ≤ 1, define aµ : f 7→ µ(af ) for f ∈ C(Q, R) and µ ∈ C(Q, R)0 . Then |aµ| = a|µ|. C Given f ∈ C(Q, R)+ , infer that (aµ)+ (f ) = sup{µ(ag) : 0 ≤ g ≤ f } ≤ sup µ[0, af ] = µ+ (af ) = aµ+ (f ). Furthermore, µ+ = (aµ + (1 − a)µ)+ ≤ (aµ)+ + ((1 − a)µ)+ ≤ aµ+ + (1 − a)µ+ = µ+ . Consequently, (aµ)+ = aµ+ , whence the claim follows. B 10.8.16. De Branges Lemma. Let A be a subalgebra of C(Q, R) containing constant functions. Take µ ∈ ext(A⊥ ∩ BC(Q, R)0 ). Then the restriction of each member of A to the support of µ is a constant function. C If µ = 0 then supp(µ) = ∅, and there is nothing to be proven. If µ 6= 0 then, certainly, kµk = 1. Take a ∈ A. Since the subalgebra A contains constant functions, it suffices to settle the case in which 0 ≤ a ≤ 1 and q ∈ supp(µ) ⇒ 0 < a(q) < 1. Put µ1 := aµ and µ2 := (1 − a)µ. It is clear that µ1 + µ2 = µ and the functionals µ1 and µ2 are both nonzero. Moreover, kµk ≤ kµ1 k + kµ2 k

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= sup µ(af ) + sup µ((1 − a)g) = kf k≤1

kgk≤1

sup

µ(af + (1 − a)g) ≤ kµk,

kf k≤1,kgk≤1

because it is obvious that aBC(Q, R) + (1 − a)BC(Q, R) ⊂ BC(Q, R) . Thus, kµk = kµ1 k + kµ2 k. Since µ = kµ1 k

µ2 µ1 + kµ2 k , kµ1 k kµ2 k

and µ1 , µ2 ∈ A⊥ , conclude that µ1 = kµ1 kµ. By 10.8.15, a|µ| = |aµ| = |µ1 | = kµ1 k |µ|. Consequently, |µ|((a − kµ1 k1)g) = 0 for all g ∈ C(Q, R). Using 10.8.13 and 10.8.14, infer that the function a is constant on the support of µ. B 10.8.17. Stone–Weierstrass Theorem. Let A be a subalgebra of the algebra C(Q, R). Suppose that A contains constant functions and separates the points of Q. Then A is dense in C(Q, R). C Proceeding by way of contradiction, assume the contrary. By the Absolute Bipolar Theorem, the subspace A⊥ (coincident with A◦ ) of C(Q, R)0 is nonzero. Using the Alaoglu–Bourbaki Theorem, observe that A⊥ ∩ BC(Q,R)0 is a nonempty absolutely convex weakly compact set. Thus, by the Kre˘ın–Milman Theorem, the set has an extreme point, say, µ. Undoubtedly, µ is a nonzero functional. By the de Branges Lemma the support of µ fails to contain two distinct points, since A separates the points of Q. The support of µ is not a singleton, since µ annihilates constant functions. Thus, supp(µ) is empty. But then µ is zero (cf. 10.8.13). We arrive at a contradiction completing the proof. B 10.8.18. Corollary. The closure of a subalgebra of C(Q, R) is a vector sublattice of C(Q, R). C Using the Stone–Weierstrass Theorem, find a polynomial pn satisfying |pn (t) − |t| | ≤ 1 /2n for all t ∈ [−1, 1]. Then |pn (0)| ≤ 1 /2n . Therefore, the polynomial pn (t) := pn (t) − pn (0) maintains the inequality |pn (t) − |t| | ≤ 1 /n when −1 ≤ t ≤ 1. By construction, pn lacks the constant term. Now, if a function a lies in a subalgebra A of C(Q, R) and kak ≤ 1, then |pn (a(q)) − |a(q)| | ≤ 1 /n (q ∈ Q). Moreover, the function q 7→ pn (a(q)) is clearly a member of A. B

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10.8.19. Remark. Corollary 10.8.18 (together with 10.8.8) completely describes all closed subalgebras of C(Q, R). In turn, as the proof prompts, the claim of 10.8.18 is immediate on providing some sequence of polynomials which converges uniformly to the function t 7→ |t| on the interval [−1, 1]. It takes no pains to demonstrate 10.8.17, with 10.8.18 available. 10.8.20. Tietze–Urysohn Theorem. Let Q0 be a compact subset of a compact set Q and f0 ∈ C(Q0 , R). Then there is a function f in C(Q, R) such that f Q0 = f0 . C Suppose that Q0 6= ∅ (otherwise, there is nothing to prove). Consider the ◦ identical embedding ι : Q0 → Q and the bounded linear operator ι : C(Q, R) → ◦ ◦ C(Q0 , R) acting by the rule ιf := f ◦ ι. We have to show that ι is an epimorphism. ◦ It is beyond a doubt that im ι is a subalgebra of C(Q0 , R) separating the points of Q0 and containing constant functions. In virtue of 10.8.17 it is thus sufficient ◦ (and, clearly, necessary) to examine the closure of im ι. ◦ Consider the monoquotient ι of the operator ι and the coset mapping ϕ : ◦ C(Q, R)0 = 0 O ι. Given f ∈ C(Q, R), put g := (f ∧ sup |f (Q0 )|1) ∨ (− sup |f (Q0 )|1). By definition f Q0 = g Q0 ; i.e., f := ϕ(f ) = ϕ(g). Consequently, kgk ≥ kf k. Furthermore,   ◦ kf k = inf khkC(Q,R) : ι(h − f ) = 0 = inf khkC(Q,R) : h Q = f Q 0 0  ≥ inf kh Q0 kC(Q,R) : h Q0 = f Q0 = sup |f (Q0 )| = kgk ≥ kf k. Therefore, ◦

◦

kιf k = kιgk = kιgkC(Q0 ,R) = kg ◦ ιkC(Q0 ,R) = sup |g(Q0 )| = kgk = kf k; ◦

i.e., ι is an isometry. Successively applying 5.5.4 and 4.5.15, infer first that coim ι is a Banach space and second that im ι is closed in C(Q0 , R). It suffices to observe ◦ that im ι = im ι. B 10.9. Radon Measures 10.9.1. Definition. Let Š be a locally compact topological space. Put K(Š) := K(Š, F ) := {f ∈ C(Š, F) : supp(f ) is compact}. If Q is compact in Š then let K(Q) := KŠ (Q) := {f ∈ K(Š) : supp(f ) ⊂ Q}. The space K(Q) is furnished with the norm k · k∞ . Given E ∈ Op (Š), put K(E) := ∪ {K(Q) : Q b E}. (The notation Q b E for a subset E of Š means that Q is compact and Q lies in the interior of E relative to Š.)

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10.9.2. The following statements are valid: (1) if Q b Š and f ∈ C(Q, F) then f ∂Q = 0 ⇔ (∃ g ∈ K(Q)) g Q = f ; moreover, K(Q) is a Banach space; (2) let Q, Q1 , and Q2 be compact sets and Q b Q1 ×Q2 ; the linear span in the space C(Q, F ) of the restrictions to Q of the functions like u1 · u2 (q1 , q2 ) := u1 ⊗ u2 (q1 , q2 ) := u1 (q1 )u2 (q2 ) with us ∈ K(Qs ) is dense in C(Q, F ); (3) if Š is compact then K(Š) = C(Š, F); if Š fails to be compact and is embedded naturally into C(Š· , F), with Š· := Š ∪ {∞} the Alexandroff compactification of Š, then the space K(Š) is dense in the hyperplane {f ∈ C(Š· , F ) : f (∞) = 0}; (4) the mapping E ∈ Op (Š) 7→ K(E) ∈ Lat (K(Š)) preserves suprema; (5) for E 0 , E 00 ∈ Op (Š) the following sequence is exact: ι

σ

(E ,E ) (E ,E ) 00 = 0 O (E 0 ∩ E 00 ) −−−−−→ K(E 0 ) × K(E 00 ) −−−−−→ K(E 0 ∪ E 00 )0 = 0 O , 0

00

0

00

with ι(E 0 ,E 00 ) f := (f, −f ), and σ(E 0 ,E 00 ) (f, g) := f + g. C (1): The boundary ∂Q of Q is at the same time the boundary of the exterior of Q, the set int(Š \ Q). (2): The set under study is a subalgebra. Apply 9.3.13 and 10.8.17 (cf. 11.8.2). (3): It may be assumed that F = R. The claim will then follow from 10.8.8 since K(Š) is an order ideal separating the points of Š· (cf. 10.8.11). (4): Clearly, K(sup ∅) = K(∅) = 0. If E ⊂ Op (Š) and E is filtered upward then, for f ∈ K(∪E ), observe that supp(f ) ⊂ E for some E ∈ E by the compactness of supp(f ). Whence K(∪E ) = ∪{K(E) : E ∈ E }. To conclude, take E1 , . . . , En ∈ Op (Š) and f ∈ K(E1P∪ . . . ∪ En ). In accordance P with 9.4.18 there are some n n ψk ∈ K(Ek ) such that k=1 ψk = 1. Moreover, f = k=1 ψk f and supp(f ψk ) ⊂ Ek (k := 1, . . . , n). (5): This is straightforward from (4). B 10.9.3. Definition. A functional µ, a member of K(Š, F)# , is a measure (or, F-measure) on Š, in symbols, µ ∈ M (Š) := M (Š, F); if amply, a Radon 0 µ K(Q) ∈ K(Q) whenever Q b Š. The following notation is current: Z

Z f dµ :=

Z f dµ :=

f (x) dµ(x) := µ(f )

(f ∈ K(Š)).

Š

The scalar µ(f ) is the integral of f with respect to µ. In this connection µ is also called an integral.

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Chapter 10

10.9.4. Examples. (1) For q ∈ Š the Dirac measure f 7→ f (q) (f ∈ K(Š)) presents a Radon measure. It is usually denoted by the symbol δq and called the deltafunction at q. Suppose also that Š is furnished with group structure so that the taking of an inverse q ∈ Š 7→ q −1 ∈ Š and the group operation (s, t) ∈ Š × Š 7→ st ∈ Š are continuous; i.e., Š is a locally compact group. By δ we denote δe where e is the unity of Š (recall the concurrent terms: “identity,” “unit,” and “neutral element,” all meaning the same: es = se = s for all s ∈ Š). The nomenclature pertinent to addition is routinely involved in abelian (commutative) groups. Given a ∈ Š, acknowledge the operation of (left or right) translation or shift by a in K(Š) (in fact, every function is shifted in Š × F): (a τ f )(q) := a f (q) := f (a−1 q),

(τa f )(q) := fa (q) := f (qa−1 )

with f ∈ K(Š) and q ∈ Š. Clearly, a τ , τa ∈ L (K(Š)). A propitious circumstance of paramount importance is the presence of a nontrivial measure on Š, a member of M (Š, R), which is invariant under left (or right) translations. All (left)invariant Radon measures are proportional. Each nonzero (left)invariant positive Radon measure is a (left) Haar measure (rarely Haar integral). In the case of right translations, the term “(right) Haar measure” is in common parlance. In the abelian case, we speak only of a Haar measure and even of Haar measure, neglecting the necessity of scaling. The familiar Lebesgue measure on RN is Haar measure on the abelian group RN . That is why the conventional notation of Lebesgue integration is retained in the case of an abstract Haar measure. In particular, the left-invariance condition is written down as Z Z −1 f (a x) dx = f (x) dx (f ∈ K(Š), a ∈ Š). Š

Š

(2) Let M (Š) := (K(Š), k · k∞ )0 . A member of M (Š) is a bounded Radon measure. It is clear that a bounded measure belongs to the space C(Š· , F )0 (cf. 10.9.2 (2)). (3) Given µ ∈ M (Š), put µ∗ (f ) = µ(f ∗ )∗ , where f ∗ (q) := f (q)∗ for q ∈ Š and f ∈ K(Š). The measure µ∗ is the conjugate of µ. Distinction between µ∗ and µ is perceptible only if F = C. In case µ = µ∗ , we speak of a real C-measure. It is clear that µ = µ1 + iµ2 , where µ1 and µ2 are uniquely-determined real C-measures. In turn, each real C-measure is generates by two R-measures (also called real measures), members of M (Š, R), because K(Š, C) coincides with the complexification K(Š, R) ⊕ iK(Š, R) of K(Š, R). The real R-measures obviously constitute a Kantorovich space. Moreover, the integral with respect to such

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measure serves as (pre)integral. So, an opportunity is offered to deal with the corresponding Lebesgue extension and spaces of scalar-valued or vector-valued functions (cf. 5.5.9 (4) and 5.5.9 (5)). We seize the opportunity without much ado and specification. To every Radon measure µ we assign the positive measure |µ| that is defined for f ∈ K(Š, R), f ≥ 0, by the rule |µ|(f ) := sup{|µ(g)| : g ∈ K(Š, F), |g| ≤ f }. Observe that, in current usage, the word “measure” often means a positive measure, whereas a “general” measure is referred to as a signed measure or a charge. If µ and ν are measures and |µ| ∧ |ν| = 0 then µ and ν are called disjoint or independent (of one another). A measure ν is absolutely continuous with respect to µ on condition that ν is independent of every measure independent of µ. Such a measure ν may be given as ν = f µ, where f ∈ L1,loc (µ) and the measure f µ having density f with respect to µ acts by the rule (f µ)(g) := µ(f g) (g ∈ K(Š)) (this is the Radon–Nikod´ym Theorem). (4) Given Š0 ∈ Op (Š) and µ ∈ M (Š), consider the restriction µŠ0 := µ K(Š0 ) of µ to K(Š0 ). The restriction operator µ 7→ µŠ0 from M (Š) to M (Š0 ), viewed as depending on a subset of Š, meets the agreement condition: if Š00 ⊂ Š0 ⊂ Š and µ ∈ M (Š) then µŠ00 = (µŠ0 )Š00 . This situation is verbalized as follows: “The mapping M : E ∈ Op (Š) 7→ M (E) and the restriction operator (referred to jointly as the functor M ) defines a presheaf (of vector spaces over Š).” It stands to reason to convince oneself that the (values of the) restriction operator need not be an epimorphism. (5) Let E ∈ Op (Š) and µ ∈ M (Š). It is said that E lacks µ or is void of µ or that Š \ E supports or carries µ if µE = 0. By 10.9.2 (4) there is a least closed set supp(µ) supporting µ, the support of µ. It may be shown that supp(µ) = supp(|µ|). The above definition agrees with 10.8.12. The Dirac measure δq is a unique Radon measure supported at {q} to within scaling. (6) Let Šk be a locally compact space and µk ∈ M (Šk ) (k := 1, 2). There is a unique measure µ on the product Š1 × Š2 such that Z

Z u1 (x)u2 (y) dµ(x, y) =

Š1 ׊2

Z u1 (x) dµ1 (x)

Š1

u2 (y) dµ2 (y)

Š2

with uk ∈ K(Šk ). The next designations are popular: µ1 × µ2 := µ1 ⊗ µ2 := µ. Using 10.9.2 (4), infer that for f ∈ K(Š1 × Š2 ) the value µ1 × µ2 (f ) may be calculated by repeated integration (this is the Fubini Theorem).

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(7) Let G be a locally compact group, and µ, ν ∈ M (G). For f ∈ K(G) the function f_(s, t) := f (st) is continuous and |(µ × ν)(f_)| ≤ kµk kνk kf k∞ . This defines the Radon measure µ ∗ ν(f ) := (µ × ν)(f_) (f ∈ K(G)), the convolution of µ and ν. Using vector integrals, obtain the presentations: Z Z Z µ∗ν = δs ∗ δt dµ(s)dν(t) = δs ∗ ν dµ(s) = µ ∗ δt dν(t). G×G

G

G

The space M (G) of bounded measures with convolution as multiplication, exemplifies a Banach convolution algebra. This algebra is commutative if and only if G is an abelian group. In that event the space L1 (G), constructed with respect to Haar measure m, also possesses a natural structure of a convolution algebra (namely, that of a subalgebra of M (G)). The algebra (L1 (G), ∗) is the group algebra of G. Thus, for f, g ∈ L1 (G), the definitions of convolution for functions and measures agree with one another (cf. 9.6.17): (f ∗ g)dm = f dm ∗ gdm. By analogy, the convolution of µ ∈ M (G) and f ∈ L1 (G) is defined as (µ ∗ f )dm := µ ∗ (f dm); i.e., as the density of the convolution of µ and f dm with respect to Haar measure dm. In particular, Z Z f ∗ g = (δx ∗ g)f (x) dm(x) = τx (g)f (x) dm(x). G

G

Wendel Theorem. Let T ∈ B(L1 (G)). Then the following statements are equivalent: (i) there is a measure µ ∈ M (G) such that T f = µ ∗ f for f ∈ L1 (G); (ii) T commutes with translations: T τa = τa T for a ∈ G, where τa is a unique bounded extension to L1 (G) of translation by the element a in K(G); (iii) T (f ∗ g) = (T f ) ∗ g for f, g ∈ L1 (G); (iv) T (f ∗ ν) = (T f ) ∗ ν for ν ∈ M (G) and f ∈ L1 (G). 10.9.5. Definition. The spaces K(Š) and M (Š) are set in duality (induced by the duality bracket K(Š) ↔ K(Š)# ). In this case, the space M (Š) is furnished with the topology σ(M (Š), K(Š)), usually called vague. The space K(Š) is furnished with the Mackey topology τK(Š) := τ (K(Š), M (Š)) (thereby, in particular, (K(Š), τK(Š) )0 = M (Š)). The space of bounded measures M (Š) is usually considered with the dual norm: kµk := sup{|µ(f )| : kf k∞ ≤ 1, f ∈ K(Š)} (µ ∈ M (Š)). 10.9.6. The topology τK(Š) is strongest among all locally convex topologies making the identical embedding of K(Q) to K(Š) continuous for every Q with Q b Š (i.e., τK(Š) is the so-called inductive limit topology (cf. 9.2.15)).

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C If τ is the inductive limit topology and µ ∈ (K(Š), τ )0 then by definition µ ∈ M (Š), because µ ◦ ιK(Q) is continuous for Q b Š. In turn, for µ ∈ M (Š) the set VQ := {f ∈ K(Q) : |µ(f )| ≤ 1} is a neighborhood of zero in K(Q). From the definition of τ , infer that ∪ {VQ : Q b Š} = {f ∈ K(Š) : |µ(f )| ≤ 1} is a neighborhood of zero in τ . Thus, µ ∈ (K(Š), τ )0 and τ is compatible with duality. Therefore, τ ≤ τK(Š) . On the other hand, if p is a seminorm in the mirror of the Mackey topology then p is the supporting function of a subdifferential lying in M (Š). Consequently, the restriction q := p ◦ ιK(Q) of p to K(Q) is lower semicontinuous anyway. By the Gelfand Theorem (in view of the barreledness of K(Q)) the seminorm q is continuous. Consequently, the identical embedding ιK(Q) : K(Q) → (K(Š), τK(Š) ) is continuous and τ ≥ τK(Š) by the definition of inductive limit topology. B 10.9.7. A subset A of K(RN ) is bounded (in the inductive limit topology), if sup kAk∞ < +∞ and, moreover, the supports of the members of A lie in a common compact set. C Suppose to the contrary that, for some Q such that Q b RN , the inclusion A ⊂ K(Q) fails. In other words, for n ∈ N there are some qn ∈ RN and an ∈ A satisfying an (qn ) 6= 0 and |qn | > n. Put B := {n|an (qn )|−1 δqn : n ∈ N} and observe that this set of Radon measures is vaguely bounded and so the seminorm p(f ) := sup{|µ|(|f |) : µ ∈ B} is continuous. Moreover, p(an ) ≥ n|an (qn )|−1 δqn (|an |) = n, which contradicts the boundedness property of A. B 10.9.8. Remark. Let (fn ) ⊂ K(RN ). The notation fn  0 symbolizes the K

proposition (∃ Q b RN )(∀ n) supp(fn ) ⊂ Q & kfn k∞ → 0. From 10.9.7 it is immediate that µ ∈ K(RN )# is a Radon measure if µ(fn ) → 0 whenever fn  0. K

Observe also that the same holds for every locally compact Š countable at infinity or σ-compact, i.e. for Š presenting the union of countably many compact sets. 10.9.9. Remark. There is a sequence (pn ) of real-valued positive polynomials on R such that the measures pn dx converge vaguely to δ as n → +∞. Considering products of the measures, arrive at some polynomials Pn on the RN , with (Pn dx) converging vaguely to δ (here, as usual, dx := dx1 × . . . × dxN is the Lebesgue measure on RN ). Now, take f ∈ K(RN ) such that f is of class C (m) in some neighborhood of a compact set Q (i.e., f has continuous derivatives up to order m on Q). Arranging the convolutions (f ∗Pn ), obtain a sequence of polynomials which approximates not only f but also the derivatives of f up to order m uniformly on Q. The possibility of such reqularization is referred to as the Generalized Weierstrass Theorem in RN (cf. 10.10.2 (4)). 10.9.10. Measure Localization Pronciple. Let E be an open cover of Š and let (µE )E∈E be a family of Radon measures, µE ∈ M (E). Assume further

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that, for every pair (E 0 , E 00 ) of the members of E , the restrictions of µE 0 and µE 00 to E 0 ∩ E 00 coincide. Then there is a unique measure µ on Š whose restriction to E equals µE for all E ∈ E . C Using 10.9.2 (5), arrange the sequence X X σ ι K(E) −→ K(Š) → 0, K(E 0 ∩ E 00 ) −→ 0 00 {E ,E } 0 00 E ,E ∈E, E 0 6=E 00

E∈E

where ι is generated by summation of the coordinate identical embeddings ι(E 0,E 00 ) and σ is the standard addition. Every direct sum is always assumed to be furnished with the inductive limit topology (cf. 10.9.6). Examine the exactness of the sequence. Since K(Š) = ∪QbŠ K(Q) and in view of 10.9.2 (4), it suffices to settle the case of a finite cover by checking exactness at the second term. Thus, proceeding by induction, suppose that for every n-element cover {E1 , . . . , En } (n ≥ 2) the following sequence is exact: ιn

Kn −→

n Y

σ

n K(Ek ) −→ K(E1 ∪ . . . ∪ En ) → 0,

k=1

where ιn is the “restriction” of ι to Kn , the symbol σn stands for addition and Y Kn := K(Ek ∩ El ). k
By hypothesis, im ιn = ker σn . If σn+1 (fe, fn+1 ) = 0 where fe:= (f1 , . . . , fn ), then σn fe = −fn+1 and fn+1 ∈ K((E1 ∪ . . . ∪ En ) ∩ En+1 ). Since σn is an epimorphism by 10.9.2 (5), there are some θk ∈ K(Ek ∩ En+1 ) such that σn θ = −fn+1 for θ := (θ1 , . . . , θn ). Whence (fe − θ) ∈ ker σn , and by hypothesis there is a member κ of Kn such that ιn κ = fe − θ. Clearly, Kn+1 = Kn ×

n Y

K(Ek ∩ En+1 )

k=1

(to within isomorphism), κ := (κ, θ1 , . . . , θn ) ∈ Kn+1 , and ιn+1 κ = (fe, fn+1 ). Passing to the diagram prime (cf. 7.6.13), arrive at the exact sequence Y Y σ0 ι0 0 → M (Š) −→ M (E) −→ M (E 0 ∩ E 00 ). E∈E

The proof is complete. B

{E 0 ,E 00 } E 0,E 00 ∈E ,E 0 6=E 00

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10.9.11. Remark. In topology, a presheaf recoverable from its patches or local data in the above manner is a sheaf. In this regard, the claim of 10.9.10 is verbalized as follows: the presheaf Š 7→ M (Š) of Radon measures is a sheaf or, putting it more categorically, the functor M is a sheaf (of vector spaces over Š, cf. 10.9.4 (4)). The Spaces D() and D 0 ()

10.10.

10.10.1. Definition. A compactly-supported smooth function f : RN → F is a test function; in symbols, f ∈ D(RN ) := D(RN , F). Given Q b RN and Š ∈ N N Op (R ), designate D(Q) := f ∈ D(R ) : supp(f ) ⊂ Q and D(Š) := ∪ {D(Q) : Q b Š}. 10.10.2. The following statements are valid: (1) D(Q) = 0 ⇔ int Q = ∅; (2) given Q b RN , put kf kn,Q :=

X

k∂ α f kC(Q) :=

X

sup |(∂ α1 . . . ∂ αn f )(Q)|

α∈(Z+)N α1 +...+αN ≤n

|α|≤n

for a function f smooth in a neighborhood of Q (as usual, Z+ := N ∪ {0}); the multinorm MQ := {k · kn,Q : n ∈ N} makes D(Q) into a Fr´echet space; (3) the space of smooth functions C∞ (Š) := E (Š) on Š in Op (RN ) with the multinorm MŠ := {k · kn,Q : n ∈ N, Q b Š} is a Fr´echet space; moreover, D(Š) is dense in C∞ (Š); (4) if Q1 b RN , Q2 b RM and Q b Q1 × Q2 ; then the linear span in D(Q) of the restrictions to Q of the functions like f1 f2 (q1 , q2 ) := f1 ⊗f2 (q1 , q2 ) := f1 (q1 )f2 (q2 ), with qk ∈ Qk and fk ∈ D(Qk ), is dense in D(Q); (5) the mapping E ∈ Op (Š) 7→ D(E) ∈ Lat (D(Š)) preserves suprema: D(E 0 ∩ E 00 ) = D(E 0 ) ∩ D(E 00 ), D(E 0 ∪ E 00 ) = D(E 0 ) + D(E 00 ); D(∪E ) = L (∪ {D(E) : E ∈ E })

(E ⊂ Op (Š)).

Moreover, the next sequence is exact (cf. 10.9.2 (5)): ι

σ

(E ,E ) (E ,E ) 00 = 0 O D(E 0 ∩ E 00 ) −−−−−→ D(E 0 ) × D(E 00 ) −−−−−→ D(E 0 ∪ E 00 )0 = 0 O . 0

00

0

00

C (1) and (2) are obvious. (3): Choose a sequence (Qm )m∈N such that Qm b Š, Qm b Qm+1 and ∪m∈N Qm = Š. The multinorm {k · kn,Qm : n ∈ N, m ∈ N} is countable and

202

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equivalent to MŠ . A reference to 5.4.2 yields the metrizability of C∞ (Š). The completeness of C∞ (Š) raises no doubts. To show that D(Š) is dense in C∞ (Š), consider the truncator set Tr(Š) := {ψ ∈ D(Š) : 0 ≤ ψ ≤ 1}. Make Tr (Š) a direction on letting ψ1 ≤ ψ2 ⇔ supp(ψ1 ) ⊂ int{ψ2 = 1}. It is clear that, for f ∈ C∞ (Š), the net (ψf )ψ∈Tr (Š) approximates f as is needed. (4): Take q 0 ∈ RN and q 00 ∈ RM . Let a(q 0 , q 00 ) := a0 (q 0 )a00 (q 00 ), where a0 and a00 are mollifiers on RN and RM , respectively. Given f ∈ D(Q), n ∈ N, and ε > 0, choose χ from the condition kf − f ∗ aχ kn,Q ≤ ε /2 . Using the equicontinuity property of the family F := {∂ α f (q)τq (aχ ) : |α| ≤ n, q ∈ Q1 × Q2 }, find finite sets 0 and 00 , with 0 ⊂ Q1 and 00 ⊂ Q2 , so that the integral of each function in F be approximated to within 1 /2 (N + 1)−n ε by a Riemann sum of it using the points of 0 ׁ00 . This yields a function f , a member of the linear span of D(Q1 )×D(Q2 ), which was sought; i.e., kf − f kn,Q ≤ ε. (5): Check this as in 10.9.2 (4), on replacing 9.4.18 with 9.6.19 (2). B 10.10.3. Remark. The Generalized Weierstrass Theorem may be applied to the demonstration of 10.10.2 (4), when combined with due truncation providing that the constructed approximation has compact support. 10.10.4. Definition. A functional u, a member of D(Š, F )# , is a distribution or a generalized function whenever u |D(Q) ∈ D 0 (Q) := D(Q)0 for all Q b Š. This is expressed in writing as u ∈ D 0 (Š) := D 0 (Š, F ). Sometimes a reference is appended to the nature of the ground field F. The usual designations are as follows: hu, f i := hf | ui := u(f ). Often we use the most suggestive and ubiquitous symbol Z f (x)u(x) dx := u(f )

(f ∈ D(Š)).

10.10.5. Examples. (1) Let g ∈ L1,loc (RN ) be a locally integrable function. The mapping Z ug (f ) :=

f (x)g(x) dx (f ∈ D(Š))

determines a distribution. A distribution of this type is regular. To denote a regular distribution ug , a more convenient symbol g is also employed. In this connection, we write D(Š) ⊂ D 0 (Š) and ug = | gi. (2) Every Radon measure is a distribution. Each positive distribution u (i.e., such that f ≥ 0 ⇒ u(f ) ≥ 0) is determined by a positive measure.

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(3) A distribution u is said to has order at most m, if to every Q such that Q b RN there corresponds a number tQ satisfying |u(f )| ≤ tQ kf km,Q

(f ∈ D(Q)).

The notions of the order of a distribution and of a distribution of finite order are understood in a matter-of-fact fashion. Evidently, it is false that every distribution has finite order. (4) Let α be a multi-index, α ∈ (Z+ )N ; and let u be a distribution, u ∈ D 0 (Š). Given f ∈ D(Š), put (∂ α u)(f ) := (−1)|α| u(∂ α f ). The distribution ∂ α u is the derivative of u (of order α). We also speak of generalized differentiation, of derivatives in the distribution sense, etc. and use the conventional symbols of differential calculus. A derivative (of nonzero order) of a Dirac measure is not a measure. At the same time δ ∈ D 0 (R) is the derivative of the Heaviside function δ (−1) := H, where H : R → R is the characteristic function of R+ . If a derivative of a (regular) distribution u is some regular distribution ug , then g is a weak derivative of u or a generalized derivative of u in the Sobolev sense. For a test function, such derivative coincides with its ordinary counterpart. (5) Given u ∈ D 0 (Š), put u∗ (f ) := u(f ∗ )∗ . The distribution u∗ is the conjugate of u. The presence of the involution ∗ routinely justifies speaking of real distributions and complex distributions (cf. 10.9.3 (3)). (6) Let E ∈ Op (Š) and u ∈ D 0 (Š). For f ∈ D(E), the scalar u(f ) is easily determined. This gives rise to the distribution uE , a member of D 0 (E), called the restriction of u to E. The functor D 0 is clearly a presheaf. Given u ∈ D 0 (Š) and E ∈ Op (Š), say that E lacks or is void of u, if uE = 0. By 10.10.4 (5), if the members of a family of open subsets of Š are void of u then so is the union of the family. The complement (to RN ) of the greatest open set void of u is the support of u, denoted by supp(u). Observe that supp(∂ α u) ⊂ supp(u). Moreover, a distribution with compact support (= compactly-supported distribution) has finite order. (7) Let u ∈ D 0 (Š) and f ∈ C∞ (Š). If g ∈ D(Š) then f g ∈ D(Š). Put (f u)(g) := u(f g). The resulting distribution f u is the product of f and u. Consider the truncator direction Tr (Š). If there is a limit limψ∈Tr (Š) u(f ψ) then say that u applies to f . It is clear that a compactly-supported distribution u applies to every function in C∞ (Š). Moreover, u ∈ E 0 (Š) := C∞ (Š)0 . In turn, each element u of E 0 (Š) (cf. 10.10.2 (3)) uniquely determines a distribution with compact support, which is implicit in the notation u ∈ D 0 (Š). α If f ∈ C∞ (Š) and ∂ f supp(u) = 0 for all α, |α| ≤ m, where u is a compactlysupported distribution of order at most m, then it is easy that u(f ) = 0. In con-

204

Chapter 10

sequence, only linear combinations of a Dirac measure and its derivatives are supported at a singleton. CB (8) Let Š1 , Š2 ∈ Op (RN ) and uk ∈ D 0 (Šk ). There is a unique distribution u on Š1 × Š2 such that u(f1 f2 ) = u1 (f1 )u2 (f2 ) for all fk ∈ D(Šk ). The distribution is denoted by u1 × u2 or u1 ⊗ u2 . Using 10.10.2 (4), infer that for f ∈ D(Š1 × Š2 ) the value u(f ) of u at f appears from successive application of u1 and u2 . Strictly speaking, u(f ) = u2 (y ∈ Š2 7→ u1 (f ( · , y))) = u1 (x ∈ Š1 7→ u2 (f (x, · ))). More suggestive designations prompt the Fubini Theorem: ZZ f (x, y)(u1 × u2 )(x, y) dxdy Š1 ׊2

 Z =

 Z f (x, y)u1 (x) dx u2 (y) dy =

 Š2

 Z

Š1

 Z f (x, y)u2 (y) dy  u1 (x) dx.

 Š1

Š2

It is worth noting that supp(u1 × u2 ) = supp(u1 ) × supp(u2 ). +

(9) Let u, v ∈ D 0 (RN ). Given f ∈ D(RN ), put f := f ◦ +. It is +

clear that f ∈ C∞ (RN × RN ). Say that the distributions u and v convolute or admit convolution or are convolutive provide that the product u × v applies to +

the function f ⊂ C∞ (RN × RN ) for each f in D(RN ). It is easy (cf. 10.10.10) +

that the resulting linear functional f 7→ (u × v)( f ) (f ∈ D(RN )) is a distribution called the convolution of u and v and denoted by u ∗ v. It is beyond a doubt that the convolutions of functions (cf. 9.6.17) and measures on RN (cf. 10.9.4 (7)) are particular cases of the convolution of distributions. In some classes, every two distributions convolute. For instance, the space E 0 (RN ) of compactly-supported distributions with convolution as multiplication presents an (associative and commutative) algebra with unity the delta-function δ. Further, ∂ α u = ∂ α δ ∗ u and ∂ α (u ∗ v) = ∂ α u ∗ v = u ∗ ∂ α v. Moreover, the following remarkable equality, the Lions Theorem of Supports, holds: co (supp(u ∗ v)) = co (supp(u)) + co (supp(v)).

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It is worth emphasizing that the pairwise convolutivity of distributions fails in general to guarantee the associativity of convolution (for instance, (1 ∗ δ 0 ) ∗ δ (−1) = 0 whereas 1 ∗ (δ 0 ∗ δ (−1) ) = 1, with 1 := 1R ). Each distribution u convolutes with a test function f , yielding some regular distribution (u ∗ f )(x) = u(τx (f e )), where f e:= fe is the reflection of f ; i.e., f e(x) := f (−x) (x ∈ RN ). The operator u∗ : f 7→ u ∗ f from D(RN ) to C∞ (RN ) is continuous and commutes with translations: (u∗)τx = τx u∗ for x ∈ RN . It is easily seen that the above properties are characteristic of u∗; i.e., if an operator T , a member of L (D(RN ), C∞ (RN )), is continuous and commutes with translations, then there is a unique distribution u such that T = u∗; namely, u(f ) := (T 0 δ)(fe) for f ∈ D(RN ) (cf. the Wendel Theorem). 10.10.6. Definition. The spaces D(Š) and D 0 (Š) are set in duality (induced by the duality bracket D(Š) ↔ D(Š)# ). Moreover, D 0 (Š) is furnished with the topology of the distribution space σ(D 0 (Š), D(Š)), and D(Š) is furnished with the topology of the test function space, the Mackey topology τD := τD(Š) := τ (D(Š), D 0 (Š)). 10.10.7. Let Š ∈ Op (RN ). Then (1) τD is the strongest of the locally convex topologies making the identical embedding of D(Q) into D(Š) continuous for all Q b Š (i.e., τD is the inductive limit topology); (2) a subset A of D(Š) is bounded if and only if A lies in D(Q) for some Q such that Q b Š and is bounded in D(Q); (3) a sequence (fn ) converges to f in (D(Š), τD ) if and only if there is a compact set Q such that Q b Š, supp(fn ) ⊂ Q, supp(f ) ⊂ Q and (∂ α fn ) converges to ∂ α f uniformly on Q for every multi-index α (in symbols, fn  f ); (4) an operator T , a member of L (D(Š), Y ) with Y a locally convex space, is continuous if and only if T fn → 0 provided that fn  0; (5) a delta-like sequence (bn ) serves as a (convolution) approximate unity in D(RN ) as well as in D 0 (RN ); i.e., bn ∗ f  f (in D(RN )) and bn ∗ u → u (in D 0 (RN )) for f ∈ D(RN ) and u ∈ D 0 (RN ). C (1): This is established in much the same way as 10.9.6; and (2), by analogy with 10.9.7 using the presentation of Š as the union Š = ∪n∈N Qn , where Qn b Qn+1 for n ∈ N. (3): Note that a convergent countable sequence is bounded, and apply 10.10.7 (2) (cf. 10.9.8). (4): In virtue of 10.10.7 (1) the continuity of T amounts to that of the restriction T D(Q) for all Q b Š. By 10.10.2 (2) the space D(Q) is metrizable. It remains to refer to 10.10.7 (3).

206

Chapter 10

(5): It is clear that all of the supports supp(bn ∗ f ) lie in some compact neighborhood about supp(f ). Furthermore, for g ∈ C(RN ), it is evident that bn ∗ g → g uniformly on compact subsets of RN . Applying this to ∂ α f and considering (3), infer that bn ∗ f  f . On account of 10.10.5 (9), observe the following: u(fe) = (u ∗ f )(0) = lim(u ∗ (bn ∗ f ))(0) n

= lim((u ∗ bn ) ∗ f )(0) = lim(bn ∗ u)(fe) n

n

for f ∈ D(R ). B N

10.10.8. Remark. In view of 10.10.7 (3), for Š ∈ Op (RN ) and m ∈ Z+ it is of(m) ten convenient to consider the space D (m) (Š) := C0 (Š) comprising all compactlysupported functions f whose derivatives ∂ α f are continuous for all |α| ≤ m. The space D (m) (Q) := {f ∈ D (m) (Š) : supp(f ) ⊂ Q} for Q b Š is furnished with the norm k·km,Q making it into a Banach space. In that event, D (m) (Š) is endowed with the inductive limit topology. Thus, D (0) (Š) = K(Š) and D(Š) = ∩m∈N D (m) (Š). For a sequence (fn ) to converge in D (m) (Š) means to converge uniformly with all derivatives up to order m on some Q such that Q b Š and supp(fn ) ⊂ Q for all sufficiently large n. Note that D (m) (Š)0 comprises all distributions of order at most m. Correspondingly, [ D 0F (Š) := D (m) (Š)0 m∈N

is the space of finite-order distributions. 10.10.9. Let Š ∈ Op (RN ). Then (1) the space D(Š) is barreled; i.e., each barrel, a closed absorbing absolutely convex subset, is a neighborhood of zero; (2) every bounded closed set in D(Š) is compact; i.e., D(Š) is a Montel space; (3) every absolutely convex subset of D(Š), absorbing each bounded set, is a neighborhood of zero; i.e., D(Š) is a bornological space; (4) the test functions are dense in the distribution space. C (1): Given a barrel V in D(Š), observe that VQ := V ∩ D(Q) is a barrel in D(Q) for all Q b Š. Thus, VQ is a neighborhood of zero in D(Q) (cf. 7.1.8). (2): Such a set lies in D(Q) for some Q b Š by 10.10.7 (2). In virtue of 10.10.2 (2), D(Q) is metrizable. Applying 4.6.10 and 4.6.11 proves the claim. (3): This follows from the barreledness of D(Q) for Q b Š. (4): Let g ∈ | D(Š)i◦ , with the polar takenR with respect to D(Š) ↔ D 0 (Š). It is clear that uf (g) = 0 for f ∈ D(Š); i.e., g(x)f (x) dx = 0. Thus, g = 0. It remains to refer to 10.5.9. B

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10.10.10. Schwartz Theorem. Let (uk )k∈N be a sequence of distributions. Assume that for every f in D(Š) there is a sum u(f ) :=

∞ X

uk (f ).

k=1

Then u is a distribution and ∂αu =

∞ X

∂ α uk

k=1

for every multi-index α. C The continuity of u is guaranteed by 10.10.9 (1). Furthermore, for f ∈ D(Š) by definition (cf. 10.10.5 (4)) α



∂ u(f ) = u (−1)

|α| α



∂ f =

∞ X



uk (−1)

|α| α



∂ f =

k=1

∞ X

∂ α uk (f ). .

k=1

10.10.11. Theorem. The functor D 0 is a sheaf. C It is immediate (cf. 10.9.10 and 10.9.11). B 10.10.12. Remark. The possibility of recovering a distribution from local data, the Distribution Localization Principle stated in 10.10.11, admits clarification in view of the paracompactness of RN . Namely, consider an open cover E of Š and a distribution u ∈ D 0 (Š) with local data (uE )E∈E . Take a countable P∞ (locally finite) partition of unity (ψk )k∈N subordinate to E . Evidently, u = k=1 ψk uk , where uk := uEk and supp(ψk uk ) ⊂ Ek (k ∈ N). 10.10.13. Theorem. Each distribution u on Š of order at most m may be expressed as sum of derivatives of Radon measures: X u= ∂ α µα , |α|≤m

where µα ∈ M (Š). C To begin with, assume that u has compact support. Let Q with Q b Š be a compact neighborhood of supp(u). By hypothesis (cf. 10.10.5 (7) and 10.10.8) |u(f )| ≤ t

X |α|≤m

for some t ≥ 0.

k∂ α f k∞

(f ∈ D(Q))

208

Chapter 10 Using 3.5.7 and 3.5.3, from 10.9.4 (2) obtain X X u=t να ◦ ∂ α = t (−1)|α| ∂ α να |α|≤m

|α|≤m

for a suitable family (να )|α|≤m , where να ∈ |∂|(k · k∞ ). Passing to the general case and invoking the Countable Partition Theorem, find a partition of unity (ψk )k∈N with ψk ∈ D(Š) such that some neighborhoods Qk of supp(ψk ) compose a locally finite cover of Š (cf. 10.10.12). For each of the distributions (ψk u)k∈N it is already proven that X ψk u = ∂ α µk,α , |α|≤m

where every µk,α is a Radon measure on Š and supp(µk,α ) ⊂ Qk . From the Schwartz Theorem it is immediate that the sum µα (f ) :=

∞ X

µk,α (f )

k=1

exists for all f ∈ K(Š). Moreover, the resulting distribution µα is a Radon measure. Once again appealing to 10.10.10, infer that ! ∞ ∞ ∞ X X X X X X u= ψk u = ∂ α µk,α = ∂α µk,α = ∂ α µα , k=1

k=1 |α|≤m

|α|≤m

k=1

|α|≤m

which was required. B 10.10.14. Remark. The claim of 10.10.13 is often referred to as the theorem on the general form of a distribution. Further abstraction and clarification are available. For instance, it may be verified that a compactly-supported Radon measure serves as a derivative (in the distribution sense) of suitable order of some continuous function. This enables us to view each distribution as a result of generalized differentiation of a conventional function. 10.11. The Fourier Transform of a Distribution 10.11.1. Let χ be a nonzero functional over the space L1 (RN ) := L1 (RN , C ). The following statements are equivalent: (1) χ is a character of the group algebra (L1 (RN ), ∗); i.e., χ 6= 0, χ ∈ L1 (RN )0 and χ(f ∗ g) = χ(f )χ(g) (f, g ∈ L1 (RN )) (in symbols, χ ∈ X(L1 (RN )), cf. 11.6.4);

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(2) there is a unique vector t in RN such that Z χ(f ) = fb(t) := (f ∗ et )(0) :=

f (x)ei(x,t) dx

RN

for all f ∈ L1 (RN ). C (1) ⇒ (2): Suppose that χ(f )χ(g) 6= 0. If x ∈ RN then χ(δx ∗ f ∗ g) = χ(δx ∗ f )χ(g) = χ(δx ∗ g)χ(f ). Put ψ(x) := χ(f )−1 χ(δx ∗ f ). This soundly defines some continuous mapping ψ : RN → C. Moreover, ψ(x + y) −1

= χ(f ∗ g)

χ(δx+y ∗ (f ∗ g)) = χ(f )−1 χ(g)−1 χ(δx ∗ f ∗ δy ∗ g)

= χ(f )−1 χ(δx ∗ f )χ(g)−1 χ(δy ∗ g) = ψ(x)ψ(y) for x, y ∈ RN ; i.e., ψ is a (unitary) group character: ψ ∈ X(RN ). Calculus shows that ψ = et for some (obviously, unique) t ∈ RN . Further, by the properties of the Bochner integral 

 Z

χ(f )χ(g) = χ(f ∗ g) = χ 

(δx ∗ g)f (x) dx RN

Z

Z χ(δx ∗ g)f (x) dx =

= RN

Z f (x)χ(g)ψ(x) dx = χ(g)

RN

f (x)ψ(x) dx.

RN

Thus, Z χ(f ) =

f (x)ψ(x) dx (f ∈ L1 (RN )).

RN

(2) ⇒ (1): Given t ∈ RN and treating f , g and f ∗ g as distributions, infer that f[ ∗ g(t) = uf ∗g (et ) Z Z Z Z = f (x)g(y)et (x + y) dxdy = f (x)et (x) dx g(y)et (y) dy RN RN

RN

RN

210

Chapter 10 = uf (et )ug (et ) = fb(t)b g (t). .

10.11.2. Remark. The essential steps of the above argument remain valid for every locally compact abelian group G, and so there is a one-to-one correspondence between the character space X(L1 (G)) of the group algebra and the set X(G) comprising all (unitary) group characters of G. Recall that such character is a continuous mapping ψ : G → C satisfying |ψ(x)| = 1,

ψ(x + y) = ψ(x)ψ(y)

(x, y ∈ G).

b := X(G) becomes a commutative Endowed with pointwise multiplication, the set G group. In virtue of the Alaoglu–Bourbaki Theorem, X(L1 (G)) is locally compact b may be treated as a locally compact in the weak topology σ((L1 (G))0 , L1 (G)). So, G abelian group called the character group of G or the dual group of G. Each element b 7→ qb(q) ∈ C of the dual group G. b The resulting q in G defines the character b qb : qb ∈ G bb embedding of G into G is surprisingly an isomorphism of the locally compact abelian bb groups G and G (the Pontryagin–van Kampen Duality Theorem). 10.11.3. Definition. For a function f in L1 (RN ), the mapping fb : RN → C, defined by the rule fb(t) := fb(t) := (f ∗ et )(0), is the Fourier transform of f . 10.11.4. Remark. By way of taking convenient liberties, we customarily use the term “Fourier transform” expansively. First, it is retained not only for the operN ator F : L1 (RN ) → C R acting by the rule F f := fb but also for its modifications (cf. 10.11.13). Second, the Fourier transform F is identified with each operator Fθ f := fb ◦ θ, where θ is an automorphism (= isomorphism with itself) of RN . Among the most popular are the functions: θ(x) := ∼ (x) := −x, θ(x) := 2π (x) := 2πx and θ(x) := −2π (x) := −2πx (x ∈ RN ). In other words, the Fourier transform is usually defined by one the following formulas: F∼ f (t) =

Z

f (x)e−i(x,t) dx,

RN

F2π f (t) =

Z

f (x)e2πi(x,t) dx,

RN

F−2π f (t) =

Z RN

f (x)e−2πi(x,t) dx.

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Since the character groups of isomorphic groups are also isomorphic, there are grounds to using the same notation fb for generally distinct functions F f , F∼ f , and F±2π f . The choice of the symbol b for F2π (F−2π ) dictates the denotation ∨ for F−2π (F2π ) (cf. 10.11.12). 10.11.5. Examples. (1) Let f : R0 = 0 O R be the characteristic function of the interval [−1, 1]. Clearly, fb(t) = 2t−1 sin t. Observe that if kπ ≥ t0 > 0 then Z

Z |fb(t)| dt ≥

[t0 ,+∞)

≥

∞ X

|fb(t)| dt =

∞ X

Z |fb(t)| dt

n=k [nπ,(n+1)π]

[kπ,+∞)

∞

Z

n=k [nπ,(n+1)π]

X 1 2| sin t| dt = 4 = +∞. (n + 1)π (n + 1)π n=k

Thus, fb ∈ / L1 (R). kfbk∞

(2) For f ∈ L1 (RN ) the function fb is continuous, with the inequality ≤ kf k1 holding.

C The continuity of fb follows from the Lebesgue Dominated Convergence Theorem; and the boundedness of fb, from the obvious estimate Z |fb(t)| ≤

|f (x)| dx = kf k1

(t ∈ RN ). .

RN

(3) If f ∈ L1 (RN ), then |fb(t)| → 0 as |t| → +∞ (= the Riemann– Lebesgue Lemma). / The claim is obvious for compactly-supported step functions. It suffices to refer to 5.5.9 (6) and the containment F ∈ B(L1 (RN ), l∞ (RN )). . (4) Let f ∈ L1 (RN ), ε > 0, and fε (x) := f (εx) (x ∈ RN ). Then fbε (t) = ε−N fb(t /ε ) (t ∈ RN ). R R / fbε (t) = f (εx)et (x) dx = ε−N f (εx)et/ε (εx) dεx = ε−N fb(t /ε ) . RN

RN

N b d b (5) F (f ∗ ) = (F∼ f )∗ , τd x f = ex f , and ex f = τx f for all f ∈ L1 (R ) and x ∈ RN .

212

Chapter 10

C We will only check the first equality. Since a∗ b = (ab∗ )∗ for a, b ∈ C; on using the properties of conjugation and integration, given t ∈ RN , infer that  ∗ Z Z
∗ F (f ∗ )(t) = f (x)ei(x,t) dx =  f (x) ei(x,t) dx RN

RN

∗

 Z

f (x)e−i(x,t) dx = (F∼ f )∗ (t). .

= RN

(6) If f, g ∈ L1 (RN ) then Z

Z

f[ ∗ g = fbgb;

fbg = RN

f gb. RN

C The first equality is straightforward from 10.11.1. The second, the multiplication formula, follows on applying the Fubini Theorem: Z Z Z b fg = f (x)et (x) dxg(t) dt RN

RN RN





Z =

Z g(t)et (x) dt f (x) dx =

 RN

Z

RN

f gb. . RN

(7) If fb, f and g belong to L1 (RN ) then (fbg)b = f e∗ gb. C Given x ∈ RN , observe that Z Z Z (fbg)b(x) = g(t)fb(t)et (x) dt = g(t)f (y)et (y)et (x) dydt RN

RN RN

Z Z =

f (y)g(t)et (x + y) dtdy RN RN

Z =

Z f (y − x)b g (y) dy = f e∗ g(x). .

f (y)b g (x + y) dy = RN

RN

(8) If f ∈ D(RN ) and α ∈ (Z+ )N then F (∂ α f ) = i|α| tα F f, F2π (∂ α f ) = (2πi)|α| tα F2π f,

∂ α (F f ) = i|α| F (xα f ); ∂ α (F2π f ) = (2πi)|α| F2π (xα f )

Duality and Its Applications

213

(these equalities take the rather common liberty of designating xα := tα := (·)α : αN y ∈ RN 7→ y1α1 · . . . · yN ). C It suffices (cf. 10.11.4) to examine only the first row. Since ∂ α et = i|α| tα et ; therefore,
F (∂ α f )(t) = et ∗ ∂ α f (0)
= ∂ α et ∗ f (0) = i|α| tα (et ∗ f )(0) = i|α| tα fb(t). By analogy, on differentiating under the integral sign, infer that Z Z ∂ ∂ i(x,t) f (x)ix1 ei(x,t) dx = F (ix1 f )(t). . (F f )(t) = f (x)e dx = ∂t1 ∂t1 RN

RN


N (9) If fN (x) := exp −1 /2 |x|2 for x ∈ RN , then fbN = (2π) /2 fN . C It is clear that N Z Y

fbN (t) =

eitk xk e−

1

/2 |x|2k

dxk

(t ∈ RN ).

k=1 R

Consequently, the matter reduces to the case N = 1. Now, given y ∈ R, observe that Z fb1 (y) =

e

−1 /2 x2 ixy

e

Z dx =

R

e−

1

/2 (x−iy)2 −1 /2 y 2

dx

R

Z = f1 (y)

1

e−

/2 (x−iy)2

dx.

R

To calculate the integral A that we are interested in, consider (concurrently oriented) straight lines λ1 and λ2 parallel to the real axis R in the complex plane C R ' R2 . Applying
the Cauchy Integral Theorem to the holomorphic function 2 f (z) := exp − z /2 (z ∈ C ) and a rectangular with vertices on λ1 and λ2 and R R properly passing to the limit, conclude that λ1 f (z) dz = λ2 f (z) dz. Whence it follows that Z A=

e R

−1 /2 (x−iy)2

Z dx = R

e−

1

/2 x 2

dx =

√

2π. .

214

Chapter 10

10.11.6. Definition. The Schwartz space is the set of tempered (or rapidly decreasing) functions, cf. 10.11.17 (2),  S (RN ) := f ∈ C∞ (RN ) : (∀α, β ∈ (Z+ )N ) |x| → +∞ ⇒ xα ∂ β f (x) → 0 with the multinorm {pα,β : α, β ∈ (Z+ )N }, where pα,β (f ) := kxα ∂ β f k∞ . This space is treated as a subspace of the space of all functions from RN to C. 10.11.7. The following statements are valid: (1) S (RN ) is a Fr´echet space; (2) the operators of multiplication by a polynomial and differentiation are continuous endomorphisms of S (RN ); (3) the topology of S (RN ) may equivalently be given by the multinorm {pn : n ∈ N}, where pn (f ) :=

X

k(1 + | · |2 )n ∂ α f k∞

(f ∈ S (RN ))

|α|≤n

(as usual, |x| stands for the Euclidean norm of a vector x in RN ); (4) D(RN ) is dense in S (RN ); furthermore, the identical embedding of D(R ) into S (RN ) is continuous and S (RN )0 ⊂ D 0 (RN ); N

(5) S (RN ) ⊂ L1 (RN ). C We will check (4), because the other claims are easier. Take f ∈ S (RN ) and let ψ be a truncator in D(RN ) such that B ⊂ {ψ = 1}. Given x ∈ RN and ξ > 0, put ψξ (x) := ψ(ξx), and fξ = ψξ f. Evidently, fξ ∈ D(RN ). Let ε > 0 and α, β ∈ (Z+ )N . It is an easy matter to show that sup{k∂ γ (ψξ −1)k∞ : γ ≤ β, γ ∈ (Z+ )N } < +∞ for 0 < ξ ≤ 1. Considering that xα ∂ β f (x) → 0 as |x| → +∞, find r > 1 satisfying |xα ∂ β ((ψξ (x) − 1)f (x))| < ε whenever |x| > r. Moreover, fξ (x) − f (x) = (ψ(ξx) − 1)f (x) = 0 for |x| ≤ ξ −1 . Therefore, pα,β (fξ − f ) = sup |xα ∂ β ((ψξ (x) − 1)f (x))| |x|>ξ −1

≤ sup |xα ∂ β ((ψξ (x) − 1)f (x))| < ε |x|>r

for ξ ≤ r−1 . Thus, pα,β (fξ − f ) → 0 as ξ → 0; i.e., fξ → f in S (RN ). The required continuity of the identical embedding raises no doubt. B

Duality and Its Applications

215

10.11.8. The Fourier transform is a continuous endomorphism of S (RN ). C Given f ∈ D(RN ), from 10.11.5 (8), 10.11.5 (2) and the H¨older inequality obtain ktα fbk∞ = k(∂ α f )bk∞ ≤ k∂ α f k1 ≤ Kk∂ α f k∞ . Thus, ktα ∂ β fbk∞ = ktα (xβ f )bk∞ ≤ K 0 k∂ α (xβ f )k∞ . Whence it is easily seen that fb ∈ S (RN ) and the restriction of F to D(Q) with Q b RN is continuous. It remains to refer to 10.10.7 (4) and 10.11.7 (4). B 10.11.9. Theorem. The repeated Fourier transform in the space S (RN ) is proportional to the reflection.
C Let f ∈ S (RN ) and g(x) := fN (x) = exp −1 /2 |x|2 . From 10.11.8 and 10.11.7 derive that fb, f , g ∈ L1 (RN ) and so, by 10.11.5 (7), (fbg)b = f e∗ gb. Put gε (x) := g(εx) for x ∈ RN and ε > 0. Then for the same x, in view of 10.11.5 (4) find Z Z Z y 1 b g(εt)f (t)et (x) dt = N f (y − x)b g dy = f (εy − x)b g (y) dy. ε ε RN

RN

RN

Using 10.11.5 (9) and the Lebesgue Dominated Convergence Theorem as ε → 0, infer that Z g(0)

Z fb(t)et (x) dt = f (−x)

RN N

= (2π)

/2

gb(y) dy

RN

Z f (x)

e−

1

/2 |x|2

dx = (2π)N f (−x).

RN

Finally, F 2 f = (2π)N f e. B 2 10.11.10. Corollary. F2π is the reflection and (F2π )−1 = F−2π . C Given f ∈ S (RN ) and t ∈ RN , deduce that

f (−t) = (2π)

N

Z e RN

i(x,t)

Z fb(x) dx = RN

= (F2π (F2π f ))(t). Since F2π f e= F−2π f , the proof is complete. B

e2πi(x,t) fb(2πx) dx

216

Chapter 10

10.11.11. Corollary. S (RN ) is a convolution algebra (= an algebra with convolution as multiplication). C For f , g ∈ S (RN ) the product f g is an element of S (RN ) and so fbgb ∈ S (RN ). From 10.11.5 (6) infer that F2π (f ∗ g) ∈ S (RN ) and, consequently, by 10.11.10, f ∗ g = F−2π (F2π (f ∗ g)) ∈ S (RN ). B 10.11.12. Inversion Theorem. The Fourier transform F := F2π is a topological automorphism of the Schwartz space S (RN ), with convolution carried into pointwise multiplication. The inverse transform F−1 equals F−2π , with pointwise multiplication carried into convolution. Moreover, the Parseval identity holds: Z

∗

Z

RN

(f, g ∈ S (RN )).

fbgb ∗

fg = RN

C In view of 10.11.10 and 10.11.5 (5), only the sought identity needs examining. Moreover, given f and g, from 10.11.5 (7) and 10.11.7 (4), obtain (fbg)(0) = (fe ∗ gb)(0). Using 10.11.5 (5), conclude that Z

f g ∗ = (F(F−1 f )g ∗ )b(0) = ((F−1 f )e∗ Fg ∗ )(0)

RN

Z =

∗

Z

Ff (Fg )edx = RN

∗

Z

Ff F∼ (g ) dx = RN

Ff (Fg)∗ . .

RN

10.11.13. Remark. In view of 10.11.9, the theorem on the repeated Fourier transform, the following mutually inverse operators 1 Ff (t) = (2π)N /2

Z

f (x)ei(x,t) dx;

RN

F

−1

1 f (x) = (2π)N /2

Z

f (t)e−i(x,t) dt

RN

are considered alongside F. In this case, an analog of 10.11.12 is valid on condition N that convolution is redefined as f ∗ g := (2π)− /2 f ∗g (f, g ∈ L1 (RN )). The merits −1 of F and F are connected with some simplification of 10.11.5 (8). In the case of F, a similar goal is achieved by introducing the differential operator Dα := (2πi)−|α| ∂ α with α ∈ (Z+ )N .

Duality and Its Applications

217

10.11.14. Plancherel Theorem. The Fourier transform in the Schwartz space S (RN ) is uniquely extendible to an isometric automorphism of L2 (RN ). C Immediate from 10.11.12 and 4.5.10 since S (RN ) is dense in L2 (RN ). B 10.11.15. Remark. The extension, guaranteed by 10.11.14, retains the previous name and notation. Rarely (in search of emphasizing distinctions and subtleties) one speaks of the Fourier–Plancherel transform or the L2 -Fourier transform. It that event it stands to reason to specify the understanding of the integral formulas for Ff and F−1 f with f ∈ L2 (RN ) which are treated as the results of appropriate passage to the limit in L2 (RN ). 10.11.16. Definition. Let u ∈ S 0 (RN ) := S (RN )0 . Such a distribution u is referred to as a tempered distribution (variants: a distribution of slow growth, a slowly increasing distribution, etc.). The space S 0 (RN ) of tempered distributions is furnished with the weak topology σ(S 0 (RN ), S (RN )) and is sometimes called the Schwartz space (as well as S (RN )). 10.11.17. Examples. (1) Lp (RN ) ⊂ S 0 (RN ) for 1 ≤ p ≤ +∞. C Let f ∈ Lp (RN ), ψ ∈ S (RN ), p < +∞ and 1 /p0 + 1 /p = 1. Using H¨older inequality, for suitable positive K, K 0, and K 00 successively infer that 1 /p

 Z kψkp0 ≤  B

|ψ|p 

1 /p

 Z  +

(1 + |x|2 )N (1 + |x|2 )−N ψ(x) p dx 

R N\ B

 1 /p

  ≤ K 0 kψk∞ + k(1 + | · |2 )N ψk∞ 

Z

dx   (1 + |x|2 )N p

≤ K 00 p1 (ψ).

R N\ B

Once again using the H¨ older inequality, observe that Z |uf (ψ)| = |hψ | f i| = f ψ ≤ kf kp kψkp0 ≤ Kp1 (ψ). N R

The case p = +∞ raises no doubts. B (2) S (RN ) is dense in S 0 (RN ). C Follows from 10.11.7 (4), 10.11.17 (1), 10.11.7 (5), and 10.10.9 (4). B

218

Chapter 10 (3) Let µ ∈ M (RN ) be a tempered Radon measure; i.e., Z

d|µ|(x) < +∞ (1 + |x|2 )n

RN

for some n ∈ N. Evidently, µ is a tempered distribution. (4) If u ∈ S 0 (RN ), f ∈ S (RN ) and α ∈ (Z+ )N then f u ∈ S 0 (RN ) and ∂ α u ∈ S 0 (RN ) in virtue of 10.11.7 (2). By a similar argument, putting Dα u(f ) := (−1)|α| uDα f for f ∈ S (RN ), infer that Dα u ∈ S 0 (RN ) and Dα u = (2πi)−|α| ∂ α u. (5) Every compactly-supported distribution is tempered. C In accordance with 10.10.5 (7) such u in D 0 (RN ) may be identified with a member of E 0 (RN ). Since the topology of S (RN ) is stronger than that induced by the identical embedding in C∞ (RN ), conclude that u ∈ S 0 (RN ). B (6) Let u ∈ S 0 (RN ). If f ∈ S (RN ) then u convolutes with f and u ∗ f ∈ S (RN ). In may be shown that u also convolutes with every distribution v, a member of E 0 (RN ), and u ∗ v ∈ S 0 (RN ). (7) Take u ∈ D 0 (RN ) and x ∈ RN . Let τx u := (τ−x )0 u = u ◦ τ−x be the corresponding translation of u. A distribution u is periodic (with period x) if τx u = u. Every periodic distribution is tempered. Periodicity is preserved under differentiation and convolution. 0 N ∈ N) and for every f ∈ S (RN ) there is a sum P∞(8) If un ∈ S (R ) (u P ∞ u(f ) := n=1 un (f ), then u ∈ S 0 (RN ) and ∂ α u = n=1 ∂ α un (cf. 10.10.10).

10.11.18. Theorem. Each tempered distribution is the sum of derivatives of tempered measures. C Let u ∈ S 0 (RN ). On account of 10.11.7 (3) and 5.3.7, for some n ∈ N and K > 0, observe that |u(f )| ≤ K

X

(1 + | · |2 )n ∂ α f ∞

(f ∈ S (RN )).

|α|≤n

From 3.5.3 and 3.5.7, for some µα ∈ M (RN ), obtain u(f ) =

X

µα (1 + | · |2 )n ∂ α f




(f ∈ S (RN )).

|α|≤n

Let να := (−1)|α| (1 + | · |2 )n µα . Then να is tempered and u =

P

|α|≤n

∂ α να . .

Duality and Its Applications

219

10.11.19. Definition. For u ∈ S 0 (RN ), the distribution Fu acting as hf | Fui = hFf | ui (f ∈ S (RN )) is the Fourier transform or, amply, the Fourier–Schwartz transform of u. 10.11.20. Theorem. The Fourier–Schwartz transform F is a unique extension of the Fourier transform in S (RN ) to a topological automorphism of S 0 (RN ). The inverse F−1 of F is a unique continuous extension of the inverse Fourier transform in S (RN ). C The Fourier–Schwartz transform in S 0 (RN ) is the dual of the Fourier transform in S (RN ). It remains only to appeal to 10.11.7 (5), 10.11.12, 10.11.17 (2), and 4.5.10. B Exercises 10.1. Give examples of linear topological spaces and locally convex spaces as well as constructions leading to them. 10.2. Prove that a Hausdorff topological vector space is finite-dimensional if and only if it is locally compact. 10.3. Characterize weakly continuous sublinear functionals. 10.4. Prove that the weak topology of a locally convex space is normable or metrizable if and only if the space is finite-dimensional. 10.5. Describe weak convergence in classical Banach spaces. 10.6. Prove that a normed space is finite-dimensional if and only if its unit sphere, comprising all norm-one vectors, is weakly closed. 10.7. Assume that an operator T carries each weakly convergent net into a norm convergent net. Prove that T has finite rank. 10.8. Let X and Y be Banach spaces and let T be a linear operator from X to Y . Prove that T is bounded if and only if T is weakly continuous (i.e., continuous as a mapping from (X, σ(X, X 0 )) to (Y, σ(Y, Y 0 ))). 10.9. Let k · k1 and k · k2 be two norms making X into a Banach space. Assume further that (X, k · k1 )0 ∩ (X, k · k2 )0 separates the points of X. Prove that these norms are equivalent. 10.10. Let S act from Y 0 to X 0 . When does S serve as the dual of some operator from X to Y ? 10.11. What is the Mackey topology τ (X, X # )? Q 10.12. Let (Xξ )ξ∈„ be a family of locally convex spaces, and let X := ξ∈„ Xξ be the product of the family. Validate the presentations: σ(X, X 0 ) =

Y ξ∈„

σ(Xξ , Xξ0 );

τ (X, X 0 ) =

Y

τ (Xξ , Xξ0 ).

ξ∈„

10.13. Let X and Y be Banach spaces and let T be a element of B(X, Y ) satisfying im T = Y . Demonstrate that Y is reflexive provided so is X.

220

Chapter 10

10.14. Show that the spaces (X 0 )00 and (X 00 )0 coincide. 10.15. Prove that the space c0 has no infinite-dimensional reflexive subspaces. 10.16. Let p be a continuous sublinear functional on Y , and let T ∈ L (X, Y ) be a continuous linear operator. Establish the following inclusion of the sets of extreme points: ext T 0 (∂p) ⊂ T 0 (ext ∂p). 10.17. Let p be a continuous seminorm on X and let X be a subspace of X. Prove that f ∈ ext(X ◦ ∩ ∂p) if and only if the next equality holds: X = cl X + {p − f ≤ 1} − {p − f ≤ 1}. 10.18. Prove that the absolutely convex hull of a totally bounded subset of a locally convex space is also totally bounded. 10.19. Establish that barreledness is preserved under passage to the inductive limit. What happens with other linear topological properties?

Chapter 11 Banach Algebras

11.1. The Canonical Operator Representation 11.1.1. Definition. An element e of an algebra A is called a unity element if e 6= 0 and ea = ae = a for all a ∈ A. Such an element is obviously unique and is also referred to as the unity or the identity or the unit of A. An algebra A is unital provided that A has unity. 11.1.2. Remark. Without further specification, we only consider unital algebras over a basic field F. Moreover, for simplicity, it is assumed that F := C, unless stated otherwise. In studying a representation of unital algebras, we naturally presume that it preserves unity. In other words, given some algebras A1 and A2 , by a representation of A1 in A2 we henceforth mean a morphism, a multiplicative linear operator, from A1 to A2 which sends the unity element of A1 to the unity element of A2 . For an algebra A without unity, the process of unitization or adjunction of unity is in order. Namely, the vector space Ae := A × C is transformed into an algebra by putting (a, λ)(b, µ) := (ab + µa + λb, λµ), where a, b ∈ A and λ, µ ∈ C. In the normed case, it is also taken for granted that k(a, λ)kAe := kakA + |λ|. 11.1.3. Definition. An element ar in A is a right inverse of a if aar = e. An element al of A is a left inverse of a if al a = e. 11.1.4. If an element has left and right inverses then the latter coincide. / ar = (al a)ar = al (aar ) = al e = al . 11.1.5. Definition. An element a of an algebra A is called invertible, in writing a ∈ Inv(A), if a has a left and right inverse. Denote a−1 := ar = al . The element a−1 is the inverse of a. A subalgebra (with unity) B of an algebra A is called pure or full or inverse-closed in A if Inv(B) = Inv(A) ∩ B.

222

Chapter 11

11.1.6. Theorem. Let A be a Banach algebra. Given a ∈ A, put La : x 7→ ax (x ∈ A). Then the mapping LA := L : a 7→ La

(a ∈ A)

is a faithful operator representation. Moreover, L(A) is a pure closed subalgebra of B(A) and L : A → L(A) is a topological isomorphism. C Clearly, L(ab) : x 7→ Lab (x) = abx = a(bx) = La (Lb x) = (La)(Lb)x for x, a, b ∈ A; i.e., L is a representation (because the linearity of L is obvious). If La = 0 then 0 = La(e) = ae = a, so that L is a faithful representation. To prove the closure property of the range L(A), consider the algebra Ar coinciding with A as a vector space and equipped with the reversed multiplication ab := ba (a, b ∈ A). Let R := LAr , i.e. Ra := Ra : x 7→ xa for a ∈ A. Check that L(A) is in fact the centralizer of the range R (A), i.e. the closed subalgebra Z(im R) := {T ∈ B(A) : T Ra = Ra T (a ∈ A)}. Indeed, if T ∈ L(A), i.e. T = La for some a ∈ A, then La Rb (x) = axb = Rb (La (x)) = Rb La (x) for all b ∈ A. Hence, T ∈ Z(R (A)). If, in turn, T ∈ Z(R (A)) then, putting a := T e, find La x = ax = (T e)x = Rx (T e) = (Rx T )e = (T Rx )e = T (Rx e) = T x for all x ∈ A. Consequently, T = La ∈ L(A). Thus, L(A) is a Banach subalgebra of B(A). For T = La there is a T −1 in B(A). Put b := T −1 e and observe that ab = La b = T b = T T −1 e = e. Moreover, ab = e ⇒ aba = a ⇒ T (ba) = La ba = aba = a = La e = T e. Whence ba = e, because T is a monomorphism. Thus, L(A) is a subalgebra of A. By the definition of a Banach algebra, the norm is submultiplicative, providing kLk = sup{kLak : kak ≤ 1} = sup{kabk : kak ≤ 1, kbk ≤ 1} ≤ 1. Using the Banach Isomorphism Theorem, conclude that L is a topological isomorphism (i.e., L−1 is a continuous operator from L(A) onto A). B 11.1.7. Definition. The representation LA , constructed in 11.1.6, is the canonical operator representation of A.

Banach Algebras

223

11.1.8. Remark. The presence of the canonical operator representation allows us to confine the subsequent exposition to studying Banach algebras with norm-one unity. For such an algebra A the canonical operator representation LA implements an isometric embedding of A into the endomorphism algebra B(A) or, in short, an isometric representation of A in B(A). In this case, LA implements an isometric isomorphism between the algebras A and L(A). The same natural terminology is used for studying representations of arbitrary Banach algebras. Observe immediately that the canonical operator representation of A, in particular, justifies the use of the symbol λ in place of λe for λ ∈ C, where e is the unity of A (cf. 5.6.5). In other words, henceforth the isometric representation λ 7→ λe is considered as the identification of C with the subalgebra Ce of A. 11.2. The Spectrum of an Element of an Algebra 11.2.1. Definition. Let A be a Banach algebra and a ∈ A. A scalar λ in C is a resolvent value of a, in writing λ ∈ res(a)), if (λ − a) ∈ Inv(A). The resolvent 1 R (a, λ) of a at λ is R (a, λ) := λ−a := (λ − a)−1 . The set Sp(a) := C \ res(a) is the spectrum of a, with a point of Sp(a) a spectral value of a. When it is necessary, more detailed designations like SpA (a) are in order. 11.2.2. For a ∈ A the equalities hold: SpA (a) = SpL(A) (La ) = Sp(La ); LR (a, λ) = R (La , λ)

(λ ∈ res(a) = res(La )). /.

11.2.3. Gelfand–Mazur Theorem. The field of complex numbers is up to isometric isomorphism the sole Banach division algebra (or skew field); i.e., each complex Banach algebra with norm-one unity and invertible nonzero elements has an isometric representation on C. C Consider ‰ : λ 7→ λe, with e the unity of A and λ ∈ C. It is clear that ‰ represents C in A. Take a ∈ A. By virtue of 11.2.2 and 8.1.11, Sp(a) 6= ∅. Consequently, there is λ ∈ C such that the element (λ − a) is not invertible; i.e., a = λe by hypothesis. Hence, ‰ is an epimorphism. Moreover, k‰(λ)k = kλek = |λ| kek = |λ| so that ‰ is an isometry. B 11.2.4. Shilov Theorem. Let A be a Banach algebra and let B be a closed (unital) subalgebra of A. Then SpB (b) ⊃ SpA (b), for all b ∈ B.

∂ SpA (b) ⊃ ∂ SpB (b)

224

Chapter 11 C If b := λ − b ∈ Inv(B) then surely b ∈ Inv(A). Whence resB (b) ⊂ resA (b);

i.e., SpB (b) = C \ resB (b) ⊃ C \ resA (b) = SpA (b). If λ ∈ ∂ SpB (b) then b ∈ ∂ Inv(B). Therefore,

−1 there is a sequence (bn ), bn ∈ Inv(B), convergent to b. Putting t := supn∈N bn , deduce that

−1

−1

−1

−1

bn − b−1

= bn (1 − bn b−1

≤ t2 bn − bm . m m ) = bn (bm − bn )bm In other words, if t < +∞ then there is a limit a := lim b−1 n in B. Multiplication is jointly continuous, and so ab = ba = 1; i.e., b ∈ Inv(B). Since Inv(B) is open by the Banach Inversion Stability Theorem and 11.1.6, arrive at a contradiction to the containment b ∈ ∂ Inv(B). Therefore, it may be assumed (on dropping to a subsequence, if need be) that

−1 −1

−1

bn . Then

bn → +∞. Put an := b−1 n



ban = (b − bn )an + bn an



−1 −1

bn bn → 0. ≤ b − bn kan k + b−1 n Whence it follows that b is not invertible. Indeed, in the opposite case for a := b it would hold that



1 = kan k = aban ≤ kak ban → 0.

−1

Finally, conclude that λ − b does not belong to Inv(A); i.e., λ ∈ SpA (b). Since λ is a boundary point of a larger set SpB (b); undoubtedly, λ ∈ ∂ SpA (b). B 11.2.5. Corollary. If SpB (b) lacks interior points then SpB (b) = SpA (b). / SpB (b) = ∂ SpB (b) ⊂ ∂ SpB (b) ⊂ ∂ SpA (b) ⊂ SpA (b) ⊂ SpB (b) . 11.2.6. Remark. The Shilov Theorem is often referred to as the Unremovable Spectral Boundary Theorem or Spectral Permanence Theorem and verbalized as follows: “A boundary spectral value is unremovable.” 11.3. The Holomorphic Functional Calculus in Algebras 11.3.1. Definition. Let a be an element of a Banach algebra A, and let h ∈ H (Sp(a)) be a germ of a holomorphic function on the spectrum of a. Put I 1 h(z) Ra h := dz. 2πi z−a The element Ra h of A is the Riesz–Dunford integral of h. If, in particular, f ∈ H(Sp(a)) is a function holomorphic in a neighborhood about the spectrum of a, then f (a) := Ra f .

Banach Algebras

225

11.3.2. Gelfand–Dunford The Riesz–Dunford integral Ra represents the algebra of germs of holomorphic functions on the spectrum of an element a of a Banach P∞ algebra P A in A. Moreover, if f (z) := n=0 cn z n (in a neighborhood of Sp(a)) then ∞ f (a) := n=0 cn an . From 11.2.3, 8.2.1, and 11.2.2 obtain (LRa h)(b) = LRa h b = (Ra h)b I 1 1 = h(z)R (a, z) dzb = h(z)R (a, z) bdz 2πi 2πi I I 1 1 = h(z)R (La , z) bdz = h(z)R (La , z) dzb 2πi 2πi = RLa h(b) I

for all b ∈ A. In particular, im L includes the range of RLa (H (Sp(a))). Therefore, the already-proven commutativity of the diagram

H (Sp(a)) @ @ R RLa @ a @ ? R @ L B(A)  A implies that the following diagram also commutes:

H (Sp(a)) @ @ R RLa @ a @ ? −1 @ R -A L(A) L It remains to appeal to 11.1.6 and the Gelfand–Dunford Theorem in an operator setting. B 11.3.3. Remark. All that we have established enables us to use in the sequel the rules of the holomorphic functional calculus which were exposed in 8.2 for the endomorphism algebra B(X), with X a Banach space.

226

Chapter 11 11.4. Ideals of Commutative Algebras

11.4.1. Definition. Let A be a commutative algebra. A subspace J of A is an ideal of A, in writing J C A, provided that AJ ⊂ J. 11.4.2. The set J(A) of all ideals of A, ordered by inclusion, is a complete lattice. Moreover, supJ(A) E = supLat(A) E ,

inf J(A) E = inf Lat(A) E ,

for every subset E of J(A); i.e., J(A) is embedded into the complete lattice Lat(A) of all subspaces of A with preservation of suprema and infima of arbitrary subsets. C Clearly, 0 is the least ideal, whereas A is the greatest ideal. Furthermore, the intersection of ideals and the sum of finitely many ideals are ideals. It remains to refer to 2.1.5 and 2.1.6. B 11.4.3. Let J0 C A. Assume further that ϕ : A → A/J0 is the coset mapping of A onto the quotient algebra A := A/J0 . Then J C A ⇒ ϕ(J) C A;

J C A ⇒ ϕ−1 (J) C A.

C Since by definition ab := ϕ(ϕ−1 (a)ϕ−1 (b)) for a, b ∈ A, the operator ϕ is multiplicative: ϕ(ab) = ϕ(a)ϕ(b) for a, b ∈ A. Whence successively derive ϕ(J) ⊂ Aϕ(J) = ϕ(A)ϕ(J) ⊂ ϕ(AJ) ⊂ ϕ(J); −1

ϕ

(J) ⊂ Aϕ−1 (J) ⊂ ϕ−1 (ϕ(A)J) = ϕ−1 (AJ) ⊂ ϕ−1 (J). .

11.4.4. Let J C A and J 6= 0. The following conditions are equivalent: (1) A 6= J; (2) 1 ∈ / J; (3) no element of J has a left inverse. CB 11.4.5. Definition. An ideal J of A is called proper if J is other than A. A maximal element of the set of proper ideals (ordered by inclusion) is a maximal ideal. 11.4.6. A commutative algebra is a division algebra if and only if it has no proper ideals other than zero. CB 11.4.7. Let J be a proper ideal of A. Then (J is maximal) ⇔ (A/J is a field).

Banach Algebras

227

C ⇒: Let J C A/J. Then, by 11.4.3, ϕ−1 (J) C A. Since, beyond a doubt, J ⊂ ϕ−1 (J); therefore, either J = ϕ−1 (J) and 0 = ϕ(J) = ϕ(ϕ−1 (J)) = J, or A = ϕ−1 (J) and J = ϕ(ϕ−1 (J)) = ϕ(A) = A/J in virtue of 1.1.6. Consequently, A/J has no proper ideals other than zero. It remains to refer to 11.4.6. ⇐: Let J0 C A and J0 ⊂ J. Then, by 11.4.3, ϕ(J0 ) C A/J. In virtue of 11.4.6, either ϕ(J0 ) = 0 or ϕ(J0 ) = A/J. In the first case, J0 ⊂ ϕ−1 ◦ ϕ(J0 ) ⊂ ϕ−1 (0) = J and J = J0 . In the second case, ϕ(J0 ) = ϕ(A); i.e., A = J0 + J ⊂ J0 + J0 = J0 ⊂ A. Thus, J is a maximal ideal. B 11.4.8. Krull Theorem. Each proper ideal is included in some maximal ideal. C Let J0 be a proper ideal of an algebra A. Assume further that E is the set comprising all proper ideals J of A such that J0 ⊂ J. In virtue of 11.4.2 each chain E0 in E has a least upper bound: sup E = ∪ {J : J ∈ E0 }. By 11.4.4 the ideal sup E0 is proper. Thus, E is inductive and the claim follows from the Kuratowski–Zorn Lemma. B 11.5.

Ideals of the Algebra C(Q, C)

11.5.1. Minimal Ideal Theorem. Let J be an ideal of the algebra C(Q, C) of complex continuous functions on a compactum Q. Assume further that Q0 := ∩ {f −1 (0) : f ∈ J}; J0 := {f ∈ C(Q, C) : int f −1 (0) ⊃ Q0 }. Then J0 C C(Q, C) and J0 ⊂ J. C Let Q1 := cl(Q \ f −1 (0)) for a function f , a member of J0 . By hypothesis, Q1 ∩ Q0 = ∅. To prove the containment f ∈ J it is necessary (and, certainly, sufficient) to find u ∈ J satisfying u(q) = 1 for all q ∈ Q1 . Indeed, in that event uf = f . With this in mind, observe first that for q ∈ Q1 there is a function fq in J such that fq (q) 6= 0. Putting gq := fq∗ fq , where as usual fq∗ : x 7→ fq (x)∗ is the conjugate of fq , observe that gq ≥ 0 and, moreover, gq (q) > 0. It is also clear that gq ∈ J for q ∈ Q1 . The family (Uq )q∈Q1 , with Uq := {x ∈ Q1 : gq (x) > 0}, is an open cover of Q1 . Using a standard compactness argument, choose a finite subset {q1 , . . . , qn } of Q1 such that Q1 ⊂ Uq1 ∪ . . . ∪ Uqn . Put g := gq1 + . . . + gqn . Undoubtedly, g ∈ J and g(q) > 0 for q ∈ Q1 . Let h0 (q) := g(q)−1 for q ∈ Q1 . By the Tietze–Urysohn Theorem, there is a function h in C(Q, R) satisfying h Q1 = h0 . Finally, let u := hg. This u is a sought function. We have thus demonstrated that J0 ⊂ J. Moreover, J0 is an ideal of C(Q, C) for obvious reasons. B

228

Chapter 11

11.5.2. For every closed ideal J of the algebra C(Q, C ) there is a unique compact subset Q0 of Q such that J = J(Q0 ) := {f ∈ C(Q, C) : q ∈ Q0 ⇒ f (q) = 0}. C Uniqueness follows from the Urysohn Theorem. Define Q0 as in 11.5.1. Then, surely, J ⊂ J(Q0 ). Take f ∈ J(Q0 ) and, given n ∈ N, put   Un := |f | ≤ 1 /2n , Vn := |f | ≥ 1 /n . Once again using the Urysohn Theorem, find hn ∈ C(Q, R) satisfying 0 ≤ hn ≤ 1 with hn U = 0 and hn V = 1. Consider fn := f hn . Since n

n

int fn−1 (0) ⊃ int Un ⊃ Q0 , therefore, from 11.5.1 derive fn ∈ J. It suffices to observe that fn → f by construction. B 11.5.3. Maximal Ideal Theorem. A maximal ideal of C(Q, C ) has the form J(q) := J({q}) = {f ∈ C(Q, C) : f (q) = 0}, with q a point of Q. C Follows from 11.5.2, because the closure of an ideal is also an ideal. B 11.6. The Gelfand Transform 11.6.1. Let A be a commutative Banach algebra, and let J be a closed ideal of A other than A. Then the quotient algebra A/J, endowed with the quotient norm, is a Banach algebra. If ϕ : A → A/J is the coset mapping then ϕ(1) is the unity of A/J, the operator ϕ is multiplicative and kϕk = 1. C Given a, b ∈ A, from 5.1.10 (5) derive kϕ(a)ϕ(b)kA/J = inf{ka0 b0 kA : ϕ(a0 ) = ϕ(a), ϕ(b0 ) = ϕ(b)} ≤ inf{ka0 kA kb0 kA : ϕ(a0 ) = ϕ(a), ϕ(b0 ) = ϕ(b)} = kϕ(a)kA/J kϕ(b)kA/J . In other words, the norm of A/J is submultiplicative. Consequently, kϕ(1)k ≥ 1. Furthermore, kϕ(1)kA/J = inf{kakA : ϕ(a) = ϕ(1)} ≤ k1kA = 1; i.e., kϕ(1)k = 1. Whence, in particular, the equality kϕk = 1 follows. The remaining claims are evident. B

Banach Algebras

229

11.6.2. Remark. The message of 11.6.1 remains valid for a noncommutative Banach algebra A under the additional assumption that J is a bilateral ideal of A; i.e., J is a subspace of A satisfying the condition AJA ⊂ J. 11.6.3. Let χ : A → C be a nonzero multiplicative linear functional on A. Then χ is continuous and kχk = χ(1) = 1 (in particular, χ is a representation of A in C). C Since χ 6= 0; therefore, 0 6= χ(a) = χ(a1) = χ(a)χ(1) for some a in A. Consequently, χ(1) = 1. If now a in A and λ in C are such that |λ| > kak, then λ−a ∈ Inv(A) (cf. 5.6.15). So, 1 = χ(1)χ(λ−a)χ((λ−a)−1 ). Whence χ(λ−a) 6= 0; i.e., χ(a) 6= λ. Thus, |χ(a)| ≤ kak and kχk ≤ 1. Since kχk = kχk k1k ≥ |χ(1)| = 1, conclude that kχk = 1. B 11.6.4. Definition. A nonzero multiplicative linear functional on an algebra A is a character of A. The set of all characters of A is denoted by X(A), furnished with the topology of pointwise convergence (induced in X(A) by the weak topology σ(A0 , A)) and called the character space of A. 11.6.5. The character space is a compactum. C It is beyond a doubt that X(A) is a Hausdorff space. By virtue of 11.6.3, X(A) is a σ(A0 , A)-closed subset of the ball BA0 . The latter is σ(A0 , A)-compact by the Alaoglu–Bourbaki Theorem. It remains to refer to 9.4.9. B 11.6.6. Ideal and Character Theorem. Each maximal ideal of a commutative Banach algebra A is the kernel of a character of A. Moreover, the mapping χ 7→ ker χ from the character space X(A) onto the set M(A) of all maximal ideals of A is one-to-one. C Let χ ∈ X(A) be a character of A. Clearly, ker χ C A. From 2.3.11 it follows that the monoquotient χ : A/ ker χ → C of χ is a monomorphism. In view of 11.6.1, χ(1) = χ(1) = 1; i.e., χ is an isomorphism of A/ ker χ and C. Consequently, A/ ker χ is a field. Using 11.4.7, infer that ker χ is a maximal ideal; i.e., ker χ ∈ M(A). Now, let m ∈ M(A) be some maximal ideal of A. It is clear that m ⊂ cl m, cl m C A, and 1 ∈ / cl m (because 1 ∈ Inv(A), and the last set is open by the Banach Inversion Stability Theorem and 11.1.6). Therefore, the ideal m is closed. Consider the quotient algebra A/m and the coset mapping ϕ : A → A/m. In view of 11.4.7 and 11.6.1, A/m is a Banach field. By the Gelfand–Mazur Theorem, there is an isometric representation ψ : A/m → C. Put χ := ψ ◦ ϕ. It is evident that χ ∈ X(A) and ker χ = χ−1 (0) = ϕ−1 (ψ −1 (0)) = ϕ−1 (0) = m. To complete the proof, it suffices to show that the mapping χ 7→ ker χ is oneto-one. Indeed, let ker χ1 = ker χ2 for χ1 , χ2 ∈ X(A). By 2.3.11 χ1 = λχ2 for some λ ∈ C. Furthermore, by 11.6.3, 1 = χ1 (1) = λχ2 (1) = λ. Finally, χ1 = χ2 . B

230

Chapter 11

11.6.7. Remark. Theorem 11.6.6 makes it natural to furnish M(A) with the inverse image topology translated from X(A) to M(A) by the mapping χ 7→ ker χ. In this regard, M(A) is referred to as the compact maximal ideal space of A. In other words, the character space and the maximal ideal space are often identified along the lines of 11.6.6. 11.6.8. Definition. Let A be a commutative Banach algebra and let X(A) be the character space of A. Given a ∈ A and χ ∈ X(A), put b a(χ) := χ(a). The resulting function b a : χ 7→ b a(χ), defined on X(A), is the Gelfand transform of a. The mapping a 7→ b a, with a ∈ A is the Gelfand transform of A, denoted by GA (or b ). 11.6.9. Gelfand Transform Theorem. The Gelfand transform GA : a 7→ b a is a representation of a commutative Banach algebra A in the algebra C(X(A), C ). Moreover, Sp(a) = Sp(b a) = b a(X(A)), kb ak = r(a), with a ∈ A and r(a) standing for the spectral radius of a. C The implications a ∈ A ⇒ b a ∈ C(X(A), C ), b 1 = 1 and a, b ∈ A ⇒ b b ab = b ab follow from definitions and 11.6.3. The linearity of GA raises no doubts. Consequently, the mapping GA is a representation. To begin with, take λ ∈ Sp(a). Then λ − a is not invertible, and so the ideal Jλ−a := A(λ − a) is proper in virtue of 11.4.4. By the Krull Theorem, there is a maximal ideal m of A satisfying the condition m ⊃ Jλ−a . By Theorem 11.6.6, m = ker χ for a suitable character χ. In particular, χ(λ − a) = 0; i.e., λ = λχ(1) = χ(λ) = χ(a) = b a(χ). Consequently, λ ∈ Sp(b a). If, in turn, λ ∈ Sp(b a) then (λ − b a) is not invertible in C(X(A), C); i.e., there is a character χ ∈ X(A) such that λ = b a(χ). In other words, χ(λ − a) = 0. Thus, the assumption λ − a ∈ Inv(A) leads to the following contradiction: 1 = χ(1) = χ((λ − a)−1 (λ − a)) = χ((λ − a)−1 )χ(λ − a) = 0. Hence, λ ∈ Sp(a). Finally, Sp(a) = Sp(b a). Using the Beurling–Gelfand formula (cf. 11.3.3 and 8.1.12), infer that r(a) = sup{|λ| : λ ∈ Sp(a)} = sup{|λ| : λ ∈ Sp(b a)} = sup{|λ| : λ ∈ b a(X(A))} = sup{|b a(χ)| : χ ∈ X(A)} = kb ak, what was required. B

Banach Algebras

231

11.6.10. The Gelfand transform of a commutative Banach algebra A is an isometric embedding if and only if ka2 k = kak2 for all a ∈ A. C ⇒: The mapping t 7→ t2 , viewed as acting on R+ , and the inverse of the mapping on R+ are both increasing. Therefore, from 10.6.9 obtain ka2 k = kb a2 kC(X(A),C ) = sup |b a2 (χ)| = sup |χ(a2 )| χ∈X(A)

χ∈X(A)

= sup |χ(a)χ(a)| = sup |χ(a)|2 χ∈X(A)

χ∈X(A)

!2 =

sup |χ(a)|

= kb ak2 = kak2 .

χ∈X(A) 1

Gelfand formula, r(a) = lim kan k /n . In particular, observe that

2n ⇐:

By the n

a = kak2 ; i.e., r(a) = kak. By 10.6.9, r(a) = kb ak, completing the proof. B 11.6.11. Remark. It is interesting sometimes to grasp situations in which the Gelfand transform of A is faithful but possibly not isometric. The kernel of the Gelfand transform GA is the intersection of all maximal ideals, called the radical of A. Therefore, the condition for GA to be a faithful representation of A in C(X(A), C ) reads: “A is semisimple” or, which is the same, “The radical of A is trivial.” 11.6.12. Theorem. For an element a of a commutative Banach algebra A the following diagram of representations commutes: H (Sp(a)) = H (Sp(b a)) @ @ R Ra @ ^a @ ? R GA -@ C(X(A), C) A Moreover, f (b a) = f ◦ b a = f (b a) for f ∈ H(Sp(a)). C Take χ ∈ X(A). Given z ∈ res(a), observe that     1 1 1 1 χ (z − a) = 1 ⇒ χ = = . z−a z−a χ(z − a) z − χ(a) In other words, R\ (a, z)(χ) =

d 1 1 1 (χ) = = (χ) = R (b a, z)(χ). z−a z−b a(χ) z−b a

232

Chapter 11

Therefore, appealing to the properties of the Bochner integral (cf. 5.5.9 (6)) and given f ∈ H(Sp(a)), infer that 

1 2πi

I



fd (a) = GA ◦ Ra f = GA f (z)R (a, z) dz I I 1 1 f (z)GA (R (a, z)) dz = f (z)R\ (a, z)) dz = 2πi 2πi I 1 = f (z)R (b a, z) dz = R^a (f ) = f (b a). 2πi Furthermore, given χ ∈ X(A) and using the Cauchy Integral Formula, derive the following chain of equalities f ◦b a(χ) = f (b a(χ)) = f (χ(a))  I I  1 f (z) 1 f (z) dz = χ dz = 2πi z − χ(a) 2πi z−a I  1 f (z) = χ dz = fd (a)(χ) = f (b a)(χ). . 2πi z−a 11.6.13. Remark. The theory of the Gelfand transform may be naturally generalized to the case of a commutative Banach algebra A without unity. Retain Definitions 11.6.4 and 11.6.8 verbatim. A character χ in X(A) generates some character χe in X(Ae ) by the rule χe (a, λ) := χ(a) + λ (a ∈ A, λ ∈ C). The set X(Ae ) \ {χe : χ ∈ X(A)} is a singleton consisting of the sole element χ∞ (a, λ) := λ (a ∈ A, λ ∈ C ). The space X(A) is locally compact (cf. 9.4.19), because the mapping χ ∈ X(A) 7→ χe ∈ X(Ae )\{χ∞ } is a homeomorphism. Moreover, ker χ∞ = A × 0. Consequently, the Gelfand transform of a commutative Banach algebra without unity represents it in the algebra of continuous complex functions defined on a locally compact space and vanishing at infinity. Given the group algebra (L1 (RN ), ∗), observe that by 10.11.1 and 10.11.3 the Fourier transform coincides with the Gelfand transform, which in turn entails the Riemann–Lebesgue Lemma as well as the multiplication formula 10.11.5 (6). 11.7.

The Spectrum of an Element of a C ∗ -Algebra

11.7.1. Definition. An element a of an involutive algebra A is called hermitian if a∗ = a. An element a of A is called normal if a∗ a = aa∗ . Finally, an element a is called unitary if aa∗ = a∗ a = 1 (i.e. a, a∗ ∈ Inv(A) with a−1 = a∗ and a∗−1 = a).

Banach Algebras

233

11.7.2. Hermitian elements of an involutive algebra A compose a real subspace of A. Moreover, for every a in A there are unique hermitian elements x, y ∈ A such that a = x + iy. Namely, 1 1 x = (a + a∗ ), y = (a − a∗ ). 2 2i ∗ Moreover, a = x − iy. C Only the claim of uniqueness needs examining. If a = x1 + iy1 then, using the properties of involution (cf. 6.4.13), proceed as follows: a∗ = x∗1 + (iy1 )∗ = x∗1 − iy1∗ = x1 − iy1 . Thus, x1 = x and y1 = y. B 11.7.3. The unity of A is a hermitian element of A. / 1∗ = 1∗ 1 = 1∗ 1∗∗ = (1∗ 1)∗ = 1∗∗ = 1 . 11.7.4. a ∈ Inv(A) ⇔ a∗ ∈ Inv(A). Moreover, involution and inversion commute. C For a ∈ Inv(A) by definition aa−1 = a−1 a = 1. Consequently, a−1∗ a∗ = a∗ a−1∗ = 1∗ . Using 11.7.3, infer that a∗ ∈ Inv(A) and a∗−1 = a−1∗ . Repeating this argument for a := a∗ , complete the proof. B 11.7.5. Sp(a∗ ) = Sp(a)∗ . CB 11.7.6. The spectrum of a unitary element of a C ∗ -algebra is a subset of the unit circle. C By Definition 6.4.13, ka2 k = ka∗ ak ≤ ka∗ k kak for an arbitrary a. In other words, kak ≤ ka∗ k. Therefore, from a = a∗∗ infer that kak = ka∗ k. If now a is a unitary element, a∗ = a−1 ; then kak2 = ka∗ ak = ka−1 ak = 1. Consequently, kak = ka∗ k = ka−1 k = 1. Whence it follows that Sp(a) and Sp(a−1 ) both lie within the unit disk. Furthermore, Sp(a−1 ) = Sp(a)−1 . B 11.7.7. The spectrum of a hermitian element of a C ∗ -algebra is real. C Take a in A arbitrarily. From the Gelfand–Dunford Theorem in an algebraic setting derive !∗ ∞ ∞ ∞ n X X X a (an )∗ (a∗ )n = = = exp(a∗ ). exp(a)∗ = n! n! n! n=0 n=0 n=0 If now h = h∗ is a hermitian element of A then, applying the holomorphic functional calculus to a := exp(ih), deduce that a∗ = exp(ih)∗ = exp((ih)∗ ) = exp(−ih∗ ) = exp(−ih) = a−1 . Consequently, a is a unitary element of A, and by 11.7.6 the spectrum Sp(a) of A is a subset of the unit circle T. If λ ∈ Sp(h) then by the Spectral Mapping Theorem (also cf. 11.3.3) exp(iλ) ∈ Sp(a) ⊂ T. Thus, 1 = | exp(iλ)| = | exp(i Re λ − Im λ)| = exp(− Im λ). Finally, Im λ = 0; i.e., λ ∈ R. B

234

Chapter 11

11.7.8. Definition. Let A be a C ∗ -algebra. A subalgebra B of A is called a C -subalgebra of A if b ∈ B ⇒ b∗ ∈ B. In this event, B is considered with the norm induced from A. ∗

11.7.9. Theorem. Every closed C ∗ -subalgebra of a C ∗ -algebra is pure. C Let B be a closed (unital) C ∗ -subalgebra of a C ∗ -algebra A and b ∈ B. If b ∈ Inv(B) then it is easy that b ∈ Inv(A). Let now b ∈ Inv(A). From 11.7.4 derive b∗ ∈ Inv(A). Consequently, b∗ b ∈ Inv(A) and the element (b∗ b)−1 b∗ is a left inverse of b. By virtue of 11.1.4, it means that b−1 = (b∗ b)−1 b∗ . Consequently, to complete the proof it suffices to show only that (b∗ b)−1 belongs to B. Since b∗ b is hermitian in B, the inclusion holds: SpB (b∗ b) ⊂ R (cf. 11.7.7). Using 11.2.5, infer that SpA (b∗ b) = SpB (b∗ b). Since 0 ∈ / SpA (b∗ b); therefore, b∗ b ∈ Inv(B). Finally, b ∈ Inv(B). B 11.7.10. Corollary. Let b be an element of a C ∗ -algebra A and let B be some closed C ∗ -subalgebra of A with b ∈ B. Then SpB (b) = SpA (b). /. 11.7.11. Remark. In view of 11.7.10, Theorem 11.7.9 is often referred to as the Spectral Purity Theorem for a C ∗ -algebra. It asserts that the concept of the spectrum of an element a of a C ∗ -algebra is absolute, i.e. independent of the choice of a C ∗ -subalgebra containing a. 11.8.

The Commutative Gelfand–Na˘ımark Theorem

11.8.1. The Banach algebra C(Q, C ) with the natural involution f 7→ f ∗ , where f ∗ (q) := f (q)∗ for q ∈ Q, is a C ∗ -algebra. / kf ∗ f k = sup{|f (q)∗ f (q)| : q ∈ Q} = sup{|f (q)|2 : q ∈ Q} = (sup |f (Q)|)2= kf k2 . 11.8.2. Stone–Weierstrass Every unital C ∗ -subalgebra of the C ∗ -algebra C(Q, C ), which separates the points of Q, is dense in C(Q, C). C Let A be such subalgebra. Since f ∈ A ⇒ f ∗ ∈ A; therefore, f ∈ A ⇒ Re f ∈ A and so the set Re A := {Re f : f ∈ A} is a real subalgebra of C(Q, R). It is beyond a doubt that Re A contains constant functions and separates the points of Q. By the Stone–Weierstrass Theorem for C(Q, R), the subalgebra Re A is dense in C(Q, R). It remains to refer to 11.7.2. B

Banach Algebras

235

11.8.3. Definition. A representation of a ∗-algebra agreeing with involution ∗ is a ∗-representation. In other words, if (A, ∗) and (B, ∗) are involutive algebras and R : A → B is a multiplicative linear operator, then R is called a ∗-representation of A in B whenever the following diagram commutes: R

A−→B ∗↓ ↓∗ R A−→B If R is also an isomorphism then R is a ∗-isomorphism of A and B. In the presence of norms in the algebras, the naturally understood terms “isometric ∗-representation” and “isometric ∗-isomorphism” are in common parlance. 11.8.4. Commutative Gelfand–Na˘ımark Theorem. The Gelfand transform of a commutative C ∗ -algebra A implements an isometric ∗-isomorphism of A and C(X(A), C ). C Given a ∈ A, observe that 1

ka2 k = k(a2 )∗ a2 k

/2

1

= ka∗ aa∗ ak

/2

= ka∗ ak = kak2 .

In virtue of 11.6.10 the Gelfand transform GA is thus an isometry of A and a closed b of C(X(A), C). Undoubtedly, A b separates the points of X(A) and subalgebra A contains constant functions. By virtue of 11.6.9 and 11.7.7, b h(X(A)) = Sp(h) ⊂ R for every hermitian ∗ element h = h of A. Now, take an arbitrary element a of A. Using 11.7.2, write a = x + iy, where x and y are hermitian elements. The containments χ(x) ∈ R and χ(y) ∈ R hold for every character χ, a member of X(A). With this in mind, successively infer that GA (a)∗ (χ) = b a∗ (χ) = b a(χ)∗ = χ(a)∗ = χ(x + iy)∗

= (χ(x) + iχ(y))∗ = χ(x) − iχ(y) = χ(x − iy) = χ(a∗ ) = ab∗ (χ) = GA (a∗ )(χ) (χ ∈ X(A)). Consequently, the Gelfand transform GA is a ∗-representation and, in particular, b is a C ∗ -subalgebra of C(X(A), C ). It remains to appeal to 11.8.2 and conclude A b = C(X(A), C ). B that A 11.8.5. Assume that R : A → B is a ∗-representation of a C ∗ -algebra A in a C ∗ -algebra B. Then kRak ≤ kak for a ∈ A.

236

Chapter 11

C Since R(1) = 1; therefore, R(Inv(A)) ⊂ Inv(B). Hence, SpB (R(a)) ⊂ SpA (a) for a ∈ A. Whence it follows from the Beurling–Gelfand formula that the inequality rA (a) ≥ rB (R(a)) holds for the spectral radii. If a is a hermitian element of A then R(a) is a hermitian element of B, because R(a)∗ = R(a∗ ) = R(a). If now A0 is the least closed C ∗ -subalgebra containing a and B0 is an analogous subalgebra containing R(a), then A0 and B0 are commutative C ∗ -algebras. Therefore, from Theorems 11.8.4 and 11.6.9 obtain kR(a)k = kR(a)kB0 = kGB0 (R(a))k = rB0 (R(a)) = rB (R(a)) ≤ rA (a) = rA0 (a) = kGA0 (a)k = kak. Given a ∈ A, it is easy to observe that a∗ a is a hermitian element. Thus, kR(a)k2 = kR(a)∗ R(a)k = kR(a∗ a)k ≤ ka∗ ak = kak2 . . 11.8.6. Spectral Theorem. Let a be a normal element of a C ∗ -algebra A, with Sp(a) the spectrum of a. There is a unique isometric ∗-representation Ra of C(Sp(a), C) in A such that a = Ra (ISp(a) ). C Let B be the least closed C ∗ -subalgebra of A containing a. It is clear that the algebra B is commutative by the normality of a (this algebra presents the closure of the algebra of all polynomials in a and a∗ ). Moreover, by 11.7.10, Sp(a) = SpA (a) = SpB (a). The Gelfand transform b a := GB (a) of a acts from X(B) onto Sp(a) by 11.6.9 and is evidently one-to-one. Since X(B) and Sp(a) are compact sets; on using 9.4.11, conclude that b a is a homeomorphism. Whence it is immediate ◦

a implements an isometric ∗-isomorphism between that the mapping R : f 7→ f ◦ b C(Sp(a), C ) and C(X(B), C ). Using Theorem 11.3.2 and the connection between the Gelfand transform and the Riesz–Dunford integral which is revealed in 11.6.12, for the identity mapping observe that ◦ b a = R^a IC = IC ◦ b a = IC ^a(X(B)) ◦ b a = IC Sp(a) ◦ b a = ISp(a) ◦ b a = R(ISp(a) ). Now, put ◦

Ra := GB−1 ◦ R. Clearly, Ra is an isometric embedding and a ∗-representation. Moreover, ◦

Ra (ISp(a) ) = GB−1 ◦ R(ISp(a) ) = GB−1 (b a) = a. Uniqueness for such representation Ra is guaranteed by 11.8.5 and the Stone– Weierstrass Theorem implies that the C ∗ -algebra C(Sp(a), C) is its least closed (unital) C ∗ -subalgebra containing ISp(a) . B

Banach Algebras

237

11.8.7. Definition. The representation Ra : C(Sp(a), C) → A of 11.8.6 is the continuous functional calculus (for a normal element a of A). The element Ra (f ) with f ∈ C(Sp(a), C) is usually denoted by f (a). 11.8.8. Remark. Let f be a holomorphic function in a neighborhood about the spectrum of a normal element a of some C ∗ -algebra A; i.e., f ∈ H(Sp(a)). Then the element f (a) of A was defined by the holomorphic functional calculus. Retain the symbol f for the restriction of f to the set Sp(a). Then,
using the continuous functional calculus, define the element Ra (f ) := Ra f Sp(a) of A. This element, as mentioned in 11.8.7, is also denoted by f (a). The use of the designation is by far not incidental (and sound in virtue of 11.6.12 and 11.8.6). Indeed, it would be weird to deliberately denote by different symbols one and the same element. This circumstance may be expressed in visual form. Namely, let · Sp(a) stand for the mapping sending a germ h, a member of H (Sp(a)), to its restriction to Sp(a); i.e., let h Sp(a) at a point z stand for the value of h at z (cf. 8.1.21). It is clear that · Sp(a) : H (Sp(a)) → C(Sp(a), C). The above connection between the continuous and holomorphic functional calculuses for a normal element a of the C ∗ -algebra A may be expressed as follows: The next diagram commutes ·|Sp(a)H (Sp(a)) C(Sp(a), C ) @ @ Ra @ Ra @ R ? @ A 11.9.

Operator ∗-Representations of a C ∗ -Algebra

11.9.1. Definition. Let A be a (unital) Banach algebra. An element s in A0 is a state of A, in writing s ∈ S(A), if ksk = s(1) = 1. For a ∈ A, the set N (a) := {s(a) : s ∈ S(A)} is the numeric range of a. 11.9.2. The numeric range of a positive function, a member of C(Q, C ), lies in R+ . C Let a ≥ 0 and ksk = s(1) = 1. We have to prove that s(a) ≥ 0. Take z ∈ C and ε > 0 such that the disk Bε (z) := z + εD includes a(Q). Then ka − zk ≤ ε and, consequently, |s(a − z)| ≤ ε. Hence, |s(a) − z| = |s(a) − s(z)| ≤ ε; i.e., s(a) ∈ Bε (z). Observe that ∩ {Bε (z) : Bε (z) ⊃ a(Q)} = cl co(a(Q)) ⊂ R+ .

238

Chapter 11

Thus, s(a) ∈ R+ . B 11.9.3. Lemma. Let a be a hermitian element of a C ∗ -algebra. Then (1) Sp(a) ⊂ N (a); (2) Sp(a) ⊂ R+ ⇔ N (a) ⊂ R+ . C Let B be the least closed C ∗ -subalgebra, of the algebra A under study, which contains a. It is evident that B is a commutative algebra. By virtue of 11.6.9, the Gelfand transform b a := GB (a) provides b a(X(B)) = SpB (a). In view of 11.7.10, SpB (a) = Sp(a). In other words, for λ ∈ Sp(a) there is a character χ of B satisfying the condition χ(a) = λ. By 11.6.3, kχk = χ(1) = 1. Using 7.5.11, find a normpreserving extension s of χ onto A. Then s is a state of A and s(a) = λ. Finally, Sp(a) ⊂ N (a) (in particular, if N (a) ⊂ R+ then Sp(a) ⊂ R+ ). Now, let s stand for an arbitrary state of A. It is clear that the restriction s B is a state of B. It is an easy matter to show that b a maps X(B) onto Sp(a) in a one-to-one fashion. Consequently, B may be treated as the algebra C(Sp(a), C ). From 11.9.2 derive s(a) = s B (a) ≥ 0 for b a ≥ 0. Thus, Sp(a) ⊂ R+ ⇒ N (a) ⊂ R+ , which ends the proof. B 11.9.4. Definition. An element a of a C ∗ -algebra A is called positive if a is hermitian and Sp(a) ⊂ R+ . The set of all positive elements of A is denoted by A+ . 11.9.5. In each C ∗ -algebra A the set A+ is an ordering cone. C It is clear that N (a+b) ⊂ N (a)+N (b) and N (αa) = αN (a) for a, b ∈ A and α ∈ R+ . Hence, 11.9.3 ensures the inclusion α1 A+ + α2 A+ ⊂ A+ for α1 , α2 ∈ R+ . Thus, A+ is a cone. If a ∈ A+ ∩ (−A+ ) then Sp(a) = 0. Since a is a hermitian element, from Theorem 11.8.6 deduce that kak = 0. B 11.9.6. To every hermitian element a of a C ∗ -algebra A there correspond some elements a+ and a− of A+ such that a = a+ − a− ;

a+ a− = a− a+ = 0.

C Everything is immediate from the continuous functional calculus. B 11.9.7. Kaplansky–Fukamija Lemma. An element a of a C ∗ -algebra A is positive if and only if a = b∗ b for some b in A. C ⇒: Let a ∈√A+ ; i.e., a = a∗ and Sp(a) ⊂ R+ . Then (cf. 11.8.6) there is a square root b := a. Moreover, b = b∗ and b∗ b = a. ⇐: If a = b∗ b then a is hermitian. Therefore, in view of 11.9.6, it may be assumed that b∗ b = u − v, where uv = vu = 0 with u ≥ 0 and v ≥ 0 (in the ordered vector space (AR , A+ )). Straightforward calculation yields the equalities (bv)∗ bv = v ∗ b∗ bv = vb∗ bv = v(u − v)v = (vu − v 2 )v = −v 3 .

Banach Algebras

239

Since v ≥ 0, it follows that v 3 ≥ 0; i.e., (bv)∗ bv ≤ 0. By Theorem 5.6.22, Sp((bv)∗ bv) and Sp(bv(bv)∗ ) may differ only by zero. Therefore, bv(bv)∗ ≤ 0. In virtue of 11.7.2, bv = a1 + ia2 for suitable hermitian elements a1 and a2 . It is evident that a21 , a22 ∈ A+ and (bv)∗ = a1 − ia2 . Using 11.9.5 twice, arrive at the estimates
0 ≥ (bv)∗ bv + bv(bv)∗ = 2 a21 + a22 ≥ 0. By 11.9.5, a1 = a2 = 0; i.e., bv = 0. Hence, −v 3 = (bv)∗ bv = 0. The second appeal to 11.9.5 shows v = 0. Finally, a = b∗ b = u − v = u ≥ 0; i.e., a ∈ A+ . B 11.9.8. Every state s of a C ∗ -algebra A is hermitian; i.e., s(a∗ ) = s(a)∗

(a ∈ A).

C By Lemmas 11.9.7 and 11.9.3, s(a∗ a) ≥ 0 for all a ∈ A. Putting a := a + 1 and a := a + i, successively infer that 0 ≤ s((a + 1)∗ (a + 1)) = s(a∗ a + a + a∗ + 1) ⇒ s(a) + s(a∗ ) ∈ R; 0 ≤ s((a + i)∗ (a + i)) = s(a∗ a − ia + ia∗ + 1) ⇒ i(−s(a) + s(a∗ )) ∈ R. In other words, Im s(a) + Im s(a∗ ) = 0;

Re(−s(a)) + Re s(a∗ ) = 0.

Whence it follows that s(a∗ ) = Re s(a∗ ) + i Im s(a∗ ) = Re s(a) − i Im s(a) = s(a)∗ . . 11.9.9. Let s be a state of a C ∗ -algebra A. Given a, b ∈ A, denote (a, b)s := s(b∗ a). Then ( · , · )s is a semi-inner product on A. C From 11.9.8 derive (a, b)s = s(b∗ a) = s((a∗ b)∗ ) = s(a∗ b)∗ = (b, a)∗s . Hence, ( · , · )s is a hermitian form. Since a∗ a ≥ 0 for a ∈ A in virtue of 11.9.7, (a, a)s = s(a∗ a) ≥ 0 by 11.9.3. Consequently, ( · , · )s is a positive-definite hermitian form. B 11.9.10. GNS-Construction Theorem. To every state s of an arbitrary C ∗ -algebra A there correspond a Hilbert space (Hs , ( · , · )s ), an element xs in Hs and a ∗-representation Rs : A → B(Hs ) such that s(a) = (Rs (a)xs , xs )s for all a ∈ A and the set {Rs (a)xs : a ∈ A} is dense in Hs .

240

Chapter 11

∗ C In virtue of 11.9.9, putting (a, b)p s := s(b a) for a, b ∈ A, obtain a preHilbert space (A, ( · , · )s ). Let ps (a) := (a, a)s stand for the seminorm of the space. Assume that ϕs : A → A/ ker ps is the coset mapping of A onto the Hausdorff pre-Hilbert space A/ ker ps associated with A. Assume further that ιs : A/ ker ps → Hs is an embedding (for instance, by the double prime mapping) of A/ ker ps onto a dense subspace of the Hilbert space Hs associated with (A, ( · , · )s ) (cf. 6.1.10 (4)). The inner product in Hs retains the previous notation ( · , · )s . Therefore, in particular,

(ιs ϕs a, ιs ϕs b)s = (a, b)s = s(b∗ a)

(a, b ∈ A).

Given a ∈ A, consider (the image under the canonical operator representation) La : b 7→ ab (b ∈ A). Demonstrate first that there are unique bounded operators La and Rs (a) making the following diagram commutative: ϕs

ι

s A−→A/ ker ps −→H s ↓ Rs (a) La ↓ ↓ La ϕs ιs A−→A/ ker ps −→H s

A sought operator La is a solution to the equation Xϕs = ϕs La . Using 2.3.8, observe next that the necessary and sufficient condition for solvability of the equation in linear operators consists in invariance of the subspace ker ps under La . Thus, examine the inclusion La (ker ps ) ⊂ ker ps . To this end, take an element b of ker ps , i.e. ps (b) = 0. By definition and the Cauchy–Bunyakovski˘ı–Schwarz inequality, deduce that 0 ≤ (La b, La b)s = (ab, ab)s = s((ab)∗ ab) = s(b∗ a∗ ab) = (a∗ ab, b)s ≤ ps (b)ps (a∗ ab) = 0; i.e., La b ∈ ker ps . Uniqueness for La is provided by 2.3.9, since ϕs is an epimorphism. Observe also that ϕs is an open mapping (cf. 5.1.3). Whence the continuity of La is immediate. Therefore, in virtue of 5.3.8 the correspondence ιs ◦ La ◦ (ιs )−1 may be considered as a bounded linear operator from ιs (A/ ker ps ) to the Banach space Hs . By 4.5.10 such an operator extends uniquely to an operator Rs (a) in B(Hs ). Demonstrate now that Rs : a 7→ Rs (a) is a sought representation. By 11.1.6, Lab = La Lb for a, b ∈ A. Consequently, ϕs Lab = ϕs La Lb = La ϕs Lb = La Lb ϕs . Since Lab is a unique solution to the equation Xϕs = ϕs Lab , infer the equality Lab = La Lb , which guarantees multiplicativity for Rs . The linearity of Rs may be verified likewise. Furthermore, L1 ϕs = ϕs L1 = ϕs IA = ϕs = IA/ ker ps ϕs = 1ϕs ;

Banach Algebras

241

i.e., Rs (1) = 1. For simplicity, put ψs := ιs ϕs . Then, on account taken of the definition of the inner product on Hs (cf. 6.1.10 (4)) and the involution in B(Hs ) (cf. 6.4.14 and 6.4.5), given elements a, b, and y in A, infer that (Rs (a∗ )ψs x, ψs y)s = (ψs La∗ x, ψs y)s = (La∗ x, y)s = (a∗ x, y)s = s(y ∗ a∗ x) = s((ay)∗ x) = (x, ay)s = (x, La y)s = (ψs x, ψs La y)s = (ψs x, Rs (a)ψs y)s = (Rs (a)∗ ψs x, ψs y)s . Now, since im ψs is dense in Hs it follows that Rs (a∗ ) = Rs (a)∗ for all a in A; i.e., Rs is a ∗-representation. Let xs := ψs 1. Then Rs (a)xs = Rs (a)ψs 1 = ψs La 1 = ψs a (a ∈ A). Consequently, the set {Rs (a)xs : a ∈ A} is dense in Hs . Furthermore, (Rs (a)xs , xs )s = (ψs a, ψs 1)s = (a, 1)s = s(1∗ a) = s(a). . 11.9.11. Remark. The construction, presented in the proof of 11.9.10, is called the GNS-construction or, in expanded form, the Gelfand–Na˘ımark–Segal construction, which is reflected in the name of 11.9.10. 11.9.12. Gelfand–Na˘ımark Each C ∗ -algebra has an isometric ∗-representation in the endomorphism algebra of a suitable Hilbert space. C Let A be a C ∗ -algebra. We have to find a Hilbert space H and an isometric ∗-representation R of A in the C ∗ -algebra B(H) of bounded endomorphisms of H. For this purpose, consider the Hilbert sum H of the family of Hilbert spaces (Hs )s∈S(A) which exists in virtue of the GNS-Construction Theorem; i.e.,   Y H := ⊕ Hs = h := (hs )s∈S(A) ∈ Hs :  s∈S(A) s∈S(A)

X s∈S(A)

khs k2Hs

  < +∞ . 

Observe that the inner product of h := (hs )s∈S(A) and g := (gs )s∈S(A) is calculated by the rule (cf. 6.1.10 (5) and 6.1.9): (h, g) =

X

(hs , gs )s .

s∈S(A)

242

Chapter 11

Assume further that Rs is a ∗-representation of A on the space Hs corresponding to s in S(A). Since in view of 11.8.5 there is an estimate kRs (a)kB(Hs ) ≤ kak for a ∈ A; therefore, given h ∈ H, infer that X X X kRs (a)k2B(Hs ) khs k2Hs ≤ kak2 khs k2Hs . kRs (a)hs k2Hs ≤ s∈S(A)

s∈S(A)

s∈S(A)

Whence it follows that the expression R(a)h : s 7→ Rs (a)hs defines an element R(a)h of H. The resulting operator R(a) : h 7→ R(a)h is a member of B(H). Moreover, the mapping R : a 7→ R(a) (a ∈ A) is a sought isometric ∗-representation of A. Indeed, from the definition of R and the properties of Rs for s ∈ S(A), it follows easily that R is a ∗-representation of A in B(H). Check for instance that R agrees with involution. To this end, take a ∈ A and h, g ∈ H. Then X (R(a∗ )h, g) = (Rs (a∗ )hs , gs )s s∈S(A)

=

X

(Rs (a)∗ hs , gs )s =

s∈S(A)

X

(hs , Rs (a)gs )s

s∈S(A)

= (h, R(a)g) = (R(a)∗ h, g). Since h and g in H are arbitrary, conclude that R(a∗ ) = R(a)∗ . It remains to establish only that the ∗-representation R is an isometry, i.e. the equalities kR(a)k = kak for all a ∈ A. First, assume a positive. From the Spectral Theorem and the Weierstrass Theorem it follows that kak ∈ Sp(a). In virtue of 11.9.3 (1) there is a state s ∈ S(A) such that s(a) = kak. Using the properties of the vector xs corresponding to the ∗-representation Rs (cf. 11.9.10) and applying the Cauchy–Bunyakovski˘ı–Schwarz inequality, infer that kak = s(a) = (Rs (a)xs , xs )s ≤ kRs (a)xs kHs kxs kHs ≤ kRs (a)kB(Hs ) kxs k2Hs = kRs (a)kB(Hs ) (xs , xs )s = kRs (a)kB(Hs ) (Rs (1)xs , xs )s = kRs (a)kB(Hs ) s(1) = kRs (a)kB(Hs ) . From the estimates kR(a)k ≥ kRs (a)kB(Hs ) and kak ≥ kR(a)k, the former obvious and the latter indicated in 11.8.5, derive kak ≥ kR(a)k ≥ kRs (a)kB(Hs ) ≥ kak. Finally, take a ∈ A. By the Kaplansky–Fukamija Lemma, a∗ a is positive. So, kR(a)k2 = kR(a)∗ R(a)k = kR(a∗ )R(a)k = kR(a∗ a)k = ka∗ ak = kak2 . No further explanation is needed. B

Banach Algebras

243 Exercises

11.1. Give examples of Banach algebras and non-Banach algebras. 11.2. Let A be a Banach algebra. Take χ ∈ A# such that χ(1) = 1 and χ(Inv(A)) ⊂ Inv(C). Prove that χ is multiplicative and continuous. 11.3. Let the spectrum Sp(a) of an element a of a Banach algebra A lie in an open set U . Prove that there is a number ε > 0 such that Sp(a + b) ⊂ U for all b ∈ A satisfying kbk ≤ ε. 11.4. Describe the maximal ideal spaces of the algebras C(Q, C) and C (1) ([0, 1], C) with pointwise multiplication, and of the algebra of two-way infinite summable sequences l1 (Z) with multiplication (a ∗ b)(n) :=

∞ X

an−k bk .

k=−∞

11.5. Show that a member T of the endomorphism algebra B(X) of a Banach space X has a left inverse if and only if T is a monomorphism and the range of T is complemented in X. 11.6. Show that a member T of the endomorphism algebra B(X) of a Banach space X has a right inverse if and only if T is an epimorphism and the kernel of T is complemented in X. 11.7. Assume that a Banach algebra A has an element with disconnected spectrum (having a proper clopen part). Prove that A has a nontrivial idempotent. 11.8. Let A be a unital commutative Banach algebra and let E be some set of maximal ideals of A. Such a set E is a boundary of A if kb ak∞ = sup |b a(E)| for all a ∈ A. Prove that the intersection of all closed boundaries of A is also a boundary of A. This is the Shilov boundary of A. 11.9. Let A and B be unital commutative Banach algebras, with B ⊂ A and 1B = 1A . Prove that each maximal ideal of the Shilov boundary of B lies in some maximal ideal of A. 11.10. Let A and B be unital C ∗ -algebras and let T be a morphism from A to B. Assume further that a is a normal element of A and f is a continuous function on SpA (a). Demonstrate that SpB (T a) ⊂ SpA (a) and T f (a) = f (T a). 11.11. Let f ∈ A0 , with A a commutative C ∗ -algebra. Show that f is a positive form (i.e., f (a∗ a) ≥ 0 for a ∈ A) if and only if kf k = f (1). 11.12. Describe extreme rays of the set of positive forms on a commutative C ∗ -algebra. 11.13. Prove that the algebras C(Q1 , C) and C(Q2 , C), with Q1 and Q2 compact, are isomorphic if and only if Q1 and Q2 are homeomorphic. 11.14. Let a normal element a of a C ∗ -algebra has real spectrum. Prove that a is hermitian. 11.15. Using the continuous functional calculus, develop a spectral theory for normal operators in a Hilbert space. Describe compact normal operators. 11.16. Let T be an algebraic morphism between C ∗ -algebras, and kT k ≤ 1. Then T (a∗ ) = for all a.

(T a)∗

11.17. Let T be a normal operator in a Hilbert space H. Show that there are a hermitian operator S in H and a continuous function f : Sp(S)0 = 0 O C such that T = f (S). Is an analogous assertion valid in C ∗ -algebras? 11.18. Let A and B be C ∗ -algebras and let ρ be a ∗-monomorphism from A to B. Prove that ρ is an isometric embedding of A into B. 11.19. Let a and b be hermitian elements of a C ∗ -algebra A. Assume that ab = ba and, moreover, a ≤ b. Prove that f (a) ≤ f (b) for (suitable restrictions of) every increasing continuous scalar function f over R.

References

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Notation Index

:=, ix /, ., ix Ar , 11.1.6 A × B, 1.1.1 Bp , 5.1.1, 5.2.11 ◦

B p , 5.1.1 BT , 5.1.3 BX , 5.1.10, 5.2.11 B(X), 5.6.4, B(E , F ), 5.5.9 (2) B(X, Y ), 5.1.10 (7) C(Q, F ), 4.6.8 C (m) , 10.9.9 C∞ (Š), 10.10.2 (3) Dα , 10.11.13 F −1 , 1.1.3 (1) F (B ), 1.3.5 (1) Fp , 5.5.9 (6) F |U , 1.1.3 (5) F (U ), 1.1.3 (5) F (a1 , · ), 1.1.3 (6) F ( · , a2 ), 1.1.3 (6) F ( · , · ), 1.1.3 (6) F (X, Y ), 8.3.6 b , 10.11.2 G

b , 10.11.2 G G ◦ F , 1.1.4 H∗ , 6.1.10 (3) H(K), 8.1.13 H€ (U ), 3.1.11 IC , 8.2.10 IU , 1.1.3 (3) J(q), 11.5.3 J(Q0 ), 11.5.2 b

J C A, 11.4.1 K(E), 10.9.1 K(Q), 10.9.1 K(Š), 10.9.1 LA , 11.1.6 Lp , 5.5.9 (4), 5.5.9 (6) Lp (X), 5.5.9 LQ0 , 10.8.4 (3)



L

Q0

, 10.8.4 (4)

L∞ , 5.5.9 (5) M (Š), 10.9.4 (2) N (a), 11.9.1 Np , 5.5.9 (6) PH0 , 6.2.7 Pσ , 8.2.10 PX1 ||X2 , 2.2.9 (4) P1 ⊥ P2 , 6.2.12 R (a, λ), 11.2.1 R (T, λ), 5.6.13 S(A), 11.9.1 T 0 , 7.6.2 T ∗ , 6.4.4 kT k, 5.1.10 (7) U ◦ , 10.5.7 U ⊥ , 6.2.5 U ∈ (€ ), 3.1.1 hX |, 10.3.1 X 0 , 5.1.10 (8), 10.2.11 X 00 , 5.1.10 (8) X∗ , 2.1.4 (2) X+ , 3.2.5 X0⊥ , 7.6.8 Xσ , 8.2.10 X N , 2.1.4 (4) X # , 2.2.4

Notation Index XR , 3.7.1 X „ , 2.1.4 (4) X = X1 ⊕ X2 , 2.1.7 X ⊕ iX, 8.4.8 X1 × X2 × . . . × XN , 2.1.4 (4) (X, τ )0 , 10.2.11 X/X0 , 2.1.4 (6) (X/X0 , pX/X0 ), 5.1.10 (5) X ↔ Y , 10.3.3 X ' Y , 2.2.6 | Y i, 10.3.1 B, 9.6.14 C, 2.1.2 D, 8.1.3 F, 2.1.2 N, 1.2.16 Q, 7.4.11 R, 2.1.2 R· , 3.4.1 R+ , 3.1.2 (4)

R, 3.8.1 Re, 3.7.3 Re−1 , 3.7.4 T, 8.1.3 Z, 8.5.1 Z+ , 10.10.2 (2) Ae , 11.1.2 D (Q), 10.10.1 D (Š), 10.10.1 D 0 (Š), 10.10.4 D 0F (Š), 10.10.8 D (m) (Q), 10.10.8 D (m) (Š), 10.10.8 D (m) (Š)0 , 10.10.8 E (Š), 10.10.2 (3) E 0 (RN ), 10.10.5 (9) E ◦ T , 2.2.8 F , 10.11.4 Fp , 5.5.9 (6) F r(X, Y ), 8.5.1 F (X), 1.3.6 GA , 11.6.8 H (K), 8.1.14 K (X), 8.3.3 K (X, Y ), 6.6.1 L (X), 2.2.8 L (X, Y ), 2.2.3 Lr (X, Y ), 3.2.6 (3) L∞ , 5.5.9 (5)

257 M (Š), 10.9.3 N (µ), 10.8.11 Np (f ), 5.5.9 (4) N∞ , 5.5.9 (5) P (X), 1.2.3 (4) RT , 8.2.1 Ra h, 11.3.1 S (RN ), 10.11.6 S 0 (RN ), 10.11.16 T (X), 9.1.2 Up , 5.2.2 UM , 5.2.4 UX , 4.1.5, 5.2.4 ⊥ X , 7.6.8 0 F u, 10.11.19 M , 5.3.9 M ∼ N , 5.3.1 M  N , 5.3.1 MX , 5.1.6 Mτ , 10.2.7 N ◦ T , 5.1.10 (3) NT , 5.1.10 (3) Ra , 11.8.7 Cl(τ ), 4.1.15, 9.1.4 Im f , 5.5.9 (4) Inv(A), 11.1.5 Inv(X, Y ), 5.6.12 ƒB , 8.1.2 (4) Lat(X), 2.1.5 LCT (X), 10.2.3 M(A), 11.6.6 Op(τ ), 4.1.11, 9.1.4 Re, 2.1.2 Re f , 5.5.9 (4) Sp(a), 11.2.1 SpA (a), 11.2.1 Sp(T ), 5.6.13 T1 , 9.3.2 T2 , 9.3.5 T3 , 9.3.9 T31 /2 , 9.3.15 T4 , 9.3.11 T(X), 9.1.7 Tr (Š), 10.10.2 VT(X), 10.1.5 X(A), 11.6.4 δ, 10.9.4 (1) δ (−1) , 10.10.5 (4) δq , 10.9.4 (1) µ∗ , 10.9.4 (3)

258 µ+ , 10.8.13 µ− , 10.8.13 |µ|, 10.8.13, 10.9.4 (3) kµk, 10.9.5 µŠ0 , 10.9.4 (4) µ1 ⊗ µ2 , 10.9.4 (6) µ1 × µ2 , 10.9.4 (6) µ ∗ f , 10.9.4 (7) µ ∗ ν, 10.9.4 (7) π(U ), 10.5.1, 10.5.7 π −1 (V ), 10.5.1 −1 πF (πF (U )), 10.5.5 2π , 10.11.4 σ 0 , 8.2.9 σ(T ), 5.6.13 σ(X, Y ), 10.3.5 τ (X, Y ), 10.4.4 τa f , 10.9.4 (1) τM , 5.2.8 κσ , 8.2.10 abs pol , 10.5.7 cl U , 4.1.13 co(U ), 3.1.14 codim X, 2.2.9 (5) coim T , 2.3.1 coker T , 2.3.1 core U , 3.4.11 diam U , 4.5.3 dim X, 2.2.9 (5) dom f , 3.4.2 dom F , 1.1.2 0 = 0 OO , 3.4.2 ext V , 3.6.1 fil B , 1.3.3 fr U , 4.1.13 im F , 1.1.2 inf U , 1.2.9 int U , 4.1.13 ker T , 2.3.1 lin(U ), 3.1.14 pol , 10.5.7 rank T , 8.5.7 (2) res(a), 11.2.1 res(T ), 5.6.13 seg, 3.6.1 sup U , 1.2.9 supp(f ), 9.6.4 supp(µ), 10.8.11, 10.9.4 (5) supp(u), 10.10.5 (6) b a, 11.6.8

Notation Index aµ, 10.8.15 a τ f , 10.9.4 (1) (a, b)s , 11.9.9 c, 3.3.1 (2), 5.5.9 (3) c (E , F ), 5.5.9 (3) c0 , 5.5.9 (3) c0 (E ), 5.5.9 c0 (E , F ), 5.5.9 (3) ∂ α u, 10.10.5 (4) ∂ (p), 3.5.2 (1) |∂ |(p), 3.7.8 ∂U , 4.1.13 ∂ x (f ), 3.5.1 dp , 5.2.1 dx, 10.9.9 e, 10.9.4 (1), 11.1.1 b f , 10.11.3 f (a), 11.3.1 {f < t}, 3.8.1 {f = t}, 3.8.1 {f ≤ t}, 3.8.1 f (T ), 8.2.1 f e, 10.10.5 (9) f µ, 10.9.4 (3) f ∗ , 10.9.4 (3) f u, 10.10.5 (7) fn  f , 10.10.7 (3) fn  0, 10.9.8 K

b bg, 10.11.2 ◦

g(f ), 8.2.6 hhi, 6.3.5 lp , lp (E ), 5.5.9 (4) l∞ , l∞ (E ), 5.5.9 (2) m, 5.5.9 (2) p  q, 5.3.3 pe , 5.5.9 (5) pS , 3.8.6 p T , 5.1.3 pX/X0 , 5.1.10 (5) r(T ), 5.6.6 s, Ex. 1.19 tα , 10.11.5 (8) ug , 10.10.5 (1) u∗ , 10.10.5 (5) u ∗ f , 10.10.5 (9) u1 ⊗ u2 , 10.10.5 (8) u1 × u2 , 10.10.5 (8) hx |, 10.3.1

Notation Index x0 , 6.4.1 x00 , 5.1.10 (8) xα , 10.11.5 (8) x+ , 3.2.12 x− , 3.2.12 |x|, 3.2.12 kxkp , 5.5.9 (4) kxk∞ , 5.5.9 (2) ∼ (x), 10.11.4 ∼X0 , 2.1.4 P (6) x := x , 5.5.9 (7) e∈E e 0 x 7→ x , 6.4.1 x1 ∨ x2 , x1 ∧ x2 , 1.2.12 (x1 , x2 ), 1.2.12 hx | f i, 5.1.11 x ≤σ y, 1.2.2 x0 ⊗ y, 5.5.6 x ⊥ y, 6.2.5 | yi, 10.3.1

259 |||y|||p , 5.5.9 (6) k · k, 5.1.9 k · kn,Q , 10.10.2 (2) k · k∞ , 5.5.9 (5) k · kX , 5.1.9 k · | Xk, 5.1.9 | ·i, 10.3.1 1, 5.3.10, 10.8.4 (6) 2X , 1.2.3 (4) ∗, 6.4.13 b R , 10.9.1 , 5.5.9 (6) E h· | ·i, 10.3.1 h· |, 10.3.1 ∼, P 1.2.2 X , 2.1.4 (5) ξ∈„ ξ

Q X , 2.1.4 (4) H ξ∈„ ξ

h(z)R(z)dz, 8.1.20

b, 11.6.8

Subject Index

Absolute Bipolar Theorem, 10.5.9, 178 absolute concept, 9.4.7, 157 absolute polar, 10.5.7, 178 absolutely continuous measure, 10.9.4 (3), 190 absolutely convex set, 3.1.2 (6), 20 absolutely fundamental family of vectors, 5.5.9 (7), 73 absorbing set, 3.4.9, 28 addition in a vector space, 2.1.3, 10 adherence of a filterbase, 9.4.1, 155 adherent point, 4.1.13, 41 adherent point of a filterbase, 9.4.1, 155 adjoint diagram, 6.4.8, 92 adjoint of an operator, 6.4.5, 91 adjunction of unity, 11.1.2, 213 affine hull, 3.1.14, 22 affine mapping, 3.1.7, 21 affine operator, 3.4.8 (4), 28 affine variety, 3.1.2 (5), 20 agreement condition, 10.9.4 (4), 190 Akilov Criterion, 10.5.3, 178 Alaoglu–Bourbaki Theorem, 10.6.7, 180 Alexandroff compactification, 9.4.22, 159 algebra, 5.6.2, 73 algebra of bounded operators, 5.6.5, 74 algebra of germs of holomorphic functions, 8.1.18, 125 algebraic basis, 2.2.9 (5), 14 algebraic complement, 2.1.7, 12 algebraic dual, 2.2.4, 13 algebraic isomorphism, 2.2.5, 13 algebraic subdifferential, 7.5.8, 113 algebraically complementary subspace, 2.1.7, 12

algebraically interior point, 3.4.11, 28 algebraically isomorphic spaces, 2.2.6, 13 algebraically reflexive space, Ex. 2.8, 19 ambient space, 2.1.4 (3), 11 annihilator, 7.6.8, 116 antidiscrete topology, 9.1.8 (3), 147 antisymmetric relation, 1.2.1, 3 antitone mapping, 1.2.3, 4 approximate inverse, 8.5.9, 139 approximately invertible operator, 8.5.9, 139 approximation property, 8.3.10, 133 approximation property in Hilbert space, 6.6.10, 98 arc, 4.8.2, 54 Arens multinorm, 8.3.8, 133 ascent, Ex. 8.10, 144 Ascoli–ArzelΓ a Theorem, 4.6.10, 50 assignment operator, ix associate seminorm, 6.1.7, 81 associated Hausdorff pre-Hilbert space, 6.1.10 (4), 83 associated Hilbert space, 6.1.10 (4), 83 associated multinormed space, 10.2.7, 172 associated topology, 9.1.12, 148 associativity of least upper bounds, 3.2.10, 24 asymmetric balanced Hahn–Banach formula, 3.7.10, 34 asymmetric Hahn–Banach formula, 3.5.5, 30 Atkinson Theorem, 8.5.18, 141

Subject Index Automatic Continuity Principle, 7.5.5, 112 automorphism, 10.11.4, 203 Baire Category Theorem, 4.7.6, 52 Baire space, 4.7.2, 52 Balanced Hahn–Banach Theorem, 3.7.13, 35 Balanced Hahn–Banach Theorem in a topological setting, 7.5.10, 113 balanced set, 3.1.2 (7), 20 balanced subdifferential, 3.7.8, 34 Balanced Subdifferential Lemma, 3.7.9, 34 ball, 9.6.14, 166 Banach algebra, 5.6.3, 74 Banach Closed Graph Theorem, 7.4.7, 108 Banach Homomorphism Theorem, 7.4.4, 108 Banach Inversion Stability Theorem, 5.6.12, 76 Banach Isomorphism Theorem, 7.4.5, 108 Banach range, 7.4.18, 111 Banach space, 5.5.1, 66 Banach’s Fundamental Principle, 7.1.5, 101 Banach’s Fundamental Principle for a Correspondence, 7.3.7, 107 Banach–Steinhaus Theorem, 7.2.9, 104 barrel, 10.10.9 (1), 199 barreled normed space, 7.1.8, 102 barreled space, 10.10.9 (1), 199 base for a filter, 1.3.3, 6 basic field, 2.1.2, 10 Bessel inequality, 6.3.7, 88 best approximation, 6.2.3, 84 Beurling–Gelfand formula, 8.1.12 (2), 124 bilateral ideal, 8.3.3, 132; 11.6.2, 220 bilinear form, 6.1.2, 80 bipolar, 10.5.5, 178 Bipolar Theorem, 10.5.8, 178 Birkhoff Theorem, 9.2.2, 148 Bochner integral, 5.5.9 (6), 72 bornological space, 10.10.9 (3), 199 boundary of an algebra, Ex. 11.8, 234 boundary of a set, 4.1.13, 41 boundary point, 4.1.13, 41 bounded above, 1.2.19, 6

261 bounded below, 3.2.9, 23 bounded endomorphism algebra, 5.6.5, 74 Bounded Index Stability Theorem, 8.5.21, 142 bounded operator, 5.1.10 (7), 59 bounded Radon measure, 10.9.4 (2), 189 bounded set, 5.4.3, 66 boundedly order complete lattice, 3.2.8, 23 Bourbaki Criterion, 4.4.7, 46; 9.4.4, 156 bracketing of vector spaces, 10.3.1, 173 bra-functional, 10.3.1, 173 bra-mapping, 10.3.1, 173 bra-topology, 10.3.5, 174 B-stable, 10.1.8, 171 bump function, 9.6.19, 167 Calkin algebra, 8.3.5, 132 Calkin Theorem, 8.3.4, 132 canonical embedding, 5.1.10 (8), 59 canonical exact sequence, 2.3.5 (6), 16 canonical operator representation, 11.1.7, 214 canonical projection, 1.2.3 (4), 4 Cantor Criterion, 4.5.6, 47 Cantor Theorem, 4.4.9, 46 cap, 3.6.3 (4), 32 Cauchy–BunyakovskiΦ ß–Schwarz inequality, 6.1.5, 80 Cauchy filter, 4.5.2, 47 Cauchy net, 4.5.2, 47 Cauchy–Wiener Integral Theorem, 8.1.7, 122 centralizer, 11.1.6, 214 chain, 1.2.19, 6 character group, 10.11.2, 203 character of a group algebra, 10.11.1 (1), 201 character of an algebra, 11.6.4, 221 character space of an algebra, 11.6.4, 221 characteristic function, 5.5.9 (6), 72 charge, 10.9.4 (3), 190 ChebyshΞ ev metric, 4.6.8, 50 classical Banach space, 5.5.9 (5), 71 clopen part of a spectrum, 8.2.9, 130 closed ball, 4.1.3, 40 closed convex hull, 10.6.5, 179

262 closed correspondence, 7.3.8, 107 closed cylinder, 4.1.3, 40 closed-graph correspondence, 7.3.9, 107 closed halfspace, Ex. 3.3, 39 closed linear span, 10.5.6, 178 closed set, 9.1.4, 146 closed set in a metric space, 4.1.11, 41 closure of a set, 4.1.13, 41 closure operator, Ex. 1.11, 8 coarser cover, 9.6.1, 164 coarser filter, 1.3.6, 7 coarser pretopology, 9.1.2, 146 codimension, 2.2.9 (5), 14 codomain, 1.1.2, 1 cofinite set, Ex. 1.19, 9 coimage of an operator, 2.3.1, 15 coincidence of the algebraic and topological subdifferentials, 7.5.8, 113 coinitial set, 3.3.2, 25 cokernel of an operator, 2.3.1, 15 comeager set, 4.7.4, 52 commutative diagram, 2.3.3, 15 Commutative Gelfand–NaΦ ßmark Theorem, 11.8.4, 227 compact convergence, 7.2.10, 105 Compact Index Stability Theorem, 8.5.20, 142 compact-open topology, 8.3.8, 133 compact operator, 6.6.1, 95 compact set, 9.4.2, 155 compact set in a metric space, 4.4.1, 46 compact space, 9.4.7, 157 compact topology, 9.4.7, 157 compactly-supported distribution, 10.10.5 (6), 196 compactly-supported function, 9.6.4, 165 compactum, 9.4.17, 158 compatible topology, 10.4.1, 175 complementary projection, 2.2.9 (4), 14 complementary subspace, 7.4.9, 109 Complementation Principle, 7.4.10, 109 complemented subspace, 7.4.9, 108 complement of an orthoprojection, 6.2.10, 85 complement of a projection, 2.2.9 (4), 14 complete lattice, 1.2.13, 5 complete metric space, 4.5.5, 47 complete set, 4.5.14, 49

Subject Index completely regular space, 9.3.15, 155 completion, 4.5.13, 49 complex conjugate, 2.1.4 (2), 10 complex distribution, 10.10.5 (5), 196 complex plane, 8.1.3, 121 complex vector space, 2.1.3, 10 complexification, 8.4.8, 136 complexifier, 3.7.4, 34 composite correspondence, 1.1.4, 2 Composite Function Theorem, 8.2.8, 129 composition, 1.1.4, 2 Composition Spectrum Theorem, 5.6.22, 78 cone, 3.1.2 (4), 20 conical hull, 3.1.14, 22 conical segment, 3.1.2 (9), 20 conical slice, 3.1.2 (9), 20 conjugate distribution, 10.10.5 (5), 196 conjugate exponent, 5.5.9 (4), 69 conjugate-linear functional, 2.2.4, 13 conjugate measure, 10.9.4 (3), 189 connected elementary compactum, 4.8.5, 54 connected set, 4.8.4, 54 constant function, 5.3.10, 64; 10.8.4 (6), 182 Continuous Extension Principle, 7.5.11, 113 continuous function at a point, 4.2.2, 43; 9.2.5, 149 Continuous Function Recovery Lemma, 9.3.12, 153 continuous functional calculus, 11.8.7, 228 continuous mapping of a metric space, 4.2.2, 43 continuous mapping of a topological space, 9.2.4, 149 continuous partition of unity, 9.6.6, 166 contour integral, 8.1.20, 125 conventional summation, 5.5.9 (4), 70 convergent filterbase, 4.1.16, 42 convergent net, 4.1.17, 42 convergent sequence space, 3.3.1 (2), 25 convex combination, 3.1.14, 22 convex correspondence, 3.1.7, 21 convex function, 3.4.4, 27 convex hull, 3.1.14, 22 convex set, 3.1.2 (8), 20 convolution algebra, 10.9.4 (7), 190

Subject Index convolution of a measure and a function, 10.9.4 (7), 191 convolution of distributions, 10.10.5 (9), 197 convolution of functions, 9.6.17, 167 convolution of measures, 10.9.4 (7), 190 convolutive distributions, 10.10.5 (9), 197 coordinate projection, 2.2.9 (3), 13 coordinatewise operation, 2.1.4 (4), 11 core, 3.4.11, 28 correspondence, 1.1.1, 1 correspondence in two arguments, 1.1.3 (6), 2 correspondence onto, 1.1.3 (3), 2 coset, 1.2.3 (4), 4 coset mapping, 1.2.3 (4), 4 countable convex combination, 7.1.3, 101 Countable Partition Theorem, 9.6.20, 167 countable sequence, 1.2.16, 6 countably normable space, 5.4.1, 64 cover of a set, 9.6.1, 164 C ∗ -algebra, 6.4.13, 92 C ∗ -subalgebra, 11.7.8, 225 Davis–Figiel–Szankowski Counterexample, 8.3.14, 134 de Branges Lemma, 10.8.16, 185 decomplexification, 6.1.10 (2), 83 decomposition reduces an operator, 2.2.9 (4), 14 decreasing mapping, 1.2.3, 4 Dedekind complete vector lattice, 3.2.8, 23 deficiency, 8.5.1, 137 definor, ix delta-function, 10.9.4 (1), 188 delta-like sequence, 9.6.15, 166 δ-like sequence, 9.6.15, 166 δ-sequence, 9.6.15, 166 dense set, 4.5.10, 48 denseness, 4.5.10, 48 density of a measure, 10.9.4 (3), 190 derivative in the distribution sense, 10.10.5 (4), 196 derivative of a distribution, 10.10.5 (4), 196

263 descent, Ex. 8.10, 144 diagonal, 1.1.3 (3), 2 diagram prime, 7.6.5, 115 Diagram Prime Principle, 7.6.7, 115 diagram star, 6.4.8, 92 Diagram Star Principle, 6.4.9, 92 diameter, 4.5.3, 47 DieudonnΓ e Lemma, 9.4.18, 158 dimension, 2.2.9 (5), 14 Dini Theorem, 7.2.10, 105 Dirac measure, 10.9.4 (1), 188 direct polar, 7.6.8, 116; 10.5.1, 177 direct sum decomposition, 2.1.7, 12 direct sum of vector spaces, 2.1.4 (5), 11 directed set, 1.2.15, 6 direction, 1.2.15, 6 directional derivative, 3.4.12, 28 discrete element, 3.3.6, 26 Discrete KreΦ ßn–Rutman Theorem, 3.3.8, 26 discrete topology, 9.1.8 (4), 147 disjoint measures, 10.9.4 (3), 190 disjoint sets, 4.1.10, 41 distance, 4.1.1, 40 distribution, 10.10.4, 195 distribution applies to a function, 10.10.5 (7), 196 Distribution Localization Principle, 10.10.12, 200 distribution of finite order, 10.10.5 (3), 195 distribution size at most m, 10.10.5 (3), 195 distribution of slow growth, 10.11.16, 209 distributions admitting convolution, 10.10.5 (9), 197 distributions convolute, 10.10.5 (9), 197 division algebra, 11.2.3, 215 domain, 1.1.2, 1 Dominated Extension Theorem, 3.5.4, 30 Double Prime Lemma, 7.6.6, 115 double prime mapping, 5.1.10 (8), 59 double sharp, Ex. 2.7, 19 downward-filtered set, 1.2.15, 6 dual diagram, 7.6.5, 115 dual group, 10.11.2, 203 dual norm of a functional, 5.1.10 (8), 59 dual of a locally convex space, 10.2.11, 173

264 dual of an operator, 7.6.2, 114 duality bracket, 10.3.3, 174 duality pair, 10.3.3, 174 dualization, 10.3.3, 174 Dualization Theorem, 10.3.9, 175 Dunford–Hille Theorem, 8.1.3, 121 Dunford Theorem, 8.2.7 (2), 129 Dvoretzky–Rogers Theorem, 5.5.9 (7), 73 dyadic-rational point, 9.3.13, 154 effective domain of definition, 3.4.2, 27 Eidelheit Separation Theorem, 3.8.14, 39 eigenvalue, 6.6.3 (4), 95 eigenvector, 6.6.3, 95 element of a set, 1.1.3 (4), 2 elementary compactum, 4.8.5, 54 endomorphism, 2.2.1, 12; 8.2.1, 126 endomorphism algebra, 2.2.8, 13; 5.6.5, 74 endomorphism space, 2.2.8, 13 Enflo counterexample, 8.3.12, 134 entourage, 4.1.5, 40 envelope, Ex. 1.11, 8 epigraph, 3.4.2, 26 epimorphism, 2.3.1, 15 ε-net, 8.3.2, 132 ε-perpendicular, 8.4.1, 134 ε-Perpendicular Lemma, 8.4.1, 134 Equicontinuity Principle, 7.2.4, 103 equicontinuous set, 4.2.8, 44 equivalence, 1.2.2, 3 equivalence class, 1.2.3 (4), 4 equivalent multinorms, 5.3.1, 62 equivalent seminorms, 5.3.3, 63 estimate for the diameter of a spherical layer, 6.2.1, 84 Euler identity, 8.5.17, 141 evaluation mapping, 10.3.4 (3), 174 everywhere-defined operator, 2.2.1, 12 everywhere dense set, 4.7.3 (3), 52 exact sequence, 2.3.4, 15 exact sequence at a term, 2.3.4, 15 exclave, 8.2.9, 130 expanding mapping, Ex. 4.14, 55 extended function, 3.4.2, 26 extended real axis, 3.8.1, 35 extended reals, 3.8.1, 35 extension of an operator, 2.3.6, 16 exterior of a set, 4.1.13, 41

Subject Index exterior point, 4.1.13, 41 Extreme and Discrete Lemma, 3.6.4, 32 extreme point, 3.6.1, 31 extreme set, 3.6.1, 31 face, 3.6.1, 31 factor set, 1.2.3 (4), 4 faithful representation, 8.2.2, 126 family, 1.1.3 (4), 2 filter, 1.3.3, 6 filterbase, 1.3.1, 6 finer cover, 9.6.1, 164 finer filter, 1.3.6, 7 finer multinorm, 5.3.1, 62 finer pretopology, 9.1.2, 146 finer seminorm, 5.3.3, 63 finest multinorm, 5.1.10 (2), 58 finite complement filter, 5.5.9 (3), 68 finite descent, Ex. 8.10, 144 finite-rank operator, 6.6.8, 97; 8.3.6, 132 finite-valued function, 5.5.9 (6), 72 first category set, 4.7.1, 52 first element, 1.2.6, 5 fixed point, Ex. 1.11, 8 flat, 3.1.2 (5), 20 formal duality, 2.3.15, 18 Fourier coefficient family, 6.3.15, 89 Fourier–Plancherel transform, 10.11.15, 209 Fourier–Schwartz transform, 10.11.19, 211 Fourier series, 6.3.16, 89 Fourier transform of a distribution, 10.11.19, 211 Fourier transform of a function, 10.11.3, 203 Fourier transform relative to a basis, 6.3.15, 89 Fr∆ echet space, 5.5.2, 66 Fredholm Alternative, 8.5.6, 138 Fredholm index, 8.5.1, 137 Fredholm operator, 8.5.1, 137 Fredholm Theorem, 8.5.8, 139 frontier of a set, 4.1.13, 41 from A into/to B, 1.1.1, 1 Fubini Theorem for distributions, 10.10.5 (8), 197 Fubini Theorem for measures, 10.9.4 (6), 190 full subalgebra, 11.1.5, 213 fully norming set, 8.1.1, 120

Subject Index Function Comparison Lemma, 3.8.3, 36 function of class C (m) , 10.9.9, 192 function of compact support, 9.6.4, 165 Function Recovery Lemma, 3.8.2, 35 functor, 10.9.4 (4), 190 fundamental net, 4.5.2, 47 fundamental sequence, 4.5.2, 47 fundamentally summable family of vectors, 5.5.9 (7), 73 gauge, 3.8.6, 37 gauge function, 3.8.6, 37 Gauge Theorem, 3.8.7, 37 -correspondence, 3.1.6, 21 -hull, 3.1.11, 21 -set, 3.1.1, 20 Gelfand–Dunford Theorem in an operator setting, 8.2.3, 127 Gelfand–Dunford Theorem in an algebraic setting, 11.3.2, 216 Gelfand formula, 5.6.8, 74 Gelfand–Mazur Theorem, 11.2.3, 215 Gelfand–NaΦ ßmark–Segal construction, 11.9.11, 232 Gelfand Theorem, 7.2.2, 103 Gelfand transform of an algebra, 11.6.8, 222 Gelfand transform of an element, 11.6.8, 221 Gelfand Transform Theorem, 11.6.9, 222 general form of a compact operator in Hilbert space, 6.6.9, 97 general form of a linear functional in Hilbert space, 6.4.2, 90 general form of a weakly continuous functional, 10.3.10, 175 general position, Ex. 3.10, 39 generalized derivative in the Sobolev sense, 10.10.5 (4), 196 Generalized Dini Theorem, 10.8.6, 183 generalized function, 10.10.4, 195 Generalized Riesz–Schauder Theorem, 8.4.10, 137 generalized sequence, 1.2.16, 6 Generalized Weierstrass Theorem, 10.9.9, 192 germ, 8.1.14, 124 GNS-construction, 11.9.11, 232 GNS-Construction Theorem, 11.9.10, 231

265 gradient mapping, 6.4.2, 90 Gram–Schmidt orthogonalization process, 6.3.14, 89 graph norm, 7.4.17, 111 Graph Norm Principle, 7.4.17, 111 greatest element, 1.2.6, 5 greatest lower bound, 1.2.9, 5 Grothendieck Criterion, 8.3.11, 133 Grothendieck Theorem, 8.3.9, 133 ground field, 2.1.3, 10 ground ring, 2.1.1, 10 group algebra, 10.9.4 (7), 191 group character, 10.11.1, 202 Haar integral, 10.9.4 (1), 189 Hahn–Banach Theorem, 3.5.3, 29 Hahn–Banach Theorem in analytical form, 3.5.4, 30 Hahn–Banach Theorem in geometric form, 3.8.12, 38 Hahn–Banach Theorem in subdifferential form, 3.5.4, 30 Hamel basis, 2.2.9 (5), 14 Hausdorff Completion Theorem, 4.5.12, 48 Hausdorff Criterion, 4.6.7, 50 Hausdorff metric, Ex. 4.8, 55 Hausdorff multinorm, 5.1.8, 57 Hausdorff multinormed space, 5.1.8, 57 Hausdorff space, 9.3.5, 152 Hausdorff Theorem, 7.6.12, 117 Hausdorff topology, 9.3.5, 152 H-closed space, Ex. 9.10, 168 Heaviside function, 10.10.5 (4), 196 Hellinger–Toeplitz Theorem, 6.5.3, 93 hermitian element, 11.7.1, 224 hermitian form, 6.1.1, 80 hermitian operator, 6.5.1, 93 hermitian state, 11.9.8, 230 Hilbert basis, 6.3.8, 88 Hilbert cube, 9.2.17 (2), 151 Hilbert dimension, 6.3.13, 89 Hilbert identity, 5.6.19, 78 Hilbert isomorphy, 6.3.17, 90 Hilbert–Schmidt norm, Ex. 8.9, 144 Hilbert–Schmidt operator, Ex. 8.9, 144 Hilbert–Schmidt Theorem, 6.6.7, 96 Hilbert space, 6.1.7, 81 Hilbert-space isomorphism, 6.3.17, 90 Hilbert sum, 6.1.10 (5), 84

266 HΞ older inequality, 5.5.9 (4), 69 holey disk, 4.8.5, 54 holomorphic function, 8.1.4, 122 Holomorphy Theorem, 8.1.5, 122 homeomorphism, 9.2.4, 149 homomorphism, 7.4.1, 107 HΞ ormander transform, Ex. 3.19, 39 hyperplane, 3.8.9, 38 hypersubspace, 3.8.9, 38 ideal, 11.4.1, 217 Ideal and Character Theorem, 11.6.6, 221 ideal correspondence, 7.3.3, 106 Ideal Correspondence Lemma, 7.3.4, 106 Ideal Correspondence Principle, 7.3.5, 106 Ideal Hahn–Banach Theorem, 7.5.9, 113 ideally convex function, 7.5.4, 112 ideally convex set, 7.1.3, 101 idempotent operator, 2.2.9 (4), 14 identical embedding, 1.1.3 (3), 2 identity, 10.9.4, 188 identity element, 11.1.1, 213 identity mapping, 1.1.3 (3), 2 identity relation, 1.1.3 (3), 1 image, 1.1.2, 1 image of a filterbase, 1.3.5 (1), 7 image of a set, 1.1.3 (5), 2 image of a topology, 9.2.12, 150 image topology, 9.2.12, 150 Image Topology Theorem, 9.2.11, 150 imaginary part of a function, 5.5.9 (4), 69 increasing mapping, 1.2.3 (5), 4 independent measure, 10.9.4 (3), 190 index, 8.5.1, 137 indicator function, 3.4.8 (2), 27 indiscrete topology, 9.1.8 (3), 147 induced relation, 1.2.3 (1), 4 induced topology, 9.2.17 (1), 151 inductive limit topology, 10.9.6, 191 inductive set, 1.2.19, 6 infimum, 1.2.9, 5 infinite-rank operator, 6.6.8, 97 infinite set, 5.5.9 (3), 68 inner product, 6.1.4, 80 integrable function, 5.5.9 (4), 69 integral, 5.5.9 (4), 68

Subject Index integral with respect to a measure, 10.9.3, 188 interior of a set, 4.1.13, 41 interior point, 4.1.13, 41 intersection of topologies, 9.1.14, 148 interval, 3.2.15, 24 Interval Addition Lemma, 3.2.15, 24 invariant subspace, 2.2.9 (4), 14 inverse-closed subalgebra, 11.1.5, 213 inverse image of a multinorm, 5.1.10 (3), 58 inverse image of a preorder, 1.2.3 (3), 4 inverse image of a seminorm, 5.1.4, 57 inverse image of a set, 1.1.3 (5), 2 inverse image of a topology, 9.2.9, 150 inverse image of a uniformity, 9.5.5 (3), 160 inverse image topology, 9.2.9, 150 Inverse Image Topology Theorem, 9.2.8, 149 inverse of a correspondence, 1.1.3 (1), 1 inverse of an element in an algebra, 11.1.5, 213 Inversion Theorem, 10.11.12, 208 invertible element, 11.1.5, 213 invertible operator, 5.6.10, 75 involution, 6.4.13, 92 involutive algebra, 6.4.13, 92 irreducible representation, 8.2.2, 127 irreflexive space, 5.1.10 (8), 59 isolated part of a spectrum, 8.2.9, 130 isolated point, 8.4.7, 136 isometric embedding, 4.5.11, 48 isometric isomorphism of algebras, 11.1.8, 215 isometric mapping, 4.5.11, 48 isometric representation, 11.1.8, 214 isometric ∗-isomorphism, 11.8.3, 226 isometric ∗-representation, 11.8.3, 226 isometry into, 4.5.11, 48 isometry onto, 4.5.11, 48 isomorphism, 2.2.5, 13 isotone mapping, 1.2.3, 4 James Theorem, 10.7.5, 181 Jensen inequality, 3.4.5, 27 join, 1.2.12, 5 Jordan arc, 4.8.2, 54 Jordan Curve Theorem, 4.8.3, 54 juxtaposition, 2.2.8, 13

Subject Index Kakutani Criterion, 10.7.1, 180 Kakutani Lemma, 10.8.7, 183 Kakutani Theorem, 7.4.11 (3), 109 Kantorovich space, 3.2.8, 23 Kantorovich Theorem, 3.3.4, 25 Kaplansky–Fukamija Lemma, 11.9.7, 230 Kato Criterion, 7.4.19, 111 kernel of an operator, 2.3.1, 15 ket-mapping, 10.3.1, 173 ket-topology, 10.3.5, 174 Kolmogorov Normability Criterion, 5.4.5, 66 KreΦ ßn–Milman Theorem, 10.6.5, 179 KreΦ ßn–Milman Theorem in subdifferential form, 3.6.5, 33 KreΦ ßn–Rutman Theorem, 3.3.5, 26 Krull Theorem, 11.4.8, 219 Kuratowski–Zorn Lemma, 1.2.20, 6 K-space, 3.2.8, 23 K-ultrametric, 9.5.13, 162 last element, 1.2.6, 5 lattice, 1.2.12, 5 lear trap map, 3.7.4, 34 least element, 1.2.6, 5 Lebesgue measure, 10.9.4 (1), 189 Lebesgue set, 3.8.1, 35 Lefschetz Lemma, 9.6.3, 165 left approximate inverse, 8.5.9, 139 left Haar measure, 10.9.4 (1), 189 left inverse of an element in an algebra, 11.1.3, 213 lemma on continuity of a convex function, 7.5.1, 112 lemma on the numeric range of a hermitian element, 11.9.3, 229 level set, 3.8.1, 35 Levy Projection Theorem, 6.2.2, 84 limit of a filterbase, 4.1.16, 42 Lindenstrauss space, 5.5.9 (5), 71 Lindenstrauss–Tzafriri Theorem, 7.4.11 (3), 110 linear change of a variable under the subdifferential sign, 3.5.4, 30 linear combination, 2.3.12, 17 linear correspondence, 2.2.1, 12; 3.1.7, 21 linear functional, 2.2.4, 13 linear operator, 2.2.1, 12 linear representation, 8.2.2, 126

267 linear set, 2.1.4 (3), 11 linear space, 2.1.4 (3), 11 linear span, 3.1.14, 22 linear topological space, 10.1.3, 169 linear topology, 10.1.3, 169 linearly independent set, 2.2.9 (5), 14 linearly-ordered set, 1.2.19, 6 Lions Theorem of Supports, 10.10.5 (9), 197 Liouville Theorem, 8.1.10, 123 local data, 10.9.11, 193 locally compact group, 10.9.4 (1), 188 locally compact space, 9.4.20, 159 locally compact topology, 9.4.20, 159 locally convex space, 10.2.9, 172 locally convex topology, 10.2.1, 171 locally finite cover, 9.6.2, 164 locally integrable function, 9.6.17, 167 locally Lipschitz function, 7.5.6, 112 loop, 4.8.2, 54 lower bound, 1.2.4, 5 lower limit, 4.3.5, 45 lower right Dini derivative, 4.7.7, 53 lower semicontinuous, 4.3.3, 44 L2 -Fourier transform, 10.11.15, 209 Mackey–Arens Theorem, 10.4.5, 176 Mackey Theorem, 10.4.6, 176 Mackey topology, 10.4.4, 176 mapping, 1.1.3 (3), 1 massive subspace, 3.3.2, 25 matrix form, 2.2.9 (4), 14 maximal element, 1.2.10, 5 maximal ideal, 11.4.5, 218 maximal ideal space, 11.6.7, 221 Maximal Ideal Theorem, 11.5.3, 220 Mazur Theorem, 10.4.9, 177 meager set, 4.7.1, 52 measure, 10.9.3, 188 Measure Localization Principle, 10.9.10, 192 measure space, 5.5.9 (4), 69 meet, 1.2.12, 5 member of a set, 1.1.3 (4), 2 metric, 4.1.1, 40 metric space, 4.1.1, 40 metric topology, 4.1.9, 41 metric uniformity, 4.1.5, 40 Metrizability Criterion, 5.4.2, 64 metrizable multinormed space, 5.4.1, 64 minimal element, 1.2.10, 5

268 Minimal Ideal Theorem, 11.5.1, 219 Minkowski–Ascoli–Mazur Theorem, 3.8.12, 38 Minkowski functional, 3.8.6, 37 Minkowski inequality, 5.5.9 (4), 69 minorizing set, 3.3.2, 25 mirror, 10.2.7, 172 module, 2.1.1, 10 modulus of a scalar, 5.1.10 (4), 58 modulus of a vector, 3.2.12, 24 mollifier, 9.6.14, 166 mollifying kernel, 9.6.14, 166 monomorphism, 2.3.1, 15 monoquotient, 2.3.11, 17 Montel space, 10.10.9 (2), 199 Moore subnet, 1.3.5 (2), 7 morphism, 8.2.2, 126; 11.1.2, 213 morphism representing an algebra, 8.2.2, 126 Motzkin formula, 3.1.13 (5), 22 multimetric, 9.5.9, 161 multimetric space, 9.5.9, 161 multimetric uniformity, 9.5.9, 161 multimetrizable topological space, 9.5.10, 161 multimetrizable uniform space, 9.5.10, 161 multinorm, 5.1.6, 57 Multinorm Comparison Theorem, 5.3.2, 62 multinorm summable family of vectors, 5.5.9 (7), 73 multinormed space, 5.1.6, 57 multiplication formula, 10.11.5, 205 multiplication of a germ by a complex number, 8.1.16, 125 multiplicative linear operator, 8.2.2, 126 natural order, 3.2.6 (1), 23 negative part, 3.2.12, 24 neighborhood about a point, 9.1.1 (2), 146 neighborhood about a point in a metric space, 4.1.9, 41 neighborhood filter, 4.1.10, 41 neighborhood filter of a set, 9.3.7, 152 neighborhood of a set, 8.1.13 (2), 124; 9.3.7, 152 Nested Ball Theorem, 4.5.7, 47 nested sequence, 4.5.7, 47 net, 1.2.16, 6

Subject Index net having a subnet, 1.3.5 (2), 7 net lacking a subnet, 1.3.5 (2), 7 Neumann series, 5.6.9, 75 Neumann Series Expansion Theorem, 5.6.9, 75 neutral element, 2.1.4 (3), 11; 10.9.4, 188 Nikol 0skiΦ ß Criterion, 8.5.22, 143 Noether Criterion, 8.5.14, 140 nonarchimedean element, 5.5.9 (5), 70 nonconvex cone, 3.1.2 (4), 20 Nonempty Subdifferential Theorem, 3.5.8, 31 non-everywhere-defined operator, 2.2.1, 12 nonmeager set, 4.7.1, 52 nonpointed cone, 3.1.2 (4), 20 nonreflexive space, 5.1.10 (8), 59 norm, 5.1.9, 57 norm convergence, 5.5.9 (7), 73 normable multinormed space, 5.4.1, 64 normal element, 11.7.1, 224 normal operator, Ex. 8.17, 145 normal space, 9.3.11, 153 normalized element, 6.3.5, 88 normally solvable operator, 7.6.9, 116 normative inequality, 5.1.10 (7), 59 normed algebra, 5.6.3, 74 normed dual, 5.1.10 (8), 59 normed space, 5.1.9, 57 normed space of bounded elements, 5.5.9 (5), 70 norming set, 8.1.1, 120 norm-one element, 5.5.6, 68 nowhere dense set, 4.7.1, 52 nullity, 8.5.1, 137 numeric family, 1.1.3 (4), 2 numeric function, 9.6.4, 165 numeric range, 11.9.1, 229 numeric set, 1.1.3 (4), 2 one-point compactification, 9.4.22, 159 one-to-one correspondence, 1.1.3 (3), 2 open ball, 4.1.3, 40 open ball of RN , 9.6.16, 166 open correspondence, 7.3.12, 107 Open Correspondence Principle, 7.3.13, 107 open cylinder, 4.1.3, 40 open halfspace, Ex. 3.3, 39

Subject Index Open Mapping Theorem, 7.4.6, 108 open segment, 3.6.1, 31 open set, 9.1.4, 146 open set in a metric space, 4.1.11, 41 openness at a point, 7.3.6, 107 operator, 2.2.1, 12 operator ideal, 8.3.3, 132 operator norm, 5.1.10 (7), 58 operator representation, 8.2.2, 126 order, 1.2.2, 3 order by inclusion, 1.3.1, 6 order compatible with vector structure, 3.2.1, 22 order ideal, 10.8.11, 183 order of a distribution, 10.10.5 (3), 195 ordered set, 1.2.2, 3 ordered vector space, 3.2.1, 22 ordering, 1.2.2, 3 ordering cone, 3.2.4, 23 oriented envelope, 4.8.8, 54 orthocomplement, 6.2.5, 85 orthogonal complement, 6.2.5, 85 orthogonal family, 6.3.1, 87 orthogonal orthoprojections, 6.2.12, 86 orthogonal set, 6.3.1, 87 orthogonal vectors, 6.2.5, 85 orthonormal family, 6.3.6, 88 orthonormal set, 6.3.6, 88 orthonormalized family, 6.3.6, 88 orthoprojection, 6.2.7, 85 Orthoprojection Summation Theorem, 6.3.3, 87 Orthoprojection Theorem, 6.2.10, 85 Osgood Theorem, 4.7.5, 52 pair-dual space, 10.3.3, 174 pairing, 10.3.3, 174 pairwise orthogonality of finitely many orthoprojections, 6.2.14, 86 paracompact space, 9.6.9, 166 Parallelogram Law, 6.1.8, 81 Parseval identity, 6.3.16, 89; 10.11.12, 208 part of an operator, 2.2.9 (4), 14 partial correspondence, 1.1.3 (6), 2 partial operator, 2.2.1, 12 partial order, 1.2.2, 3 partial sum, 5.5.9 (7), 73 partition of unity, 9.6.6, 165 partition of unity subordinate to a cover, 9.6.7, 166

269 patch, 10.9.11, 193 perforated disk, 4.8.5, 54 periodic distribution, 10.11.17 (7), 211 Pettis Theorem, 10.7.4, 181 Phillips Theorem, 7.4.13, 110 Plancherel Theorem, 10.11.14, 209 point finite cover, 9.6.2, 164 point in a metric space, 4.1.1, 40 point in a space, 2.1.4 (3), 11 point in a vector space, 2.1.3, 10 pointwise convergence, 9.5.5 (6), 161 pointwise operation, 2.1.4 (4), 11 polar, 7.6.8, 116; 10.5.1, 177 Polar Lemma, 7.6.11, 116 polarization identity, 6.1.3, 80 Pontryagin–van Kampen Duality Theorem, 10.11.2, 203 poset, 1.2.2, 3 positive cone, 3.2.5, 23 positive definite inner product, 6.1.4, 80 positive distribution, 10.10.5 (2), 195 positive element of a C ∗ -algebra, 11.9.4, 230 positive form on a C ∗ -algebra, Ex. 11.11, 235 positive hermitian form, 6.1.4, 80 positive matrix, Ex. 3.13, 39 positive operator, 3.2.6 (3), 23 positive part, 3.2.12, 24 positive semidefinite hermitian form, 6.1.4, 80 positively homogeneous functional, 3.4.7 (2), 27 powerset, 1.2.3 (4), 4 precompact set, Ex. 9.16, 168 pre-Hilbert space, 6.1.7, 81 preimage of a multinorm, 5.1.10 (3), 58 preimage of a seminorm, 5.1.4, 57 preimage of a set, 1.1.3 (5), 2 preintegral, 5.5.9 (4), 68 preneighborhood, 9.1.1 (2), 146 preorder, 1.2.2, 3 preordered set, 1.2.2, 4 preordered vector space, 3.2.1, 22 presheaf, 10.9.4 (4), 190 pretopological space, 9.1.1 (2), 146 pretopology, 9.1.1, 146 primary Banach space, Ex. 7.17, 119 prime mapping, 6.4.1, 90 Prime Theorem, 10.2.13, 173

270 Principal Theorem of the Holomorphic Functional Calculus, 8.2.4, 128 product, 4.3.2, 44 product of a distribution and a function, 10.10.5 (7), 196 product of germs, 8.1.16, 125 product of sets, 1.1.1, 1; 2.1.4 (4), 11 product of topologies, 9.2.17 (2), 151 product of vector spaces, 2.1.4 (4), 11 product topology, 4.3.2, 44; 9.2.17 (2), 151 projection onto X1 along X2 , 2.2.9 (4), 14 projection to a set, 6.2.3, 84 proper ideal, 11.4.5, 218 pseudometric, 9.5.7, 161 p-sum, 5.5.9 (6), 71 p-summable family, 5.5.9 (4), 70 punctured compactum, 9.4.21, 159 pure subalgebra, 11.1.5, 213 Pythagoras Lemma, 6.2.8, 85 Pythagoras Theorem, 6.3.2, 87 quasinilpotent, Ex. 8.18, 145 quotient mapping, 1.2.3 (4), 4 quotient multinorm, 5.3.11, 64 quotient of a mapping, 1.2.3 (4), 4 quotient of a seminormed space, 5.1.10 (5), 58 quotient seminorm, 5.1.10 (5), 58 quotient set, 1.2.3 (4), 4 quotient space of a multinormed space, 5.3.11, 64 quotient vector space, 2.1.4 (6), 12 radical, 11.6.11, 223 Radon F-measure, 10.9.3, 188 Radon–Nikod∆ ym Theorem, 10.9.4 (3), 190 range of a correspondence, 1.1.2, 1 rank, 8.5.7 (2), 139 rare set, 4.7.1, 52 Rayleigh Theorem, 6.5.2, 93 real axis, 2.1.2, 10 real carrier, 3.7.1, 33 real C-measure, 10.9.4 (3), 189 real distribution, 10.10.5 (5), 196 real hyperplane, 3.8.9, 38 real measure, 10.9.4, 189 real part map, 3.7.2, 33 real part of a function, 5.5.9 (4), 69 real part of a number, 2.1.2, 10

Subject Index real subspace, 3.1.2 (3), 20 real vector space, 2.1.3, 10 realification, 3.7.1, 33 realification of a pre-Hilbert space, 6.1.10 (2), 83 realifier, 3.7.2, 33 reducible representation, 8.2.2, 127 refinement, 9.6.1, 164 reflection of a function, 10.10.5, 197 reflexive relation, 1.2.1, 3 reflexive space, 5.1.10 (8), 59 regular distribution, 10.10.5 (1), 195 regular operator, 3.2.6 (3), 23 regular space, 9.3.9, 153 regular value of an operator, 5.6.13, 76 relation, 1.1.3 (2), 1 relative topology, 9.2.17 (1), 151 relatively compact set, 4.4.4, 46 removable singularity, 8.2.5 (2), 128 representation, 8.2.2, 126 representation space, 8.2.2, 126 reproducing cone, Ex. 7.12, 119 residual set, 4.7.4, 52 resolvent of an element of an algebra, 11.2.1, 215 resolvent of an operator, 5.6.13, 76 resolvent set of an operator, 5.6.13, 76 resolvent value of an element of an algebra, 11.2.1, 215 resolvent value of an operator, 5.6.13, 76 restriction, 1.1.3 (5), 2 restriction of a distribution, 10.10.5 (6), 196 restriction of a measure, 10.9.4 (4), 190 restriction operator, 10.9.4 (4), 190 reversal, 1.2.5, 5 reverse order, 1.2.3 (2), 4 reverse polar, 7.6.8, 116; 10.5.1, 177 reversed multiplication, 11.1.6, 214 Riemann function, 4.7.7, 53 Riemann–Lebesgue Lemma, 10.11.5 (3), 204 Riemann Theorem on Series, 5.5.9 (7), 73 Riesz Criterion, 8.4.2, 134 Riesz Decomposition Property, 3.2.16, 24 Riesz–Dunford integral, 8.2.1, 126

Subject Index Riesz–Dunford Integral Decomposition Theorem, 8.2.13, 131 Riesz–Dunford integral in an algebraic setting, 11.3.1, 216 Riesz–Fisher Completeness Theorem, 5.5.9 (4), 70 Riesz–Fisher Isomorphism Theorem, 6.3.16, 89 Riesz idempotent, 8.2.11, 130 Riesz–Kantorovich Theorem, 3.2.17, 24 Riesz operator, Ex. 8.15, 145 Riesz Prime Theorem, 6.4.1, 90 Riesz projection, 8.2.11, 130 Riesz–Schauder operator, Ex. 8.11, 144 Riesz–Schauder Theorem, 8.4.8, 136 Riesz space, 3.2.7, 23 Riesz Theorem, 5.3.5, 63 right approximate inverse, 8.5.9, 139 right Haar measure, 10.9.4 (1), 189 right inverse of an element in an algebra, 11.1.3, 213 R-measure, 10.9.4 (3), 189 rough draft, 4.8.8, 54 row-by-column rule, 2.2.9 (4), 14 salient cone, 3.2.4, 23 Sard Theorem, 7.4.12, 110 scalar, 2.1.3, 10 scalar field, 2.1.3, 10 scalar multiplication, 2.1.3, 10 scalar product, 6.1.4, 80 scalar-valued function, 9.6.4, 165 Schauder Theorem, 8.4.6, 135 Schwartz space of distributions, 10.11.16, 209 Schwartz space of functions, 10.11.6, 206 Schwartz Theorem, 10.10.10, 199 second dual, 5.1.10 (8), 59 selfadjoint operator, 6.5.1, 93 semi-extended real axis, 3.4.1, 26 semi-Fredholm operator, Ex. 8.13, 145 semi-inner product, 6.1.4, 80 semimetric, 9.5.7, 161 semimetric space, 9.5.7, 161 seminorm, 3.7.6, 34 seminorm associated with a positive element, 5.5.9 (5), 70 seminormable space, 5.4.6, 66

271 seminormed space, 5.1.5, 57 semisimple algebra, 11.6.11, 223 separable space, 6.3.14, 89 separated multinorm, 5.1.8, 57 separated multinormed space, 5.1.8, 57 separated topological space, 9.3.2, 151 separated topology, 9.3.2, 151 separating hyperplane, 3.8.13, 39 Separation Theorem, 3.8.11, 38 Sequence Prime Principle, 7.6.13, 117 sequence space, 3.3.1 (2), 25 Sequence Star Principle, 6.4.12, 92 series sum, 5.5.9 (7), 73 sesquilinear form, 6.1.2, 80 set absorbing another set, 3.4.9, 28 set in a space, 2.1.4 (3), 11 set lacking a distribution, 10.10.5 (6), 196 set lacking a functional, 10.8.13, 184 set lacking a measure, 10.9.4 (5), 190 set of arrival, 1.1.1, 1 set of departure, 1.1.1, 1 set of second category, 4.7.1, 52 set supporting a measure, 10.9.4 (5), 190 set that separates the points of another set, 10.8.9, 183 set void of a distribution, 10.10.5 (6), 196 set void of a functional, 10.8.13, 184 set void of a measure, 10.9.4 (5), 190 setting in duality, 10.3.3, 174 setting primes, 7.6.5, 115 sheaf, 10.9.11, 193 shift, 10.9.4 (1), 189 Shilov boundary, Ex. 11.8, 234 Shilov Theorem, 11.2.4, 215 short sequence, 2.3.5, 16 σ-compact, 10.9.8, 192 signed measure, 10.9.4 (3), 190 simple convergence, 9.5.5 (6), 161 simple function, 5.5.9 (6), 72 simple Jordan loop, 4.8.2, 54 single-valued correspondence, 1.1.3 (3), 1 Singularity Condensation Principle, 7.2.12, 105 Singularity Fixation Principle, 7.2.11, 105 skew field, 11.2.3, 215

272 slowly increasing distribution, 10.11.16, 209 smooth function, 9.6.13, 166 smoothing process, 9.6.18, 167 Snowflake Lemma, 2.3.16, 18 space countable at infinity, 10.9.8, 192 space of bounded elements, 5.5.9 (5), 70 space of bounded functions, 5.5.9 (2), 68 space of bounded operators, 5.1.10 (7), 59 space of compactly-supported distributions, 10.10.5 (9), 197 space of convergent sequences, 5.5.9 (3), 68 space of distributions of order at most m, 10.10.8, 199 space of essentially bounded functions, 5.5.9 (5), 70 space of finite-order distributions, 10.10.8, 199 space of functions vanishing at infinity, 5.5.9 (3), 68 space of X-valued p-summable functions, 5.5.9 (6), 72 space of p-summable functions, 5.5.9 (4), 69 space of p-summable sequences, 5.5.9 (4), 70 space of tempered distributions, 10.11.16, 209 space of vanishing sequences, 5.5.9 (3), 68 Spectral Decomposition Lemma, 6.6.6, 96 Spectral Decomposition Theorem, 8.2.12, 130 Spectral Endpoint Theorem, 6.5.5, 94 Spectral Mapping Theorem, 8.2.5, 128 Spectral Purity Theorem, 11.7.11, 226 spectral radius of an operator, 5.6.6, 74 Spectral Theorem, 11.8.6, 227 spectral value of an element of an algebra, 11.2.1, 215 spectral value of an operator, 5.6.13, 76 spectrum, 10.2.7, 172 spectrum of an element of an algebra, 11.2.1, 215 spectrum of an operator, 5.6.13, 76 spherical layer, 6.2.1, 84 ∗-algebra, 6.4.13, 92

Subject Index ∗-isomorphism, 11.8.3, 226 ∗-linear functional, 2.2.4, 13 ∗-representation, 11.8.3, 226 star-shaped set, 3.1.2 (7), 20 state, 11.9.1, 229 Steklov condition, 6.3.10, 88 Steklov Theorem, 6.3.11, 88 step function, 5.5.9 (6), 72 Stone Theorem, 10.8.10, 183 Stone–Weierstrass Theorem for C(Q, C), 11.8.2, 226 Stone–Weierstrass Theorem for C(Q, R), 10.8.17, 186 Strict Separation Theorem, 10.4.8, 177 strict subnet, 1.3.5 (2), 7 strictly positive real, 4.1.3, 40 strong order-unit, 5.5.9 (5), 70 strong uniformity, 9.5.5 (6), 161 stronger multinorm, 5.3.1, 62 stronger pretopology, 9.1.2, 146 stronger seminorm, 5.3.3, 63 strongly holomorphic function, 8.1.5, 122 structure of a subdifferential, 10.6.3, 179 subadditive functional, 3.4.7 (4), 27 subcover, 9.6.1, 164 subdifferential, 3.5.1, 29 sublattice, 10.8.2, 181 sublinear functional, 3.4.6, 27 submultiplicative norm, 5.6.1, 73 subnet, 1.3.5 (2), 7 subnet in a broad sense, 1.3.5 (2), 7 subrepresentation, 8.2.2, 126 subspace of a metric space, 4.5.14, 49 subspace of a topological space, 9.2.17 (1), 151 subspace of an ordered vector space, 3.2.6 (2), 23 subspace topology, 9.2.17 (1), 151 Sukhomlinov–Bohnenblust–Sobczyk Theorem, 3.7.12, 35 sum of a family in the sense of Lp , 5.5.9 (6), 71 sum of germs, 8.1.16, 125 summable family of vectors, 5.5.9 (7), 73 summable function, 5.5.9 (4), 69 superset, 1.3.3, 6 sup-norm, 10.8.1, 181 support function, 10.6.4, 179

Subject Index

273

support of a distribution, 10.10.5 (6), 196 support of a functional, 10.8.12, 184 support of a measure, 10.9.4 (5), 190 supporting function, 10.6.4, 179 supremum, 1.2.9, 5 symmetric Hahn–Banach formula, Ex. 3.10, 39 symmetric relation, 1.2.1, 3 symmetric set, 3.1.2 (7), 20 system with integration, 5.5.9 (4), 68 Szankowski Counterexample, 8.3.13, 134 tail filter, 1.3.5 (2), 7 τ -dual of a locally convex space, 10.2.11, 173 Taylor Series Expansion Theorem, 8.1.9, 123 tempered distribution, 10.11.16, 209 tempered function, 5.1.10 (6), 58; 10.11.6, 206 tempered Radon measure, 10.11.17 (3), 210 test function, 10.10.1, 194 test function space, 10.10.1, 194 theorem on Hilbert isomorphy, 6.3.17, 90 theorem on the equation A = B, 2.3.13, 17 theorem on the equation A = B, 2.3.8, 16 theorem on the general form of a distribution, 10.10.14, 201 theorem on the inverse image of a vector topology, 10.1.6, 171 theorem on the repeated Fourier transform, 10.11.13, 209 theorem on the structure of a locally convex topology, 10.2.2, 171 theorem on the structure of a vector topology, 10.1.4, 170 theorem on topologizing by a family of mappings, 9.2.16, 151 there is a unique x, 2.3.9, 17 Tietze–Urysohn Theorem, 10.8.20, 186 topological isomorphism, 9.2.4, 149 topological mapping, 9.2.4, 149 Topological Separation Theorem, 7.5.12, 114

X X

topological space, 9.1.7, 147 topological structure of a convex set, 7.1.1, 100 topological subdifferential, 7.5.8, 113 topological vector space, 10.1.1, 169 topologically complemented subspace, 7.4.9, 108 topology, 9.1.7, 147 topology compatible with duality, 10.4.1, 175 topology compatible with vector structure, 10.1.1, 169 topology given by open sets, 9.1.12, 148 topology of a multinormed space, 5.2.8, 61 topology of a uniform space, 9.5.3, 160 topology of the distribution space, 10.10.6, 198 topology of the test function space, 10.10.6, 198 total operator, 2.2.1, 12 total set of functionals, 7.4.11 (2), 109 totally bounded, 4.6.3, 49 transitive relation, 1.2.1, 3 translation, 10.9.4 (1), 189 translation of a distribution, 10.11.17 (7), 211 transpose of an operator, 7.6.2, 114 trivial topology, 9.1.8 (3), 147 truncator, 9.6.19 (1), 167 truncator direction, 10.10.2 (5), 194 truncator set, 10.10.2, 194 twin of a Hilbert space, 6.1.10 (3), 83 twin of a vector space, 2.1.4 (2), 10 Two Norm Principle, 7.4.16, 111 two-sided ideal, 8.3.3, 132; 11.6.2, 220 Tychonoff cube, 9.2.17 (2), 151 Tychonoff product, 9.2.17 (2), 151 Tychonoff space, 9.3.15, 155 Tychonoff Theorem, 9.4.8, 157 Tychonoff topology, 9.2.17 (2), 151 Tychonoff uniformity, 9.5.5 (4), 160 T1 -space, 9.3.2, 151 T1 -topology, 9.3.2, 151 T2 -space, 9.3.5, 152 T3 -space, 9.3.9, 153 T31 /2 -space, 9.3.15, 155 T4 -space, 9.3.11, 153 ultrafilter, 1.3.9, 7 ultrametric inequality, 9.5.14, 162

274 ultranet, 9.4.4, 156 unconditionally summable family of vectors, 5.5.9 (7), 73 unconditionally summable sequence, 5.5.9 (7), 73 underlying set, 2.1.3, 10 Uniform Boundedness Principle, 7.2.5, 103 uniform convergence, 7.2.10, 105; 9.5.5 (6), 161 uniform space, 9.5.1, 159 uniformity, 9.5.1, 159 uniformity of a multinormed space, 5.2.4, 60 uniformity of a seminormed space, 5.2.2, 60 uniformity of a topological vector space, 10.1.10, 171 uniformity of the empty set, 9.5.1, 159 uniformizable space, 9.5.4, 160 uniformly continuous mapping, 4.2.5, 44 unit, 10.9.4, 188 unit ball, 5.2.11, 61 unit circle, 8.1.3, 121 unit disk, 8.1.3, 121 unit element, 11.1.1, 213 unit sphere, Ex. 10.6, 212 unit vector, 6.3.5, 88 unital algebra, 11.1.1, 213 unitary element, 11.7.1, 224 unitary operator, 6.3.17, 90 unitization, 11.1.2, 213 unity, 11.1.1, 213 unity of a group, 10.9.4 (1), 188 unity of an algebra, 11.1.1, 213 unordered sum, 5.5.9 (7), 73 unorderly summable sequence, 5.5.9 (7), 73 Unremovable Spectral Boundary Theorem, 11.2.6, 216 upper bound, 1.2.4, 4 upper envelope, 3.4.8 (3), 28

Subject Index upper right Dini derivative, 4.7.7, 53 upward-filtered set, 1.2.15, 6 Urysohn Great Lemma, 9.3.13, 154 Urysohn Little Lemma, 9.3.10, 153 Urysohn Theorem, 9.3.14, 155 2-Ultrametric Lemma, 9.5.15, 162 vague topology, 10.9.5, 191 value of a germ at a point, 8.1.21, 126 van der Waerden function, 4.7.7, 53 vector, 2.1.3, 10 vector addition, 2.1.3, 10 vector field, 5.5.9 (6), 71 vector lattice, 3.2.7, 23 vector space, 2.1.3, 10 vector sublattice, 10.8.4 (4), 182 vector topology, 10.1.1, 169 Volterra operator, Ex. 5.12, 79 von Neumann–Jordan Theorem, 6.1.9, 81 V -net, 4.6.2, 49 V -small, 4.5.3, 47 weak derivative, 10.10.5 (4), 196 weak multinorm, 5.1.10 (4), 58 weak topology, 10.3.5, 174 weak∗ topology, 10.3.11, 175 weak uniformity, 9.5.5 (6), 161 weaker pretopology, 9.1.2, 146 weakly holomorphic function, 8.1.5, 122 weakly operator holomorphic function, 8.1.5, 122 Weierstrass function, 4.7.7, 53 Weierstrass Theorem, 4.4.5, 46; 9.4.5, 157 Well-Posedness Principle, 7.4.6, 108 Wendel Theorem, 10.9.4 (7), 191 Weyl Criterion, 6.5.4, 93 X-valued function, 5.5.9 (6), 71 Young inequality, 5.5.9 (4), 69 zero of a vector space, 2.1.4 (3), 11