A nonlinear transfer technique for renorming

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1951 Aníbal Moltó ...

Author: Aníbal Moltó | José Orihuela | Stanimir Troyanski | Manuel Valdivia

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1951

Aníbal Moltó · José Orihuela Stanimir Troyanski · Manuel Valdivia

A Nonlinear Transfer Technique for Renorming

ABC

Aníbal Moltó Manuel Valdivia

José Orihuela Stanimir Troyanski

Departamento de Análisis Matemático Facultad de Matemáticas Universidad de Valencia Dr. Moliner 50 46100 Burjasot, Valencia Spain [email protected] [email protected]

Departamento de Matemáticas Universidad de Murcia Campus Espinardo 30100 Murcia Spain http://webs.um.es/joseori [email protected] [email protected]

ISBN: 978-3-540-85030-4 e-ISBN: 978-3-540-85031-1 DOI: 10.1007/978-3-540-85031-1 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008932180 Mathematics Subject Classification (2000): 46B03, 46B20, 46B26, 46T20, 54D20, 54E18, 54E25, 54E35 c 2009 Springer-Verlag Berlin Heidelberg ° This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publishing Services Printed on acid-free paper 987654321 springer.com

Preface

Banach spaces are objects with a linear structure so linear maps have been considered the natural tool for transferring good norms from one Banach space to another. It is well known that a Banach space X admits an equivalent strictly convex (rotund) norm if there is a bounded linear one-to-one operator T : X → Y where Y has such a norm. For example, J. Lindenstrauss proved that in any reﬂexive space X there is such an operator T : X → c0 (Γ) for some set Γ. F. Dashiell and J. Lindenstrauss gave an example of a strictly convex renormable space without such an operator into c0 (Γ) for any Γ. For that reason we are searching for a non linear transfer technique. We consider here locally uniformly rotund (LUR) norms, a property adding to strict convexity the coincidence of the weak and the norm topologies on the unit sphere. For these norms a class of non linear maps was not only more powerful but even more natural for this purpose, as evinced by the solution of an old open problem due to Kadec using this class of non linear maps. The scope of this technique is not restricted to that particular case but, on the contrary, oﬀers a uniﬁed method of obtaining this renorming, roughly speaking, in all cases in which this is known to be possible. We have been lecturing on these new techniques throughout the courses given in the Spring School of Paseky nad Jizerou in 1998; in the Workshop in Banach spaces, Prague 2000; and in the 28th, 30th and 31st Winter Schools of Lhota nad Rohanovem on Abstract Analysis, in 2000, 2002 and 2003, places where these notes had their genesis. We would like to thank Professors J. Lukes, J. Kottas, V. Zizler, P. Holick´ y, L. Zaj´ıˇcek, J. Tiser, M. Fabian and O. Kalenda for their invitations and their warm hospitality. Diﬀerent parts of these notes have also been presented in seminars and conferences, such as the Choquet, Godefroy, Rogalski, Saint Raymond Analysis Seminar, University Pierre and Marie Curie, Paris VI, 1999 and 2001; Laboratoire de math´ematiques pures de Bordeaux, University of Bordeaux, 1999; Functional Analysis Seminar and Analytic Topology Seminar, Mathematical Institute, Oxford University 2001 and 2002; VII Conference on Function Theory on V

VI

Preface

Inﬁnite Dimensional Spaces, UCM, Madrid in 2001; Geometry of Banach spaces, Mathematisches Forschungsinstitut Oberwolfach, Germany, 2003; Interplay between Topology and Analysis at the International Congress Massee, Borovets, Bulgaria, 2003; Spring School on Non Separable Banach Spaces, Paseky nad Jizerou in 2004, and the Contemporary Ramiﬁcations of Banach space theory conference in honour of Joram Lindenstrauss and Lior Tzafriri, Institute of Advance Studies, Hebrew University of Jerusalem, 2005. We would like to thank G. Godefroy, R. Deville, C. J. K. Batty, P. Collins, J. L. Gonz´ alez Llavona, D. Azagra, M. Jim´enez, H. K¨onig, J. Lindenstrauss, N. TomczakJaegermann, P. Kenderov, J. Lukes, M. Fabian, P. H´ ajek, V. Zizler, L. Tzafriri, T. Szankowski and M. Zippin for their excellent qualities as hosts and their grace and patience as audiences. J. Lindenstrauss deserves special gratitude for his insightful comments and encouragement with the topics presented here. Thanks are also due to I. Namioka for reading these notes, providing us with diﬀerent points of view and excellent mathematical ideas. We wish to thank R. Haydon for many helpful suggestions and for our always interesting and stimulating conversations. Last, but certainly not least, we would like to express our debt to G. Godefroy, who was the ﬁrst mathematician to suggest to us the idea of publishing all this material together, constantly encouraging us to ﬁnish our project. Therefore despite the fact that the content of these notes is new and has not been published elsewhere, they have a self-contained and uniﬁed approach to the study of the existence of local uniformly rotund norms with a new point of view. As a result we hope they are accessible for readers with a basic knowledge of Functional Analysis and Set Theoretic Topology. We study maps from a normed space X to a metric space Y which provide a LUR renorming in X. These maps are just those which satisfy two conditions that we call σ-slicely continuity and co-σ-continuity. Our main goal here is to characterize both properties, applying them as a new frame for LUR renormings. The characterization is an interplay between Functional Analysis, Optimization and Topology. We use ε-subdiﬀerentials of Lipschitz functions and apply methods of metrization theory to the study of weak topologies. For example we ﬁnd that any one-to-one operator T from X (reﬂexive, or even weakly countably determined) into c0 (Γ) satisﬁes both conditions. Nevertheless our maps can be far away from the class of linear maps even when Y is a normed space. For instance the duality map from X into its dual is σ-slicely continuous if the norm of X is Fr´echet diﬀerentiable. If in addition the dual norm is Gˆ ateaux diﬀerentiable, then the duality map is co-σ-continuous and X is LUR renormable. Murcia and Valencia, July 2007

An´ıbal Molt´ o Jos´e Orihuela Stanimir Troyanski Manuel Valdivia

Acknowledgement

The authors would like to thank all who supported the research for this work. An´ıbal Molt´ o has been partially supported by BFM2003–07540/MATE and MTM2007-064521 del Ministerio de Ciencia y Tecnolog´ıa MCIT y FEDER. Jos´e Orihuela has been partially supported by PI–55/00872/FS/01, and 00690/PI/2004 Fundaci´ on S´eneca CARM and BFM 2002–01719, MTM200508379 del Ministerio de Ciencia y Tecnolog´ıa MCIT and Ministerio de Educaci´on y Ciencia. Stanimir Troyanski has been partially supported by Grant MM–1401/04 of the NSFR of Bulgaria, by PI–55/00872/FS/01, and 00690/PI/2004 Fundaci´ on S´eneca CARM, by BFM 2002–01719, MTM2005-08379 del Ministerio de Ciencia y Tecnolog´ıa MCIT and Ministerio de Educaci´ on y Ciencia and by a research project of the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences. Manuel Valdivia has been partially supported by MEC and FEDER Project MTM2005-08210.

VII

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

σ-Continuous and Co-σ-continuous Maps . . . . . . . . . . . . . . . . . . 2.1 σ-Continuous and σ-Slicely Continuous Maps . . . . . . . . . . . . . . . 2.2 Co-σ-continuous Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 P-Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Co-σ-continuity of Maps Associated to PRI and M-Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Co-σ-continuity of the Operator of Haydon in C(Υ), Υ a Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Co-σ-continuity in Weakly Countable Determined Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Co-σ-continuous Maps in C(K). The Case of the Compact of Helly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 23 28

3

4

Generalized Metric Spaces and Locally Uniformly Rotund Renormings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Discreteness and Network Conditions . . . . . . . . . . . . . . . . . . . . . . 3.2 Fragmentability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 LUR Conditions for Pointwise Convergence Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 From the Linear to the Nonlinear Transfer Technique . . . . . . . . σ-Slicely Continuous Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Joint σ-Slicely Continuity and Characterization of σ-Slicely Continuous Maps with Values in a Metric Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Characterization of σ-Slicely Continuous Maps with Values in a LUR Normed Space . . . . . . . . . . . . . . . . . . . . . . .

31 35 38 40 49 49 53 58 64 73 73

81 95

IX

X

Contents

5

Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.1 Three Space Property for LUR Renorming . . . . . . . . . . . . . . . . . . 101 5.2 The Weakly Countably Determined Case . . . . . . . . . . . . . . . . . . . 103 5.3 Product of σ-Slicely Continuous Maps . . . . . . . . . . . . . . . . . . . . . . 104 5.4 Totally Ordered Compacta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6

Some Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.1 More on the Nonlinear Transfer Technique for Renorming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.2 Renormings of C(K) Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.3 Measures of Non Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

List of Symbols

osc (Φ A ) Id ∂ε ϕ(x | U ) ∂ϕ(x | U ) F ∩G P (V, W) Υ t+ C0 (Υ) Lε (x) Ωx 1lH c1 (Z × Λ) ∂ ϕ˜ alg A δγ α(B) β(B) χ(B) λ(B)

7 7 9 9 17 28 35 35 35 35 37 42 43 70 76 83 89 99 128 128 128 128

XI

1 Introduction

Renorming in Banach space theory involves ﬁnding isomorphisms which improve the norm. That means making the geometrical and topological properties of the unit ball of a given Banach space as close as possible to those of the unit ball in a Hilbert space. Historically the ﬁrst result in this direction is due to Clarkson [Clr36] who proved that every separable Banach space has ∞ an equivalent rotund norm. Indeed, if {fi }1 is a norm bounded sequence of linear functionals which separates the points of X then the equivalent norm given by 1/2 ∞ −i 2 2 fi (x) , x∈X (1.1) |x| = x + 1

is rotund. Let us recall that a norm · is rotund (strictly convex) if the unit sphere does not contain non-trivial segments, i.e. x = y whenever x = y = (x + y)/2 = 1. An excellent monograph of renorming theory up to 1993 is [DGZ93]. In order to have an up-to-date account of the theory we should add [Hay99], [God01] and [Ziz03]. In [Hay99] the most important properties in renorming are characterized for C(Υ), where Υ is a tree, deducing a lot of counterexamples. In this way [Hay99] ﬁxes the exact boundary of this theory. In the survey [God01] most of its results and proofs are devoted to separable and super-reﬂexive Banach spaces. The survey [Ziz03] gives an overview of the renorming theory of non-separable spaces with the classical approach. In these notes we are focused mainly on locally uniformly rotund (locally uniformly convex) renorming. Let us recall that a norm · in a normed space is locally uniformly rotund (LUR for short) if limk xk − x = 0 wheneverone 2 2 2 of the two equivalent conditions holds limk 2 xk + 2 x − xk + x = 0 or limk (xk + x) /2 = limk xk = x. Clearly every LUR norm is rotund. The converse is not true. If we construct in c0 an equivalent norm using (1.1) we get a rotund norm which is not LUR. A. Molt´ o et al., A Nonlinear Transfer Technique for Renorming. Lecture Notes in Mathematics 1951, c Springer-Verlag Berlin Heidelberg 2009

1

2

1 Introduction

The methods in these notes stem from the following result which gives a new starting point for LUR renorming. Theorem 1.1. Let X be a normed space and let F be a norming subspace of its dual. Then X admits an equivalent σ(X, F )-lower semicontinuous LUR norm if, and only if, for every ε > 0 we can write Xn,ε X= n∈N

in such way that for every x ∈ Xn,ε there exists a σ(X, F )-open half space H containing x with diam (H ∩ Xn,ε ) < ε .

This linear topological concept is a particular case of a notion introduced in [JNR92] called countable cover by sets of small local diameter, which turns out to be equivalent for Banach spaces to the notion of descriptive spaces studied by R. W. Hansell in [Han01] (see Sect. 3.2). The theorem above was proved in [MOT97] in the case where F = X ∗ . The proof was fully probabilistic and it was based on the following theorem. For a set A in a normed space X we set γ(A) = sup γk (A),

1/2 2 γk (A) = sup inf E Mm , m

k

∞

where the inﬁmum is taken over all Walsh-Paley X-valued martingales {Mn }0 such that # n∈N:

2

−1 Mn (A)

Mn − Mn−1 ≥ 1

≥k.

The quantities γk (A) measure how fast a dyadic tree must grow when it has many large branches ending at points of A. Theorem 1.2. [Tro79] A normed space X admits an

equivalent LUR norm if, and only if, for every ε > 0 we can write X = n∈N Xn,ε in such a way that Xn,ε are cones with inf γ (Xn,ε ) > ε−1 . n

Historically the theorem above is the ﬁrst characterization of LUR renormability in linear topological terms. The origin of this theorem goes back to Pisier’s renorming [Pi75] of super-reﬂexive Banach spaces with power type modulus of rotundity.

1 Introduction

3

The general case of Theorem 1.1 was proved in [Raja99], where instead of probabilistic arguments geometrical ones were applied, specially the BourgainNamioka superlemma (see, for example, [Die84, p 157]) which played an essential role there. In Sect. 4.2 we present another proof of this result where the Bourgain-Namioka superlemma is replaced by an optimization argument. An important contribution of M. Raja [Raja99] is an elegant proof to show that a rotund space in which the norm and the weak topologies coincide on the unit sphere admits a LUR renorming. Originally this result was proved in [Tro85] using Theorem 1.2. In turn, Raja’s [Raja99] approach is a variation of a method of Lancien [Lan95] based on the dentability index which is deﬁned through a modiﬁcation of the “Cantor derivation”. Namely, for a subset C of a normed space X and ε > 0 Dε (C) = {x ∈ C : diam (C ∩ H)ε for every open halfspace H of X with x ∈ H} . Using this “derivation” Lancien got a new geometrical proof of the well-known renorming result of James-Enﬂo-Pisier for super-reﬂexive Banach spaces (see [God01, Sect. 3]). It turns out that it is rather diﬃcult to apply Theorem 1.2 and even Theorem 1.1 in a straightforward way. This motivates us to build up some technique to use Theorem 1.1. The most usual technique for renorming is the so-called transfer technique designed to transfer a good convexity property from a normed space to another. The easiest example illustrating this method is the following. Theorem 1.3. Let Y be a rotund space and T be a linear bounded one-to-one operator from X into Y , then the norm |x| = xX + T xY , x ∈ X is rotund. Actually (1.1) is a particular case of the above formula for the operator from X ∞ into 2 deﬁned by x → 2−i fi (x) 1 ∈ 2 . The simple geometrical interpretation of this fact is that the sum of convex functions is strictly convex whenever one of them, at least, is strictly convex. Unfortunately it is not possible to get LUR renormings by a direct application of this technique. Indeed ∞let us ∞ consider the operator from ∞ to 2 deﬁned by x = (xi )1 → 2−i xi 1 ; it is one-to-one but ∞ is not LUR renormable. In [God82] (see also [GTWZ83] and [Fab91]) a transfer technique was developed to obtain rotund or LUR renormings by imposing compactness conditions on T : X → Y . For example we have Theorem 1.4. Let X be a dual Banach space, let Y be a LUR Banach space · and T : Y → X a bounded linear operator such that T Y = X and T BY is weak∗ -compact. Then X admits an equivalent dual LUR norm.

4

1 Introduction

In Sect. 4.1 we shall present Theorem 4.8, a reformulation of the former result in terms of our nonlinear approach to LUR renorming. In order to be able to replace rotund by LUR in Theorem 1.3 we need the following. Deﬁnition 1.5. Let Φ be a map from the metric space (X, d) into the metric space (Y, ). Φ is said to be co-σ-continuous if for every ε > 0 we can write Xn,ε X= n

and ﬁnd δn (x) > 0 for every x ∈ Xn,ε in such a way that d(x, y) < ε whenever y ∈ Xn,ε and (Φx, Φy) < δn (x). Now we can formulate the following (see [MOT97]). Theorem 1.6. Let Y be a LUR normed space and T be a bounded linear co-σ-continuous operator from the normed space X into Y , then X admits an equivalent LUR norm. In order to apply the former theorem we need the following characterization of co-σ-continuous maps. Theorem 1.7. A map Φ from a metric space (X, d) into a metric space (Y, ) is co-σ-continuous if, and only if, for every x ∈ X there exists a separable subset Zx of X such that x∈

{Zxn : n ∈ N}

d

(1.2)

whenever limn Φxn = Φx. If X is a normed space then the condition (1.2) can be replaced by x ∈ span

{Zxn : n ∈ N}

·

.

The proof of the former theorem can be found in Sect. 2.2 where co-σ-continuous maps are fully studied (see Theorem 2.32 and Proposition 2.33). Example 1.8. Let us recall that the class of Baire maps between two metric spaces is the smallest family of functions which contains all continuous functions and the pointwise limit of sequences in it. So for any Baire map Ψ between metric spaces (Y, ) and (X, d) there exists a countable family {Ψn : n ∈ N} of continuous functions such that Ψy ∈ {Ψn y : n ∈ N} for all y ∈ Y . A straightforward consequence of Theorem 1.7 is that when Φ is a one-to-one map from (X, d) into (Y, ) and Φ−1 is a Baire map then Φ is co-σ-continuous.

1 Introduction

5

From the last two theorems we now have corollaries that are easier to apply. Corollary 1.9. Let Y be a LUR normed space, let T be a bounded linear operator from the normed space X into Y such that for every x ∈ X there exists a separable subspace Zx of X with x ∈ span

{Zxn : n ∈ N}

·

whenever lim T xn − T x = 0. Then X admits an equivalent LUR norm. Actually in many cases we can require less than LUR renormability of Y in Corollary 1.9 and this fact will be a contribution developed in Sect. 3.4. To explain it let us ﬁrstly extend the notion of LUR norm. Deﬁnition 1.10. Let (X, · ) be a normed space and T a topology on it. We say that the norm · is T LUR if T − lim xk = x , k

whenever

xk + x

= lim xk = x .

lim k→∞ 2 k→∞

(1.3)

In this way we deﬁne weak LUR, weak∗ LUR and more general σ(X, F ) LUR norms if F is a subspace of X ∗ . In the case when X is a subspace of ∞ (Γ) we deﬁne pointwise LUR norm requiring that for all γ ∈ Γ lim xk (γ) = x(γ) k

whenever (1.3) holds. Clearly σ(X, F ) LUR does not imply LUR renorming in general, for example ∞ has a pointwise LUR norm which is weak∗ LUR but fails to be weakly LUR renormable [Lin72] and therefore LUR renormable. However we have (see Corollary 3.23, 3.24 and [MOTV99]) the following. Theorem 1.11. Every weakly LUR normed space is LUR renormable. Every weak∗ LUR dual norm in a dual Banach space with the Radon-Nikodym property has an equivalent dual LUR norm. ∞ ∞ By ∞ c (Γ) we denote the subspace of (Γ) containing only those x ∈ (Γ) for which # supp x ≤ ℵ0 and δγ is the projection on the γ-coordinate for γ ∈ Γ, i.e. δγ (x) := x(γ). Now we state the following.

Theorem 1.12. Let Y be a subspace of ∞ c (Γ) with a pointwise LUR norm which is pointwise lower semicontinuous, let T be a bounded linear operator from the normed space X into Y and {Xγ }γ∈Γ be a family of separable subspaces of X such that for every x ∈ X we have · x ∈ span {Xγ : γ ∈ supp T x} . Then X admits an equivalent LUR norm.

6

1 Introduction

This theorem is a generalization of a result in [FT90] where Y is the Mercourakis space c1 (Z × K) [Mer87], which is not LUR renormable since it contains a subspace isomorphic to ∞ whenever Z is an inﬁnite set. It seems surprising that it is not necessary to assume that Y is LUR renormable but it is enough that Y is pointwise LUR. In Theorem 3.46 we shall present a nonlinear version of it. For the moment let us apply Theorem 1.12 to a large class of Banach spaces X which admits some suitable linear bounded operator with range in c0 (Γ). Indeed J. Lindenstrauss [Lin65] and [Lin66] introduced the projectional resolution of the identity (PRI for short) and using it constructed in every reﬂexive Banach space a linear bounded one-to-one map T : X → c0 (Γ) for some Γ. Later this technique was extended to weakly compactly generated Banach spaces by D. Amir and J. Lindenstrauss [AL68], to weakly compactly determined spaces by L. Vaˇsak [Vas81], to weakly Lindel¨ of determined spaces by S. Argyros and S. Mercourakis [AM93, Val90, Val91, Vald90], and to duals of Asplund spaces by M. Fabian and G. Godefroy [FG88]. There exists a PRI in C(K) when K is a Corson compact and its generalization (see [AMN88] and [Val90] respectively), when K is a compact of ordinals [Alex80] and a compact topological group [Alex82]. Quite recently M. Fabian, G. Godefroy and V. Zizler [FGZ01] have obtained a PRI for Banach spaces with a uniformly Gˆ ateaux diﬀentiable norm. All these classes of Banach spaces are included in the so-called class P and, as is shown in [DGZ93, p. 236], using PRI and some hereditary properties of some complemented subspaces it is possible to construct a transﬁnite sequence of projections {Qα : 0 ≤ α ≤ µ} such that if we set Rα = (Qα+1 − Qα ) / (Qα+1 + Qα ) we have i) Q0 = 0, Qα = 0 for α > 0, Qµ = Id; ii) Qα Qβ = Qβ Qα = Qmin(α,β) ; iii) (Qα+1 − Qα ) X is separable for all α ∈ [0, µ); iv) {Rα x}0≤α<µ ∈ c0 ([0, µ)) for all x ∈ X; v) Qβ x ∈ span {Rα x : 0 ≤ α < β}

·

for all x ∈ X.

If a Banach space has such a transﬁnite sequence of projections it is easy to construct a bounded linear operator T : X → c0 ([0, µ) × N) and to ﬁnd a separable subspace Xα,n satisfying the conditions of the last theorem. Indeed we can ﬁnd for every α < µ a sequence fα,n ∈ X ∗ , fα,n ≤ 1, n ∈ N, which separates the points of Rα X. We set Xα,n = Rα X and deﬁne a bounded linear operator T : X → c0 ([0, µ) × N) by the formula T x(α, n) =

fα,n (Rα x) . n

having in mind that {Qα : 0 ≤ α ≤ µ} satisﬁes conditions i)–v) it is easy to see that T and {Xα,n : (α, n) ∈ [0, µ)× N} fulﬁll the conditions of Theorem 1.12, and therefore X is LUR renormable.

1 Introduction

7

First J. Lindenstrauss [Lin72] asked whether every strictly convex Banach space X admits a one-to-one bounded linear operator to c0 (Γ) for some Γ. Later in a joint paper with Dashiell [DL73] they constructed a strictly convex Banach space without a one-to-one bounded linear operator into c0 (Γ) for any Γ. The ﬁrst example of a LUR Banach space without a one-to-one bounded linear operator in c0 (Γ) was found by R. Deville [Dev86]. Throughout these notes some applications of the above theorems will be shown. However, in many cases the linearity of T is rather restrictive. At ﬁrst glance it seems that the linearity of T is necessary to transfer slices from Y to X. It is clear that if a map sends zero into zero and transfers slices into slices then it must be linear. Nevertheless the linearity of T can be avoided as it is shown by a comparison of Theorems 3.46 and 1.12. Taking advantage of the possibility to take additional countable splittings we can replace the linearity of T by something less restrictive. Our Theorem 1.1 motivates the following. Deﬁnition 1.13. Let A be a subset of a linear topological space X, let Φ be a map from A into a metric space (Y, ). We say that Φ is slicely continuous at x ∈ A if for every ε > 0 there exists an open half space H of X containing x with osc (Φ H∩A ) = diam Φ (H ∩ A) < ε. We say that Φ is σ-slicely continuous on A if for every ε > 0 we can write A= An,ε (1.4) n∈N

in such a way that for every x ∈ A n,ε there exists an open half space H of X containing x with osc Φ H∩An,ε = diam Φ (H ∩ An,ε ) < ε. Maps of this kind can be very far from linear. For example see the oscillation map deﬁned in Sect. 2.7, the maps deﬁned in Propositions 4.1–4.5 and Theorems 5.1, 5.13 and 5.15. Now we can reformulate Theorem 1.1 in the following way: A normed space X admits an equivalent σ(X, F )-lower semicontinuous LUR norm if and only if the identity map Id : (X, σ(X, F )) → (X, · ) is σ-slicely continuous. And consequently we have for any bounded linear operator the following. Proposition 1.14. Let T be a bounded linear operator from the normed space X into the normed space Y . Then T is σ-slicely continuous provided one of the spaces X or Y is LUR renormable. Moreover taking advantage of the nonlinear structure of the sets which satisfy (1.4) for the identity map, we can formulate our transfer result as follows. Theorem 1.15. Let X be a normed space and let F be a norming subspace of its dual. Then X admits an equivalent σ(X, F )-lower semicontinuous LUR equivalent norm if, and only if, there exists a metric space (Y, ) and a map Φ : X → Y which is σ-slicely continuous for σ(X, F ) and co-σ-continuous for the norm topology.

8

1 Introduction

Proof. If X admits an equivalent σ(X, F )-lower semicontinuous LUR equivalent norm we can take Y = X and the identity map as Φ, which according to Theorem 1.1 is σ-slicely continuous for σ(X, F ). Conversely, let Φ : X → (Y, ) be co-σ-continuous and σ-slicely continuous for σ(X, F ).

∞ Let us ﬁx ε > 0, by co-σ-continuity we have that X = n=1 Xn,ε where for every x ∈ Xn,ε there is δ(x, n, ε) > 0 so that x − y < ε whenever y ∈ Xn,ε and (Φx, Φy) < δ(x, n, ε). Let us make another decomposition deﬁning 1 Xn,p,ε := x ∈ Xn,ε : δ(x, n, ε) > p

∞ then we have Xn,ε = p=1 Xn,p,ε . Now we apply the σ-slicely continuity of

∞ the map Φ for ﬁxed p and we get another splitting of X as X = m=1 Xpm in such a way that for every m and every x ∈ Xpm we have a σ(X, F )-open half space Hx with x ∈ Hx and osc ΦH∩Xpm < 1/p. Fix m, n, p and then if x ∈ Xn,p,ε ∩ Xpm we have y − x < ε whenever y ∈ Hx ∩ Xn,p,ε ∩ Xpm . Indeed since y ∈ Hx ∩ Xpm we have (Φx, Φy) < 1/p and consequently y − x < ε since y ∈ Xn,ε and (Φx, Φy) < 1/p < δ(x, n, ε). From the construction it is clear that X = Xn,p,ε ∩ Xpm : m, n, p ∈ N , and the argument holds for every ε > 0 so the identity map from (X, σ(X, F )) into X is σ-slicely continuous, and to ﬁnish the proof it is enough to apply Theorem 1.1.

From Theorem 1.15 and Proposition 1.14 the proof of the linear transfer technique (Theorem 1.6) immediately follows. In particular, we see that if T : X → Y is a bounded linear one-to-one map with Y LUR renormable and T −1 a Baire map for the norms, then X is LUR renormable. In these notes we characterize co-σ-continuous and σ-slicely continuous maps and using Theorem 1.15 we obtain almost all known LUR renorming results as well as some new ones. Until now, LUR equivalent norms have been constructed ad hoc for each particular situation (see, for example, [Tro71], [GTWZ83], [GTWZ85], [HR90], [Fab91], [Hay99], [HJNR00] and others). Mainly they were based on the Deville Master Lemma [DGZ93, Chap. VII, Lemma 1.1.] (whose origin is in [Tro71]), distance to the unit sphere of LUR spaces, convolutions with LUR norms, and the three space problem for LUR renorming. Theorems 1.7 and 1.15 together assert that a normed space X has an equivalent LUR norm if, and only if, there is a metric d on X generating a topology ﬁner than the weak topology and such that the identity map from (X, weak) into (X, d) is σ-slicely continuous. For that reason it cannot be a surprise that the method of covers which has had such a strong inﬂuence in the problem of metrization [Fro95] of topological spaces turns out to be an important tool in LUR renorming. Let us recall some deﬁnitions to be precise on the relationships.

1 Introduction

9

Deﬁnition 1.16. A family of subsets {Dγ : γ ∈ Γ} in a topological space X

is called discrete (resp. isolated ) if for every point x ∈ X (resp. x ∈ {Dγ : γ ∈ Γ}) there is a neighbourhood U of x such that U meets at most one member of the family {Dγ : γ ∈ Γ}. When X is a linear topological space and U can be taken to be an open half space then the family is said to be slicely discrete family (resp. slicely isolated ). A family of subsets {Dγ : γ ∈ Γ} in a linear

∞ topological space X is called σ-slicely isolatedly decomposable if Dγ = n=1 Dγn for every γ ∈ Γ and n Dγ : γ ∈ Γ is slicely isolated for each n ∈ N. Deﬁnition 1.17. Let A be a subset of a linear topological space, ϕ : A → R. For x ∈ U ⊂ A and ε > 0 we denote by ∂ε ϕ(x|U ) the ε-subdiﬀerential of ϕ as a function on U , at point x ∈ U , i.e. the set of all continuous linear functionals f on X such that for all y ∈ U we have ϕ(y) ≥ ϕ(x) + f (y − x) − ε . We denote by ∂ϕ(x|U ) the subdiﬀerential of ϕ at x, i.e. ∂ε ϕ(x|U ) . ∂ϕ(x|U ) = ε>0

It seems that E. Asplund and R. Rockafellar [AR69] were the ﬁrst to apply the concept of ε-subdiﬀerentiability as a tool in nonlinear analysis. More about subdiﬀerentials can be found in [Cla90], [Phe93], [RW98]. Now we can state our characterization result for σ-slicely continuity. Theorem 1.18. Let A be a σ-bounded subset of a locally convex linear topological space X, let (Y, ) be a metric space and Φ : A → Y . The following are equivalent: i) Φ is σ-slicely continuous.

ii) for every ε > 0 we can write A = n An,ε in such a way that for all x ∈ An,ε and every Lipschitz 1 function g : ΦA → R we have (1.5) ∂ε g ◦ Φ (x|An,ε ) = ∅ iii) If {Dγ : γ ∈ Γ}is a discrete family of subsets of (Y, ) then −1 Φ (Dγ ) : γ ∈ Γ is σ-slicely isolatedly decomposable. If Y is in addition a LUR renormable space then it is enough to require (1.5) in condition ii) only for all norm one linear functionals g on Y . The former theorem is a particular case of Theorem 4.16 we study in Chap. 4. Subdiﬀerentials and Lipschitz functions in renorming theory have been recently considered in [BGV02].

10

1 Introduction

Remark 1.19. It is essential in the above theorem that Y is LUR renormable. Indeed if Y is not LUR renormable and if we consider the identity map Id : Y → Y clearly g ∈ ∂g ◦ Id(x|Y ) for all x ∈ Y and g ∈ Y ∗ but Id is not σ-slicely continuous since Y is not LUR renormable. From the proof of the above theorem we get the remarkable fact that if Φ and Ψ are σ-slicely continuous then Φ + Ψ is σ-slicely continuous and when X is a normed algebra then the product ΦΨ is σ-slicely continuous too, see Lemma 4.22 of joint σ-slicely continuity. Indeed we obtain the following. Corollary 1.20. Let X be a normed space and let Φn : X → X, n = 1, 2, . . . be a sequence of σ-slicely continuous maps such that for every x ∈ X we have ·

x ∈ span {Φn x : n = 1, 2, . . .}

. Then X admits a LUR norm.

This is a particular case of Corollary 4.23. Taking advantage of the existence of a lattice LUR norm in c0 (Γ) we can deduce from Theorem 1.18 the following. Corollary 1.21. Let Φ be a locally bounded map from a normed space X into c0 (Γ) for some Γ such that for every γ ∈ Γ the real function δγ ◦ Φ on X is non-negative and convex, where δγ is the Dirac measure on Γ at γ. Then Φ is σ-slicely continuous. This is a particular case of Corollary 4.34. We develop all these results in Chapter 4. A useful notion in topology due to Arhangel’ski˘ı [Arca92] is the following. Deﬁnition 1.22. A family N of subsets of a topological space (X, T ) is called a network if for every x ∈ X and U ∈ T with x ∈ U there exists N ∈ N with x ∈ N ⊂ U (see [Gru84]). We can now present the following. Theorem 1.23. A normed space X has an equivalent LUR norm if, and only if, there exists a metric d on X generating a topology ﬁner than the weak topology such that any of the three equivalent conditions holds:

∞ i) For every ε > 0 we can write X = n=1 Xn,ε in such a way that for all x ∈ Xn,ε and every Lipschitz 1 function g : (X, d) → R we have ∂ε g (x|Xn,ε ) = ∅ . ii) If {Dγ : γ ∈ Γ} is a d-discrete family of subsets of X then it is σ-slicely isolatedly decomposable.

∞ iii) The topology of the metric d has a network N = n=1 Nn where every Nn is a slicely isolated family.

1 Introduction

11

Conditions (i) to (iii) are equivalent to the σ-slicely continuity of the identity map on X when the metric d is used on the range, by Theorem 1.18 and Proposition 2.24. Since the metric d is ﬁner than the weak topology, Corollary 2.36 and Theorem 1.15 give the proof of the former theorem. Theorems 1.15, 1.18, and 1.23 show the relations between LUR renormability, optimizations and metrization theory. On the other hand LUR renormability is a useful tool in optimization and smooth approximation theories, namely if both X and X ∗ are LUR then the duality mapping is a homeomorphism between the unit spheres, and the Banach space X admits C (1) partitions of the unity; see, for example, [Zei90, p. 861], [Zei85, p. 400], [Cio90], [Pas78], [DGZ93, Chap. VIII], [Hay]. In order to see the intimate connection between the geometry of Banach spaces and the duality mapping in optimization theory, see, for example, [Zei85, p. 401]. In [DZ93, p. 50] there is a discussion on the well-posedness problem and LUR renorming. The method of the equivalent norms and specially LUR renormings has many applications inside Banach space theory. For example the core of Kadec’s construction [Kad66] of a homeomorphism between a Banach space with a basis and 2 is the LUR renormability of separable Banach spaces. The original Asplund proof showing that every Banach space with a separable dual is Asplund used the fact that every dual separable Banach space admits a dual LUR norm [Asp68]. J. Lindenstrauss [Lin63] proved that if X is a LUR renormable Banach space then every weakly compact convex subset K of X is the closed convex hull of its strongly exposed points. Having in mind that · span K is LUR renormable [Tro71] we obtain that the above generalization of the Krein-Milman theorem holds for any weakly compact convex subset of a Banach space. In LUR renormable Banach spaces Cepedello [Cep98] proved that any bounded norm continuous function is the pointwise limit of a sequence of diﬀerences of convex continuous functions. Motivated by all these considerations we present in the notes an up-todate account of LUR renormings inside a new frame of nonlinear maps suitable for geometric nonlinear analysis of non-separable Banach spaces. For instance, in Chap. 2 we present the solution of a problem of Kadec about the LUR renormability of the space C(H) where H is the compact of Helly. Such a compact space is a particular example of a separable Rosenthal compact space, a class widely studied [Ros74, BFT78, God80, Tod99, Tod06, HMO07], but still not clear at all for renormings of C(K). In Chap. 3 we connect LUR renormings with metrization theory proving our Theorem 3.46 as a ﬁrst nonlinear transfer result. In Chap. 4 we study deeper facts of σ-slicely continuous maps related with diﬀerentiability, presenting the Joint Continuity Lemma 4.22 which is the core for reducing weak open neighbourhoods to slices with small oscillation. In Chap. 5 we present some applications, and in particular a general frame in Sect. 5.3 from which almost all results can be obtained.

2 σ-Continuous and Co-σ-continuous Maps

In this chapter we isolate the topological setting that is suitable for our study. We ﬁrst present 2.1–2.3 to follow an understandable logical scheme nevertheless the main contribution are presented in 2.4–2.7 and our main tool will be Theorem 2.32. An important concept will be the σ-continuity of a map Φ from a topological space (X, T ) into a metric space (Y, ). The σ-continuity property is an extension of continuity suitable to deal with countable decompositions of the domain space X as well as with pointwise cluster points of sequences of functions Φn : X → Y , n = 1, 2, . . . When (X, T ) is a subset of a locally convex linear topological space we shall reﬁne our study to deal with σ-slicely continuous maps, the main object of these notes. When (X, T ) is a metric space too we shall deal with σ-continuity properties of the inverse map Φ−1 that we have called co-σ-continuity.

2.1 σ-Continuous and σ-Slicely Continuous Maps Let us remember the following deﬁnitions which will be used throughout the section. Deﬁnition 2.1. We call a map Φ, from a topological space (X, T ) into a metric

∞ space (Y, ρ), σ-continuous when for every ε > 0 we can write X = n=1 Xn,ε in such a way that, for every n ∈ N and every x ∈ Xn,ε there is a T -neighbourhood V of x with ρ − diam(Φ(V ∩ Xn,ε )) < ε. A family of subsets D in a topological space is said to be σ-discrete,

(resp. σ-isolated ) if it can be decomposed into a countable union D = Dn such that every family Dn is discrete (resp. isolated). A family of subsets {Dγ : γ ∈ Γ} in a topological space X

is called ∞ σ-discretely decomposable (resp. σ-isolatedly decomposable) if Dγ = n=1 Dγn n for every γ ∈ Γ and Dγ : γ ∈ Γ is discrete (resp. isolated) for each n ∈ N. The relationship between the former concepts is showed in the next: A. Molt´ o et al., A Nonlinear Transfer Technique for Renorming. Lecture Notes in Mathematics 1951, c Springer-Verlag Berlin Heidelberg 2009

13

14

2 σ-Continuous and Co-σ-continuous Maps

Proposition 2.2. Let (X, T ) be a topological space and let (Y, ) be a metric space. Given a map Φ : X → Y the following are equivalent: i) Φ is σ-continuous; ii) if {Dγ : γ ∈ Γ} is a discrete family of subsets in (Y, ) then −1 Φ (Dγ ) : γ ∈ Γ is σ-isolatedly decomposable in (X, T ); Proof. i)=⇒ii). If {Dγ : γ ∈ Γ} is a discrete family of subsets of (Y, ) we can ﬁrstly decompose every set Dγ by deﬁning Dγ,p := {x ∈ Dγ : B (x, 1/p) ∩ Dβ = ∅ for all β = γ, β ∈ Γ}

∞ and we have Dγ = p=1 Dγ,p . Now each family {Dγ,p : γ ∈ Γ} is 1/p-discrete.

∞ Since Φ is σ-continuous we can decompose X = n=1 Xn,1/p , in such a way that for every x ∈ Xn,1/p there exists a T -neighbourhood V of x such that − diam Φ Xn,1/p ∩ V < 1/p . (2.1) Then if we consider for n and p ﬁxed and the family −1 Φ (Dγ,p ) ∩ Xn,1/p : γ ∈ Γ it now follows that it is an isolated family. Indeed take x ∈ Φ−1 (Dγ,p )∩Xn,1/p then if V is the T -neighbourhood of x which satisﬁes (2.1) according to the choice of the sets Dγ,p , which are a 1/p-discrete family in (Y, ), we have V ∩ Φ−1 (Dβ,p ) ∩ Xn,1/p = ∅, for all β = γ, β ∈ Γ .

∞ Since Φ−1 (Dγ ) = n,p=1 Φ−1 (Dγ,p ) ∩ Xn,1/p we have shown that −1 Φ (Dγ ) : γ ∈ Γ is σ-isolatedly decomposable in (X, T ). ii)=⇒i)

∞ Let us ﬁx ε > 0. Stone’s theorem gives us a σ-discrete family B = m=1 Bm of open sets with diameter less than or equal to ε covering −1 the metric space (Y, ). By our assumption m Φ (Bm )should be a σ-isolatedly decomposable family, that is if Bm = Bγ : γ ∈ Γm then for every γ ∈ Γm ∞ Φ−1 Bγm = Xnm,γ n=1 m,γ and : γ ∈ Γm } is an isolated If n we set Xm,n,ε = N.

∞ family for n ∈

{X m,γ −1 m B : γ ∈ Γm = that X = Φ {Xn : γ ∈ Γm }, we have m,n,ε γ n=1

∞

∞ Φ−1 (Bm ) and so we have m,n=1 Xm,n,ε = m=1 Φ−1 (Bm ) = Φ−1 (Y ) = X. Moreover, if x ∈ Xm,n,ε there is a neighbourhood U of x such that if x ∈ Xnm,γ0 m,γ m,γ0 we have ⊂ U ∩ Xn = ∅ for all γ ∈ Γm , γ = γ0 . So U ∩ Xm,n,ε ∩ Xn ⊂U

Φ−1 Bγm0 and we see that diam (Φ (U ∩ Xm,n,ε )) ≤ diam Bγm0 ≤ ε.

2.1 σ-Continuous and σ-Slicely Continuous Maps

15

If the elements of a sequence of functions are σ-continuous then so are their cluster points in the topology of the pointwise convergence. Even more we have the following. Proposition 2.3. Let Φn : X → Y , n ∈ N, be a sequence of σ-continuous functions between the topological space (X, T ) and the metric space (Y, ). If Φ : X → Y veriﬁes Φx ∈ {Φn x : n ∈ N} for every x ∈ X then Φ is σ-continuous too. x, Φx) < Proof. Given ε > 0 and m ∈ N we can deﬁne Xm = {x ∈ X : (Φ

m ∞ ε/3}. Since Φm is σ-continuous there is a decomposition X = n=1 Xnm in m a neighbourhood sucha way that for any x ∈ Xn we can ﬁnd V of x such that m m If x ∈ X ∩ X we have osc Φ Xm ∩Xnm ∩V < ε osc Φm Xn ∩V < ε/3. m n

∞ ∞ m and X = m=1 Xm = m,n=1 Xm ∩ Xn and the conclusion follows.

Remember that the class of Baire maps between two metric spaces is the smallest family of functions which contains all continuous functions and the pointwise limit of sequences in it. Corollary 2.4. Any Baire map Φ from a topological space (X, T ) to a metric space (Y, ) is σ-continuous. We can also split the domain and require σ-continuity on the pieces in a very general way. Proposition 2.5. Let (X, T ) be a topological space and (Y, ) a metric space. If Φ : X → Y is a map for which there exists a σ-isolatedly decomposable cover {Dγ : γ ∈ Γ} of X such that Φ Dγ is σ-continuous for every γ ∈ Γ then Φ is σ-continuous too.

∞ γ such that for every Proof. Fix ε > 0. For each γ ∈ Γ we have Dγ = n=1 Dn,ε γ n and x ∈ Dn,ε there exists a neighbourhood V of x such that γ <ε. osc Φ V ∩Dn,ε On the other hand since {Dγ : γ ∈ Γ} is σ-isolatedly decomposable we must ∞ have Dγ = n=1 Mγ,n where for each n ∈ N the family {Mγ,n : γ ∈ Γ} is isolated. For each n and p deﬁne γ Dn,p := Mγ,n ∩ Dp,ε : γ∈Γ . Then if x ∈ Dn,p we have a neighbourhood U of x such that U ∩ Mγ,n = ∅ if, and only if, γ = γ0 where γ0 is the index with x ∈ Mγ0 ,n . Moreover there is a neighbourhood V of x with osc ΦV ∩Dγ0 < ε. Consequently p,ε γ0 γ0 < ε since Dn,p ∩ V ∩ U ⊂ V ∩ Dp,ε . osc Φ Dn,p ∩V ∩U ≤ osc Φ V ∩Dp,ε Now the choice of the sets Dn,p gives

16

2 σ-Continuous and Co-σ-continuous Maps

X=

{Dn,p : n, p ∈ N}

which ﬁnishes the proof.

Remark 2.6. The class of the σ-continuous maps is constructed with countable partitions for the continuity property. Moreover this class is stable taking cluster points of sequences (Proposition 2.3), making countable splitting again, or even σ-isolatedly decomposable splitting of the domain (Proposition 2.5). Given Φ : X → Y we say that a family B of subsets of X is a function base for Φ if, whenever V is open in Y , then Φ−1 (V ) is union of sets of B. In other words B is a function base for Φ if it is a network of the topology −1 Φ (V ) : V is open in Y . The class of maps with σ-isolated function base has been studied by Hansell [Han92] in connection with the study of LebesgueHausdorﬀ theorem [Kur66, vol 1, p. 393] as well as the descriptive set theory for non separable metric spaces. We have the following. Proposition 2.7. A map Φ from a topological space (X, T ) into the metric space (Y, ) is σ-continuous if, and only if, it has a σ-isolated function base.

∞ Proof. Let B = n=1 Bn a basis for the metrizable topology of Y such that every Bn is a discrete family on Y . If Φ is σ-continuous according to Proposition 2.2.ii) each family Φ−1 (Bn ) is σ-isolatedly decomposable. Then for each B ∈ Bn we have Φ−1 (B) =

∞

Φ−1 (B)m

m=1

and the families Φ−1 (B)m : B ∈ Bn are isolated for m, n ∈ N. Obviously

∞

Φ−1 (B)m : B ∈ Bn

m,n=1

is a function base for Φ which is σ-isolated.

∞ Conversely if Φ has a function base N = n=1

Nn and each Nn is an isolated family in X let us consider the sets Dn := {N : N ∈ Nn }, n ∈ N. For each N ∈ Nn choose xn,N ∈ N and deﬁne Φxn,N if x ∈ Dn , x ∈ N Ξn x = y if x ∈ / Dn where y is a ﬁxed element of Y . Then Ξn is constant on X \ Dn and since Nn is an isolated family we have that Ξn is locally constant on Dn so it is a σ-continuous map. Now since N is a function base for Φ we have

Φx ∈ {Ξn x : n ∈ N} for all x ∈ X so from Proposition 2.3 it follows that Φ is σ-continuous.

2.1 σ-Continuous and σ-Slicely Continuous Maps

17

Remark 2.8. In connection with measurability properties Hansell [Han92] studied functions with a σ-isolated function base of F ∩ G sets (i.e. of sets which can be written as a diﬀerence of closed sets). This class of maps is becoming central in the study of measurable selectors of upper semi-continuous multivalued functions when the domain space is not a metric space [Onc99], [Han00]. In a metric space every isolated family is σ-discretely decomposable [Han71, Lemma 2]. Indeed if {Dγ : γ ∈ Γ} is isolated in a metric space (X, d) we deﬁne Dpγ := {x ∈ Dγ : Bd (x, 1/p) ∩ Dβ = ∅ for all β ∈ Γ \ {γ}}

∞ and Dγ = p=1 Dpγ . Thanks to the triangle inequality we have now that every family Dpγ : γ ∈ Γ is discrete. Indeed if x ∈ X then Bd (x, 1/2p) meets at most one member of the family Dpγ : γ ∈ Γ . Consequently a map Φ : X → Y between metric spaces is σ-continuous if, and only if, it has a σ-discrete functions base. This class of maps has been called σ-discrete maps, and they play an important role in the study of non separable descriptive topology [Han92]. For instance when X is a metric space and Y is a normed space a map Φ : X → Y in the ﬁrst Borel class; i.e. Φ−1 (U ) is a Fσ -set for every open set U of Y , will be the pointwise limit of a sequence of continuous functions if, and only if, the map Φ is σ-discrete [Han74], [Rog88]. When X is a complete metric space Hansell [Han71] showed that any Borel measurable map Φ : X → Y is a σ-discrete map. The same assertion for an arbitrary metric space is independent of the usual axioms of the set theory [Han91]. M. Raja studied in his thesis [Ra99] the class of σ-continuous maps under the terminology of P-maps (see Sect. 2.3) and he applied them to Borel mesurability properties [Raja02] and renormings [Raj99], [Raja99] and [Raj02]. Our approach here take care for the extensions of all the good properties of these maps for the case of σ-slicely continuous maps. We are going to add two more characterizations of σ-continuous maps in the case when X is a metric space. As we will see later this case is important in our transfer technique for renorming. Let us recall the following Deﬁnition 2.9. A map Φ : X → Y is called barely continuous [Kur66, pp 395 and 419], [MN76] if for every non-empty closed set A ⊂ X the restriction Φ A has at least one point of continuity. This class of maps has been also called PC (for point of continuity) functions. The well-known Baire theorem says that Φ : X → R, for complete separable metric space X, is a Baire one map if, and only if, Φ is barely continuous. The same is true for a map Φ : X → Y where X is a complete metric space and Y is a Banach space [Ste91], [Han91]. Between arbitrary metric spaces a barely continuous map will be σ-continuous as Hansell pointed it out [Han91]. Indeed the only requirement we need on X is to be hereditarily weakly θ-reﬁnable, which means that every family of open sets in X has a σ-isolated reﬁnement

18

2 σ-Continuous and Co-σ-continuous Maps

see [Bur84, Theorem 3.7], since in that case Montgomery’s operation gives us a σ-isolated decomposable family (Lemma 6.20 in [Han92], see also [Han01]). Let us brieﬂy complete the proof. Proposition 2.10. Let (X, T ) be a hereditarily weakly θ-reﬁnable topological space and (Y, ) a metric space. If Φ : X → Y is a barely continuous map then Φ is σ-continuous. Proof. For any ε > 0 ﬁxed and any closed subset A of X there exists an open set U in X with U ∩ A = ∅ and osc (Φ U ∩A ) < ε. Consequently we can inductively deﬁne a long (transﬁnite) sequence {Uγ : γ < µ} of open subsets of X such that X= {Uγ : γ < µ}

and for every γ < µ, Mγ := Uγ \ {Uβ : β < γ} = ∅ and osc Φ Mγ < ε. According to Hansell Lemma [Han92] the family {Mγ : γ < µ} is σ-isolatedly decomposable and if we choose xγ ∈ Mγ for every γ < µ we can deﬁne Φε : X → Y by Φε x = Φxγ if x ∈ Mγ . Then from Proposition 2.5 it follows that Φε is σ-continuous and by the construction (Φε x, Φx) < ε for all x in X. Thus according to Proposition 2.3 Φ is σ-continuous.

In a similar way to in Remark 2.6 if we consider a splitting condition over the barely continuity property we get the following [JOPV93, p 248]. Deﬁnition 2.11. A map Φ from a topological space (X, T ) into a metric space (Y, ) is said to be σ-fragmented if for every ε > 0 we can decompose ∞ X as X = n=1 Xn,ε such that for every n ∈ N and every non empty subset A of Xn,ε there exists an open subset V of X such that V ∩ A = ∅ and osc (Φ V ∩A ) < ε . Given a normed space (X, · ), if d is the norm-metric, then X is said to be σ-fragmentable when the identity map Id : (X, weak) −→ (X, d) is σ-fragmented. It is almost clear that Proposition 2.10 can be extended to that case: Proposition 2.12. If (X, T ) is hereditarily weakly θ-reﬁnable and (Y, ) is a metric space then any map Φ : X → Y which is σ-fragmented is σ-continuous.

∞ Proof. Given ε > 0 and X = n=1 Xn,ε the decomposition of Deﬁnition 2.11 we can repeat the construction of Proposition 2.10 in every one of the pieces Xn,ε to deﬁne Φnε : Xn,ε → Y σ-continuous and such that (Φnε x, Φx) < ε for all x ∈ Xn,ε . Finally we glue the pieces with

2.1 σ-Continuous and σ-Slicely Continuous Maps

Φε x := Φnε x if x ∈ Xn,ε \

n−1

19

Xj,ε

j=1

and we will have a σ-continuous map Φε : X → Y with (Φε x, Φx) < ε for all x ∈ X. Then Φ is σ-continuous too.

The following result gives an approximation property that σ-continuous maps have when the domain is a metric space. Proposition 2.13. Let (X, d) and (Y, ) be metric spaces and let Φ : X → Y be a σ-continuous map. Then for every x in X there is a countable set Wx in

X such that Φx ∈ {ΦWxn : n ∈ N} whenever limn→∞ d (xn , x) = 0.

∞ Proof. Let B = n=1 Bn be a σ-discrete function base for the map Φ given by Proposition 2.7 where each family Bn is discrete in X. Let us ﬁx n ∈ N, now for every x ∈ X and every δ > 0 such that Bd (x, δ) meets at most one member of Bn we choose x(n, δ) in Bd (x, δ) ∩ {B : B ∈ Bn } when this intersection is non void. Deﬁne Wx := {x(n, 1/p) : n, p ∈ N} then let us see now that Φx ∈ {ΦWxk : k ∈ N} whenever lim d (xk , x) = 0 . k→∞

Indeed, suppose limk→∞ d (xk , x) = 0 and ﬁx ε > 0. There is an element of the function base B which contains x and it is contained in Φ−1 (B (Φx, ε)). If n is an integer and B ∈ Bn with x ∈ B ⊂ Φ−1 (B (Φx, ε)) we have δ > 0 so that Bd (x, δ) only meets the set B of Bn . If k is an integer with d (xk , x) < δ/2 we have B (xk , δ/2) ⊂ B(x, δ) and consequently B (xk , δ/2) only meets the set B of Bn and the point xk (n, δ/2) has been deﬁned with xk (n, δ/2) ∈ B ∩ B (xk , δ/2). It is not a restriction to assume that δ = 1/p for some integer p and xk (n, 1/2p) ∈ Wxk with (Φ(x), Φ (xk (n, 1/2p))) < ε since x and xk (n, 1/2p)

belong to B ⊂ Φ−1 (B (Φx, ε)). For metric spaces we have the following equivalences: Theorem 2.14. Let (X, d) and (Y, ) be metric spaces. Given a map Φ : X → Y the following are equivalent: i) Φ is σ-continuous; ii) for every x ∈ X there is a countable set Wx in X such that {ΦWxk : k ∈ N} Φx ∈ ∞

whenever {xk }1 converges to x in X; iii) Φ is σ-fragmented.

20

2 σ-Continuous and Co-σ-continuous Maps

Proof. i)=⇒ii) follows from Proposition 2.13 above. i)=⇒iii) is clear from the deﬁnition. iii)=⇒i) follows from Proposition 2.12 since metric spaces are hereditarily paracompact. It only remains to show that ii)=⇒i). If B =

∞ n=1 Bn is a σ-discrete basis of the d-topology in X and Wx = {x(n) : n ∈ N} let us choose a point xB ∈ B for every B ∈ B. For any integers m an n we can deﬁne Φm,n : X → Y by the formula Φm,n x =

Φ (xB (m)) if x ∈

B ∈ Bn ym,n if x ∈ / {B : B ∈ Bn }

where ym,n is an arbitrary point chosen for every m, n ﬁxed.

Since Bn is discrete Φm,n is locally constant on {B : B ∈ Bn } and constant on X \ {B : B ∈ Bn }. Therefore Φm,n is σ-continuous. Since B is a basis of the d-topology the property ii) ensures that Φx ∈ {Φm,n x : m, n ∈ N}

for every x in X. From Proposition 2.3 it follows that Φ is σ-continuous.

Remark 2.15. Condition ii) in Theorem 2.14 can be reformulated as follows: For every y in ΦX there exists a countable subset Wy in Y such that Φx ∈

{WΦxk : k ∈ N}

whenever (xk ) converges to x in X. Indeed the same proof works with the natural modiﬁcations in the deﬁnition of the maps Φm,n above. When the metric space (Y, ) is a normed space (Y, ·) we can make linear combinations of maps from X into Y and it is clear from the deﬁnitions that a linear combination of σ-continuous maps will be σ-continuous too. From Proposition 2.3 we get the following. Proposition 2.16. Let Φn : X → Y , n ∈ N, be a sequence of σ-continuous functions from a topological space (X, T ) into a normed space Y . If Φ : X → Y veriﬁes Φx ∈ span {Φn x : n ∈ N}

·

for every x ∈ X,

then Φ is σ-continuous . In addition, if Y is normed algebra the former condition can be replaced by Φx ∈ alg {Φn x : n ∈ N}

·

for every x ∈ X

where alg A is the subalgebra of Y generated by a subset A of Y .

2.1 σ-Continuous and σ-Slicely Continuous Maps

21

Proof. The countable set of the Q-linear combinations of functions {Φn : n ∈ N} veriﬁes the conditions of Proposition 2.3 for the map Φ itself. In the case of an algebra the Q-linear algebra generated by {Φn : n ∈ N} veriﬁes the conditions of Proposition 2.3 too.

Corollary 2.17. Let Φn : X → Y , n ∈ N, be a sequence of σ-continuous functions from a topological space (X, T ) into a normed space (Y, · ). If Φ : X → Y veriﬁes weak Φx ∈ {Φn x : n ∈ N} then Φ is σ-continuous too. Corollary 2.18. If A is a subset of a normed space X and A has a σ-isolated network for the weak topology then the identity map Id : (A, weak) → (A, · ) is σ-continuous.

∞ Proof. Let N = n=1 Nn be a σ-isolated network for the weak topology of A, where Nn is an isolated family. Choose xnN ∈ N for each N ∈ Nn and each n∈N

and let us deﬁne the map Φn x := xnN if x ∈ N ∈ Nn and Φn x := 0 since it is locally constant on if x ∈ / {N : N ∈ Nn } so it is σ-continuous

{N : N ∈ Nn } and constant on X \ {N : N ∈ Nn }. Since N is a network for the weak topology of A we also have x ∈ {Φn x : n ∈ N}

weak

and the conclusion follows from Corollary 2.17.

Remark 2.19. Let Φ a continuous map from a topological space X into (C(K), pointwise). R. Hansell has shown that if Φ has a σ-isolated function base then Φ has also a σ-isolated function base of F ∩ G sets with respect to the norm topology of C(K), and that it is a uniform limit of a sequence of “piece-wise continuous maps” for the norm topology [Han01]. The theory of these functions in this context goes back to the paper by R. Hansell [Han01], see also [JOPV93]. From Theorem 2.14 we will deduce the following Corollary 2.20. Let Φ : X → Y be a map between a metric space (X, d) into the normed space (Y, ·). If for every y ∈ ΦX there exists a countable set Wy ·

∞

in Y such that Φx ∈ span {WΦxk : k ∈ N} whenever {xk }1 converges to x in X then the map Φ is σ-continuous. In addition if Y is a normed algebra the former condition can be replaced by Φx ∈ alg {WΦxk : k ∈ N}

·

.

Proof. As in ii)=⇒i) of Theorem 2.14 we deﬁne the sequence of σ-continuous maps {Φm,n : n, m ∈ N} and it veriﬁes Φx ∈ span {Φm,n x : n, m ∈ N}

·

for every x ∈ X.

Now Proposition 2.16 proves the assertion of the corollary. When Y is an algebra simply replace span by alg.

22

2 σ-Continuous and Co-σ-continuous Maps

In this notes we are going to be mainly interested in a particular class of σ-continuous maps where the continuity property is required for a ﬁxed subbase of the topology. Indeed we shall work in a locally convex linear topological space X and we will play with the subbasis of the weak topology made up with all the open half spaces in X. Any of the terms introduced in Deﬁnition 2.1 has the corresponding one for that case and we simply add the word “slicely” as we did in Deﬁnition 1.13 for σ-slicely continuous maps. Deﬁnition 2.21. A family of subsets {Dγ : γ ∈ Γ} in a locally convex linear topological space X is called

slicely discrete (resp. slicely isolated ) if for every point x ∈ X (resp. x ∈ {Dγ : γ ∈ Γ}) there is an open half space H with x ∈ H such that H meets at most one member of the family {Dγ : γ ∈ Γ}. A family of subsets D is said to be σ-slicely discrete,

(σ-slicely isolated ) if it can be decomposed into a countable union D = Dn such that every family Dn is slicely discrete (resp. slicely isolated). A family of subsets {Dγ : γ ∈ Γ} in a locally convex linear topological space X is called σ-slicely σ-slicelyisolatedly

∞ discretely decomposable (resp. decomposable) if Dγ = n=1 Dγn for every γ ∈ Γ and Dγn : γ ∈ Γ is slicely discrete (resp. slicely isolated) for each n ∈ N. The following results follow with the same proofs we have done for σ-continuous maps with the natural modiﬁcations. Later on we will describe more properties of σ-slicely continuous maps playing with the linear structure and renorming properties when it is possible. Proposition 2.22. Let A be a subset of a locally convex linear topological space and let (Y, ) be a metric space. For a map Φ : A → Y the following assertions are equivalent: i) Φ is σ-slicely continuous. ii) If {Dγ : γ ∈ Γ} is a discrete family of subsets of (Y, ) then −1 Φ (Dγ ) : γ ∈ Γ is σ-slicely isolatedly decomposable in A.

Proof. As in Proposition 2.2.

Proposition 2.23. Let Φn : A → Y , n ∈ N be a sequence of σ-slicely continuous functions from the subset A of a locally convex linear topological space into the metric space (Y, ). If Φ : A → Y veriﬁes Φx ∈ {Φn x : n ∈ N}

for every x ∈ A then Φ is σ-slicely continuous too. Proof. As in Proposition 2.3.

2.2 Co-σ-continuous Maps

23

Proposition 2.24. A map Φ : A → (Y, ) is σ-slicely continuous if, and only if, it has a σ-slicely isolated function base. Proof. As in Proposition 2.7.

Nevertheless the extension of Proposition 2.16 needs to wait until Chap. 4 (see Lemma 4.22).

2.2 Co-σ-continuous Maps We shall need to invert the arrow in a σ-continuous map arriving to the following: Deﬁnition 2.25. A function Ψ from a metric space (Y, ) into a topological space (X, T ) is said to be co-σ-continuous if for every ε > 0 we can decompose ∞ Y as Y = n=1 Yn,ε such that for every n ∈ N and there exists every y ∈ Yn,ε a neighbourhood U of Ψy such that -diameter Yn,ε ∩ Ψ−1 (U ) < ε. Remark 2.26. If Ψ is a one-to-one map from the metric space (Y, ) into the topological space (X, T ) then Ψ is co-σ-continuous if, and only if, Ψ−1 : ΨY → Y is a σ-continuous map, nevertheless we have in general. Proposition 2.27. Let Ψ : (Y, ) → (X, T ) be a co-σ-continuous map. Then the ﬁbers of Ψ are separable subsets of Y .

∞ Proof. Given ε > 0 let Y = n=1 Yn,ε be the decomposition of Deﬁnition 2.25.

∞ If we ﬁx y ∈ Y and Z := Ψ−1 (Ψy) the ﬁber of y, then Z = n=1 Yn,ε ∩ Z and for every z, z ∈ Yn,ε ∩ Z we have (z, z ) < ε since Ψz = Ψz . If we choose zn,ε ∈ Yn,ε ∩ Z whenever this set is non void then the sequence

zn,1/p : n, p ∈ N is dense in the ﬁber Z which is -separable. The following result is analogous to Proposition 2.2 and Proposition 2.3 for σ-continuous maps and it ﬁxes the relationship with formerly studied maps by Michael [Mic82] and Hansell [Han74]. Indeed if X is a metric space then it is a Souslin-F-set in every metric embedding, i.e. it is an analytic metric space (not necessarily separable), if, and only if, it is a continuous image of a complete metric space through a map verifying ii) in the following theorem, see [Han74]. Theorem 2.28. Let (Y, ) be a metric space and let (X, T ) be a topological space. Given a map Ψ : Y → X the following assertions are equivalent: i) Ψ is co-σ-continuous. ii) If {Dγ : γ ∈ Γ} is a discrete family of subsets of (Y, ) then the family of its images {ΨDγ : γ ∈ Γ} is σ-isolatedly decomposable in (X, T ). iii) There is a sequence Φn : ΨY → Y of σ-continuous functions such that y ∈ {Φn (Ψy) : n ∈ N} for every y ∈ Y .

24

2 σ-Continuous and Co-σ-continuous Maps

Proof. i)=⇒ii) Let {Dγ : γ ∈ Γ} be a discrete family in Y , deﬁne Dγ,p := {x ∈ Dγ : B (y, 1/p) ∩ Dβ = ∅ for all β ∈ Γ \ {γ}} .

∞ Then Dγ = p=1 Dγ,p and each family {Dγ,p : γ ∈ Γ} is 1/p-discrete. Since Ψ

∞ is co-σ-continuous we can decompose Y = n=1 Yn,1/p such that for every y ∈ there exists a neighbourhood V of Ψy so that Yn,1/p −1 -diameter Yn,1/p ∩ Ψ (V ) < 1/p. Then for each n and p we consider the family Ψ Dγ,p ∩ Yn,1/p : γ ∈ Γ and we can see it is an isolated family in (X, T ). Take any x ∈ Ψ Dγ,p ∩ Yn,1/p and any y ∈ Dγ,p ∩ Yn,1/p with Ψy = x. If V is a neighbourhood of Ψy = x verifying the condition above we have V ∩ Ψ Dβ,p ∩ Yn,1/p = ∅ for all β = γ, β ∈ Γ . Indeed, for x ∈ V ∩ Ψ Dβ,p ∩ Yn,1/p and y ∈ Dβ,p ∩ Yn,1/p with Ψy = x we have y, y ∈ Ψ−1 (V ) ∩ Yn,1/p and so (y, y ) < 1/p thus it follows that Γ} is 1/p-discrete. Since Ψ (Dγ,p ) = β {Dγ,p : γ ∈

∞= γ since the family ∞ ∞ because Y = Ψ D ∩ Y Y ; and Dγ = p=1 Dγ,p we γ,p n=1 n=1 n,1/p

n,1/p ∞ have Ψ (Dγ ) = n,p=1 Ψ Dγ,p ∩ Yn,1/p and the family {Ψ (Dγ ) : γ ∈ Γ} is indeed σ-isolatedly decomposable. ii)=⇒iii) Let B be a cover of Y by sets of diameter less than ε > 0 which we can and according toStone’s theorem. Let us write

∞ do assume it is σ-discrete B = n=1 Bn where every Bn = Bjn : j ∈ Jn is a discrete family in (Y, ). According to our assumption each family ΨBjn : j ∈ Jn is σ-isolatedly de n composable so we can split it up to obtain isolated families Cj,m : j ∈ Jn

∞ n . Fix m, n ∈ N and let us deﬁne for m, n ∈ N where ΨBjn = m=1 Cj,m ε σ-continuous map Φm,n : ΨY → Y as follows:

For every Bjn ∈ Bn we choose an element yBjn ∈ Bjn . Since the sets in n Cj,m : j ∈ Jn are pairwise disjoint for every x∈

n Cj,m : j ∈ Jn

n there exists a unique j ∈ Jn with x ∈ Cj,m ⊂ ΨBjn and we deﬁne n ε Φm,n x = yBjn . Since the family Cj,m : j ∈ Jn is isolated we have that

n Φεm,n is a locally constant function on the set Cj,m : j ∈ Jn . We extend

n the function Φεm,n to X \ Cj,m : j ∈ Jn with a constant value in this set. ε Then Φm,n is σ-continuous. Moreover given y ∈ Y if y ∈ Bjn , j ∈ Jn , and n Ψy ∈ Cj,m we have y, Φεm,n Ψy < ε. Let us observe that such integers m

∞ n for every j ∈ Jn . and n exist because B is a cover and ΨBjn = m=1 Cj,m Since the above argument holds for every ε > 0 it follows that the sequence of σ-continuous functions

2.2 Co-σ-continuous Maps

25

Φ1/p m,n : m, n, p ∈ N veriﬁes the condition iii) of the statement. iii)=⇒i) Given ε > 0 we can deﬁne the sets Yn,ε := {y

∈ Y : (y, Φn ◦ Ψy) ∞ < ε/3} and according to our assumption we have Y = n=1 Yn,ε . Moreover ∞ n n where Φn Xm,ε has at since Φn is σ-continuous we get ΨY = m=1 Xm,ε every point an oscillation less than ε for every m ∈ N. Set n Ym,n,ε := y ∈ Yn,ε : Ψy ∈ Xm,ε

∞ we have again Yn,ε = m=1 Ym,n,ε . Let us choose y ∈ Ym,n,ε and V a neighbourhood of Ψy with ε n osc Φn V ∩Xm,ε < . 3 Then for y ∈ Ym,n,ε and Ψy ∈ V we have (y, y ) ≤ (y, Φn ◦ Ψy) + (Φn ◦ Ψy, Φn ◦ Ψy ) + (Φn ◦ Ψy , y ) < ε so -diameter Ym,n,ε ∩ Ψ−1 (V ) < 2ε as we claimed.

Proposition 2.29. If Ψ : (Y, ) → (X, T ) is co-σ-continuous and Φ : ΨY → Y is a selector for the multivalued map Ψ−1 : ΨY → 2Y then Φ is σ-continuous. Proof. If Φ : ΨY → Y is a selector and we take Z := Φ(ΨY ) ⊂ Y then Ψ Z is one-to-one and co-σ-continuous and the inverse map, which coincides with Φ, must be σ-continuous.

Now we can prove the following Proposition 2.30. Let (Y, ) be a metric space and let (X, T ) be a topological space. For a map Ψ : Y → X the following assertions are equivalent: i) Ψ is co-σ-continuous. ii) The ﬁbers of Ψ are separable and every selector Φ of Ψ−1 is σ-continuous. Proof. i)=⇒ii) It follows from Proposition 2.27 and Proposition 2.29. ii)=⇒i) Let us choose for every y ∈ Y a sequence {yn : n ∈ N} which is dense in the ﬁber Ψ−1 (Ψy). Let Φn : ΨY → Y the selector deﬁned by choosing yn in the ﬁber of y. Then Φn is σ-continuous and

y ∈ {Φn Ψy : n ∈ N} for all y ∈ Y . Now Theorem 2.28.iii) completes the proof.

26

2 σ-Continuous and Co-σ-continuous Maps

Remark 2.31. It was R. Hansell who ﬁrst considered maps Ψ from a space X into another space Y with the property that Ψ transforms discrete families into families with a σ-discrete basis [Han74] which were called co-σdiscrete maps; base-σ-discrete was the terminology used by E. Michael [Mic82] who also studied independently this class of maps. The emphasis was to the property of discrete families and they were used to study arbitrary non separable analytic spaces [Han92] and [Han74], as well as continuous images of some generalized metric spaces [Mic82]. In the thesis of Spahn [Spa81] it was shown that the Amir-Lindenstrauss operator T : X → c0 (Γ) for a weakly compactly generated Banach space X can be constructed with this property, i.e. it sends discrete families to σ-discretely decomposable families. L. Oncina [Onc00] showed that this is always possible when we have a PRI on a Banach space controlling the diﬀerences (Pα+1 − Pα ) X. The characterizations that we present now for co-σ-continuous maps indicate why it is so and show how these maps are the topological counterpart for the well known method of PRI in Banach spaces. Our later applications will clarify our assertions here, (see also our comments in the introduction as well as Sect. 3.4.) When X is a metric space we have the following characterization which is going to play an important role in our applications. Theorem 2.32. Let (Y, ) and (X, d) be metric spaces and let Ψ : Y → X be a map, the following assertions are equivalent: i) Ψ is co-σ-continuous. ii) For every y ∈ Y there exists a separable subset Zy in Y such that y∈

{Zyn : n ∈ N} whenever lim Ψyn = Ψy . n→∞

Proof. i)=⇒ii) Choose for every y ∈ Y a sequence {zn (y) : n ∈ N} dense in the ﬁber Ψ−1 (Ψy) and set Φn : ΨY → Y for the selector of Ψ−1 which chooses zn (y) in the ﬁber of y as in Proposition 2.30. Φn is a σ-continuous map from the metric space (ΨY, d) into the metric space (Y, ) and we can now apply n in ΨY so Proposition 2.13 to deduce the existence of a countable subset WΨy that n Φn (Ψy) ∈ : p ∈ N Φn WΨy p n whenever limp→∞ Ψyp = Ψy. If we set Zy := Ψ−1 (Ψy)∪ Φn WΨy : n∈N for y ∈ Y then Zy is a separable subset of Y . Moreover for every sequence {yp : p ∈ N} with limp→∞ Ψyp = Ψy we have Φn (Ψy) ∈

Zyp : p ∈ N for every n ∈ N ,

and consequently according to the choice of the maps Φn we have as in

Proposition 2.30 y ∈ {Φn Ψy : n ∈ N} ⊂ Zyp : p ∈ N .

2.2 Co-σ-continuous Maps

27

ii)=⇒i) It is clear that condition ii) implies Ψ−1 (Ψy) ⊂ Zy for every y ∈ Y so the ﬁbers of Ψ must be separable. Moreover if for every y ∈ Y we select a countable subsets Wy ⊂ Zy dense in Zy then for every selector Φ of Ψ−1 we have {Wyk : k ∈ N} Φ(Ψy) ∈ whenever limk→∞ Ψyk = Ψy. Then according to Remark 2.15 Φ is indeed σ-continuous. It follows from Proposition 2.30 that Ψ is co-σ-continuous.

When Y is a normed space we can take advantage of its linear structure and we have Proposition 2.33. Let A be a subset of a normed space (Y, ·) and let (X, d) be a metric space. Let Ψ : A → X be a map such that for every y ∈ A there exists a separable subset Zy in Y such that y ∈ span {Zyn : n ∈ N}

·

(2.2)

whenever limn→∞ Ψyn = Ψy. Then the map Ψ is co-σ-continuous. In addition if Y is a normed algebra the condition (2.2) can be replaced by y ∈ alg {Zyn : n ∈ N}

·

where alg D is the subalgebra of Y generated by a subset D of Y and the constant functions. Proof. As in ii)=⇒i) above we have separable ﬁbers for Ψ and every selector Φ of Ψ−1 veriﬁes · Φ(Ψy) ∈ span {Wyk : k ∈ N} whenever limk→∞ Ψyk = Ψy where Wy is a countable dense subset of Zy for every y ∈ Y . According to Corollary 2.20 Φ is σ-continuous in that case too and the assertion follows from Proposition 2.30. In the case where Y is a normed algebra it is necessary to replace span by alg and the conclusion follows.

For weak homeomorphisms we have: Corollary 2.34. If Ψ : Y → X is a weak homeomorphisms between the normed spaces Y and X then Ψ is co-σ-continuous and σ-continuous, i.e. Ψ is a homeomorphism σ-continuous in both directions. ∞

Proof. If {yn }n=1 is a sequence in Y which weakly converges to y in Y then y ∈ ·

span {yn : n ∈ N} by the Hahn-Banach theorem and from Proposition 2.33 it follows that Ψ is co-σ-continuous. The same argument is valid for Ψ−1 where applying Corollary 2.20 instead of Proposition 2.33 to show that Ψ is σ-continuous too.

28

2 σ-Continuous and Co-σ-continuous Maps

Remark 2.35. Srivatsa [Sri93] proved that any continuous map from a metric space into a normed space with the weak topology is a Baire-one function to the norm topology. Consequently a weak homeomorphism is a Baire-one homeomorphism for the norm topologies. Here we are interested only in σ-continuous maps because they are more ﬂexible dealing with renorming questions. For any metric generating a topology ﬁner than the weak topology we must have Corollary 2.36. Let X be a normed space and d a metric on X such that the d-topology is ﬁner than the weak topology. Then the identity map from X into (X, d) is co-σ-continuous. ∞

Proof. If {xn }n=1 is a sequence in X d-convergent to x then x ∈ span {xn : n ∈ N}

·

by the Hahn-Banach theorem so from Proposition 2.33 we get the conclusion.

2.3 P-Maps The following deﬁnition is an abstraction of the σ-continuity condition due to M. Raja [Ra99] and [Raja02]. In the non metrizable case it is more general than functions with σ-isolated function base and it is a very useful tool when we need to connect σ-continuity properties through non metrizable topologies. Deﬁnition 2.37. Let X and Y be sets and V ⊂ P(X), W ⊂ P(Y ) ﬁxed families of subsets of X and Y . A function Φ mapping X into Y will be said to have the property P (V, W) if there is a sequence {An : n ∈ N} of subsets of X such that for every x ∈ X and every W ∈ W with Φx ∈ W there exists a positive integer n together with V ∈ V such that x ∈ An ∩ V ⊂ Φ−1 (W ) ; i.e. the family {An ∩ V : V ∈ V, n ∈ N} is a basis for the family Φ−1 (W ) : W ∈ W}. When Y is a metric space and W is the metric topology we have (see M. Raja [Ra99] and [Raja02]). Proposition 2.38. Let (X, T ) be a topological space and let (Y, ) be a metric space. Then a map Φ : X → Y is σ-continuous if, and only if, Φ has P (T , − topology).

2.3 P-Maps

29

∞ Proof. Assume that Φ is σ-continuous and for ε = 1/p set X = n=1 Xn,ε so that for every n ∈ N and every x ∈ Xn,ε there is V ∈ T with x∈ V ∩ Xn,ε and osc Φ V ∩Xn,ε < ε. If we order the sets Xn,1/p : n, p ∈ N in a sequence {An : n ∈ N} they verify P (T , − topology). On the other hand if Φ has the property P (T , − topology) with the sets {An : n ∈ N}, for ε > 0 ﬁxed we deﬁne Xn,ε := {x ∈ An : ∃Vx ∈ T with x ∈ Vx and Φ (An ∩ Vx ) ⊂ B (Φx, ε/2)} .

∞ By our assumption on Φ we have that X = n=1 Xn,ε and for x ∈ Xn,ε we

will have osc Φ Xn,ε ∩Vx < ε as we wanted to show. With the same proof we have Proposition 2.39. Let A be a subset of a locally convex linear topological space and H be the family of open half spaces in X. If (Y, ) is a metric space a map Φ : A → Y is σ-slicely continuous if, and only if, it has the property P (H ∩ A, − topology). It is also very useful to remark the following (see M. Raja [Ra99] and [Raja02]). Proposition 2.40. Let X, Y and Z be sets with V ⊂ P(X), W ⊂ P(Y ) and Y ⊂ P(Z) families of subsets in X, Y and Z respectively. If Φ : X → Y has P (V, W) and Ψ : Y → Z has P (W, Y) then Ψ ◦ Φ : X → Z has P (V, Y). Proof. If {An : n ∈ N} is a sequence of sets in X so that {An ∩ V : V ∈ V, n ∈ N} is a basis for Φ−1 (W) and {Bm : m ∈ N} is a sequence of sets in Y so that {Bm ∩ W : W ∈ W, m ∈ N} −1 is a basis for Ψ (Y) then the family of sets An ∩ Φ−1 (Bm ) ∩ V : V ∈ V, −1 m, n ∈ N} will be a basis for Φ−1 Ψ−1 (Y) = (Ψ ◦ Φ) (Y).

Then we have as simple corollaries. Corollary 2.41. Let (X, T ) be a topological space and let (Y, ) be a metric space. Let Φ : X → Y be σ-continuous. If (Z, d) is a metric space and Ψ : Y → Z is σ-continuous then Ψ ◦ Φ : X → Z is σ-continuous too. If (W, V) is a topological space and Ξ : (W, V) → (X, T ) has P (V, T ) then Φ ◦ Ξ : W → Y is σ-continuous too. Proof. It follows from Propositions 2.38 and 2.40.

Corollary 2.42. Let A be a subset of a linear topological spaces X, let (Y, ) be a metric space and let Φ : A → Y be σ-slicely continuous. If (Z, d) is a metric space and Ψ : Y → Z is a σ-continuous map then Ψ ◦ Φ : A → Z is σ-slicely continuous too. If B is a subset of a linear topological space W and Ξ : B → A has the property P (open half spaces of W, open half spaces of X) then Φ ◦ Ξ : B → Y is σ-slicely continuous too.

30

2 σ-Continuous and Co-σ-continuous Maps

Proof. It follows from Propositions 2.38, 2.39 and 2.40.

Corollary 2.43. Let A be a subset of a locally convex linear topological space and let H be the family of all open half spaces in X. If (Y, ) is a metric space and Φ : A → Y is a σ-continuous map then Φ is σ-slicely continuous if there exists a sequence {An : n ∈ N} of subsets of A such that the family {An ∩ H : H ∈ H, n ∈ N} is a network for the topology of A. Proof. The identity map in A has the property P (A ∩ H, topology of A) and therefore Φ has P (A ∩ H, − topology) by Proposition 2.40 which is nothing else than σ-slicely continuity after Proposition 2.39.

A set of a linear topological space X is said to be radial (with respect to 0) if for every x ∈ X there exists λ > 0 such that λx ∈ A. Proposition 2.44. Let A be a radial set in a locally convex linear topological space X and let H be the family of all open half spaces in X. If for every x ∈ X, x = 0, we choose r(x) > 0 such that r(x)x ∈ A then the map Φ : X \ {0} → A deﬁned by Φx := r(x)x has the property P (H, H). Proof. Every H ∈ H can be written as H = {x ∈ X : f (x) > µ} where f is a continuous linear form in X and µ ∈ R. Let x0 ∈ X \ {0} such that Φx0 ∈ H. If q is a positive rational number such that q < r (x0 ) and f (x0 ) > µ/q, which always exists since f (x0 ) > µ/r (x0 ), and we set Aq := {y ∈ X : r(y) > q} we have x0 ∈ Aq and taking µ H = x ∈ X : f (x) > q we will have x0 ∈ Aq ∩ H , and for every y ∈ Aq ∩ H f (Φ(y)) = f (r(y)y) = r(y)f (y) >

qµ =µ, q

and so Φy ∈ H. Consequently the sequence of sets Aq : q ∈ Q+ gives us the property P (H, H) for the map Φ.

Making intersections of the sequence of sets {Aq : q ∈ Q+ } just deﬁned we have: Corollary 2.45. If A, X, H, r(x) and Φ are as above then the map Φ has the property P (weak topology, weak topology).

2.4 Co-σ-continuity of Maps Associated to PRI and M-Basis

31

Proof. Given W a weak-open set of X and x0 ∈ X such p that Φ (x0 ) ∈ W there exists H1 , H2 , . . . , Hp in H, p ∈ N, with Φ (x0 ) ∈ i=1 Hi ⊂ W . With the same notations of the proof of Proposition 2.44, for every i ∈ {1, 2, . . . , p} there is a rational qi > 0 and Hi ∈ H such that x0 ∈ A qi ∩Hi and Φ(y) ∈ Hi for x0 ∈ Aq1 ∩ Aq2 ∩ . . . ∩ Aqp ∩ (H1 ∩ H2 ∩ . . . ∩ Hp ) every y ∈ Aqi ∩ Hi . Thus and for p every y ∈ Aq1 ∩ Aq2 ∩ . . . ∩ Aqp ∩ (H1 ∩ H2 ∩ . . . ∩ Hp ) we have Φy ∈ i=1 Hi ⊂ W . Therefore the sequence of sets Aq1 ∩ Aq2 ∩ . . . ∩ Aqn : qi ∈ Q+ , n ∈ N gives us the property P (weak topology , weak topology) for the map Φ.

Now we can prove the following Proposition 2.46. Let X be a locally convex linear topological space and let A ⊂ X be a radial subset of X. If Y is a normed space and Ψ : X → Y is a homogeneous map then Ψ is σ-slicely continuous (respectively σ-continuous) if, and only if, its restriction to A is σ-slicely continuous (respectively σ-continuous ). Proof. For every x ∈ X \ {0} set r(x) > 0 with r(x)x ∈ A and denote by Φ the map from X \ {0} onto A deﬁned by Φx = r(x)x which according to Proposition 2.44 has P (H, H). Then the composition map Ψ1 x := Ψ(r(x)x) = Ψ◦Φ(x), ∀x ∈ X\{0}, is σ-slicely continuous when Ψ A is σ-slicely continuous according to Propositions 2.39 and 2.40. For every positive rational number q the map Ψq x := qΨ1 x is also σ-slicely continuous in X \ {0}. Finally since Ψ is homogeneous we have Ψx =

lim

q→1/r(x)

Ψq x for every x ∈ X \ {0} ·

consequently Ψx ∈ {Ψq x : q ∈ Q+ } for every x ∈ X \ {0}. Now Proposition 2.23 shows that ΨX\{0} is σ-slicely continuous and so is Ψ. The same argument holds for σ-continuity using Corollary 2.45 instead of Proposition 2.44 and Proposition 2.3 for the cluster point argument to ﬁnish the proof.

2.4 Co-σ-continuity of Maps Associated to PRI and M-Basis It follows from Theorem 1.6 that the only assumption we need on a linear continuous map T from a normed space X into a LUR renormable space Y to lift the LUR norm on X is the co-σ-continuity of T . We have seen in the introduction how maps T : X → c0 (Γ) can be constructed, for Banach spaces with PRI in the so called class P, being co-σ-continuous too. We shall begin this section showing that maps based on PRI with co-σ-continuous pieces

32

2 σ-Continuous and Co-σ-continuous Maps

will be co-σ-continuous too. That is the case for all “diagonal operators” based on the construction of Amir and Lindenstrauss [AL68], as well as, for instance, every M-basis in a weakly Lindel¨ of Banach spaces. Let us remember the precise deﬁnitions: Deﬁnition 2.47. Let X be a Banach space and µ the smallest ordinal such that its cardinality #µ := #[0, µ] is equal to the density character of X. A projectional resolution of the identity (PRI for short) on X is a collection {Pα : ω0 ≤ α ≤ µ} of norm one projections on X that satisfy, for every ω0 ≤ α ≤ µ the following conditions: i) Pα Pβ = Pβ Pα = Pα if ω0 ≤ α ≤ β ≤ µ; ii)

the density character Pα X is less or equal than #α; iii) {Pβ+1 X : β < α} is norm dense in Pα X; iv) Pµ = Id. A ﬁrst consequence of the existence of PRI on a Banach space X with an operator T : X → Y such that the restrictions T (Pα+1 −Pα )X are co-σ-continuous will be the co-σ-continuity of T whenever Y is comparable with the decompositions. Proposition 2.48. Let T : X → Y be a one-to-one continuous linear map between Banach spaces such that X admits a PRI {Pα : ω0 ≤ α ≤ µ} such that for every α ∈ [ω0 , µ) there is a continuous linear map Rα : Y → Y with T ◦ Sα = Rα ◦ T and T Sα X co-σ-continuous where Sα co-σ-continuous.

=

Pα+1 − Pα . Then the operator T is

Proof. Let us ﬁx for every α, ω0 ≤ α < µ, and every x ∈ Sα X a countable subset Zx,α ⊂ Sα X such that x ∈ span {Zxn ,α : n ∈ N} ∞

whenever x, xn ∈ Sα X, for all n ∈ N, and {T xn }n=1 converges to T x in Y. This is possible by the co-σ-continuity of T Sα X . If we take a point x ∈ X we know that x = limα→µ Pα x and the set {α ∈ [ω0 , µ) : Sα x = 0} is countable [Fab97, Proposition 6.2.1.]. Therefore we can deﬁne the separable piece Zx := Pω0 X ∪ {Sα x : ω0 ≤ α < µ} ∪ {ZSα x,α : Sα x = 0} and we will have now x ∈ span {Pω0 x, Sα x} ⊂ span {Zxn : n ∈ N} ,

2.4 Co-σ-continuity of Maps Associated to PRI and M-Basis ∞

33

∞

whenever {xn }n=1 is a sequence in X with {T xn }n=1 converging to T x in Y . Indeed, for every α ∈ [ω0 , µ[ such that Sα x = 0 we have T (Sα xn ) = Rα T xn and so we have lim T (Sα xn ) = lim Rα T xn = Rα T x = T (Sα x) ,

n→∞

n→∞

(2.3)

then it follows that Sα x ∈ span {ZSα xn ,α : n ∈ N} and so we have Sα x ∈ span {Zxn : n ∈ N} since for big enough n we have Sα xn = 0 because T is a one-to-one operator and (2.3).

Corollary 2.49. If a Banach space X admits a PRI {Pα : ω0 ≤ α ≤ µ} such that for every α ∈ [ω0 , µ) there exists a bounded linear one-to-one map Tα : (Pα+1 − Pα ) X → c0 (Γα ) which is co-σ-continuous then, if Γ is the disjoint union of {Γα : α ∈ [ω0 , µ) }, the “diagonal operator” T : X → c0 (Γ) deﬁned by −1 T x(γ) = Tα Tα ((Pα+1 − Pα ) x) (γ) for γ ∈ Γα , ω0 ≤ α < µ, is co-σ-continuous. Proof. T is well deﬁned (see for instance [Fab97, Proposition 6.2.2.]) and veriﬁes the conditions of the former proposition with Rα : c0 (Γ) → c0 (Γ) the

map which makes zero all coordinates outside Γα . It was Spahn [Spa81] the ﬁrst to show Corollary 2.49 for weakly compactly generated Banach spaces X. Oncina [Onc00] extended it for spaces with PRI in the so called class P, see Remark 2.31. Deﬁnition 2.50. A Markushevich basis (M-basis for short) in a Banach space X is a family of vectors {(eγ , fγ ) : γ ∈ Γ} in X × X ∗ such that i) fβ (eγ ) = 0 if β = γ and fγ (eγ ) = 1; span {eγ : γ ∈ Γ} = X; ii) iii) {Ker fγ : γ ∈ Γ} = {0}. For a given M-basis {(eγ , fγ ) : γ ∈ Γ} we can assume without loss of generality fγ ≤ 1 for all γ ∈ Γ and so we have an operator T : X → c0 (Γ) with the evaluation T x = (fγ (x))γ∈Γ for all x ∈ X. When T is co-σ-continuous? According to Theorem 1.6 if T is co-σ-continuous X must be LUR renormable, therefore it is not always true that T is co-σ-continuous since ∞ is a complemented subspace of a Banach space with M-basis [FHHMPZ01, Theorem 6.45] as Plichko showed. When the M-basis is strong; i.e. x ∈ span {eγ : fγ (x) = 0} for all x ∈ X, it follows from Proposition 2.33 that T is co-σ-continuous since we can set Zx := {eγ : fγ (x) = 0}

34

2 σ-Continuous and Co-σ-continuous Maps ∞

which is countable, and we have x ∈ span {Zxn :

n ∈ N} whenever {T xn }n=1 ∞ converges to T x in c0 (Γ) since {γ : fγ (x) = 0} ⊂ n=1 {γ : fγ (xn ) = 0} and the M-basis is strong. When the M-basis is norming; i.e. span {fγ : γ ∈ Γ} is a norming subspace for X, then from [JNR93] it follows that the Banach space X is σ-fragmentable and moreover T −1 : T X → X is σ-fragmented for the topology induced by the pointwise topology in c0 (Γ) on T X to the ·-topology. Then by Theorem 2.14 T −1 is σ-continuous from (T X, · ) into X and so T is co-σ-continuous too. We can state now this result in full generality for countably norming Mbasis; i.e. {g ∈ X ∗ : {γ ∈ Γ : g (eγ ) = 0} is countable} is a norming subspace of X. Corollary 2.51. If a Banach space X admits a countably norming M-basis {(eγ , fγ ) : γ ∈ Γ} the evaluation operator T : X → c0 (Γ); i.e. T x = (fγ (x))γ∈Γ for every x ∈ X, is co-σ-continuous. Proof. Following Plichko [Pli82], and assuming without loss of generality that the M-basis is 1-countably norming up to a renorming of X, we know that X has a projectional resolution of the identity {Pα : ω0 ≤ α ≤ µ} with a family {Γα : ω0 ≤ α ≤ µ} of increasing subsets of Γ with Pα X = span {eγ : γ ∈ Γα } and Ker Pα = span {eγ : γ ∈ / Γα }, #Γα ≤ #α, Pα∗ X ∗ ⊃ {fγ : γ ∈ Γα } and ∗ / Γα }. Thus it follows that for every α, ω0 ≤ α < µ we Ker Pα ⊃ {fγ : γ ∈ have that (Pα+1 − Pα ) X = Pα+1 X ∩ Ker Pα = span {eγ : γ ∈ Γα+1 \ Γα } . Thus the proof of Corollary 2.51 follows by induction on the density character of X and Proposition 2.48 with Rα : c0 (Γ) → c0 (Γ) the map which makes

zero the coordinates outside Γα+1 \ Γα for every α, ω0 ≤ α < µ. Remark 2.52. Countably norming M-basis can be constructed in Banach spaces with a subset M ⊂ X such that span M = X and {g ∈ X ∗ : {x ∈ M : g(x) = 0} countable} is a norming subspace of X [Vald90, Val91], in particular in C(K) where K is a Valdivia compact space. Every M-basis in a weakly-Lindel¨ of Banach space veriﬁes {γ : f (eγ ) = 0} is countable for all f ∈ X ∗ , [FHHMPZ01, Theorem 12.50] and the operator T will be co-σ-continuous too. This is the case when (BX ∗ , weak∗ ) is a Corson compact space (see also [VWZ94]). The recent book [HMVZ08] provides a complete guide for the construction of coordinates systems as well as the interplay between geometrical and topological properties with biorthogonal systems in Banach spaces.

2.5 Co-σ-continuity of the Operator of Haydon in C(Υ), Υ a Tree

35

2.5 Co-σ-continuity of the Operator of Haydon in C(Υ), Υ a Tree Let Υ be a tree in the sense of Haydon, [Hay99], i.e. a tree is a partially ordered set (Υ, ) with the property that, for every t ∈ Υ, the set {s ∈ Υ : s t} is well-ordered by . It is used normal interval notation, so that, for instance, (s, u] = {t ∈ Υ : s ≺ t u}. For convenience of notation, the tree Υ has two “imaginary” elements, not in Υ, denoted by 0 and ∞, and having the property that 0 ≺ t ≺ ∞ for all t ∈ Υ. This allows us to extend our interval notation to include expressions like (0, t] and [t, ∞). Note that, by deﬁnition, each (0, t] is well-ordered, but that [t, ∞) need not be. Let us remember that a subset Λ of a tree Υ is said to be ever branching if for every t ∈ Λ the intersection Λ ∩ [t, ∞) is not totally ordered. In what follows : Υ → R will denote a non decreasing function. Given t ∈ Υ, the symbol t+ will stand for the set of all immediate successors of t and Ft = {s ∈ t+ : (s) = (t)}. An element t of Υ is a good point for if inf s∈t+ \Ft (s) > (t). Taking into account that for every v ∈ Υ the set (0, v] is well ordered and is non decreasing on it we get # {u ∈ (0, v] : Fu ∩ (0, v] = ∅} ≤ ℵ0 .

(2.4)

For each t ∈ Υ there is a unique ordinal r(t) with the same order type as ]0, t[. Fremlin [Fre84] says that a tree is Hausdorﬀ if, whenever r(t) is a limit ordinal and (0, t ) = (0, t), we necessarily have t = t . Such a tree may be equipped with a locally compact, and Hausdorﬀ, topology which may be characterized as the coarsest for which all intervals (0, t] are clopen. We consider only trees that are Hausdorﬀ. We denote by C0 (Υ) the space of real-valued functions x on Υ, which are continuous for the locally compact topology and are such that, for all ε > 0, the set {t ∈ Υ : |x(t)| ≥ ε} is compact for that topology. The standard norm in C0 (Υ) is the sup norm · ∞ . Theorem 2.53. (Haydon [Hay99]) Let Υ a tree if there exists a non decreasing function : Υ → R which is constant on no ever-branching subset of Υ and such that every point of Υ is a good point for then there exists a LUR renorming of C0 (Υ). Since for an ordinal τ the tree Υ = [0, τ ] has not ever-branching subsets from the above theorem we will deduce the following Corollary 2.54. For any ordinal τ the space C([0, τ ]) is LUR renormable. It seems that this result was proved ﬁrst by G. Alexandrov and K. Kursten (see [Alex80]). The proof of Theorem 2.53 is based on Haydon’s operator T : C0 (Υ) → c0 (Υ). Before deﬁning T let us make some observations. Replacing by arctan we obtain a bounded function on Υ. Moreover for every non isolated point t ∈ Υ we can replace (t) by lims↑t (s) then becomes a continuous

36

2 σ-Continuous and Co-σ-continuous Maps

function on Υ. Clearly after these changes : Υ → R is a bounded non decreasing continuous function which is constant on no ever-branching subset of Υ and which has only good points. For t ∈ Υ we set ⎧ ⎨ inf {(s) − (t) : s ∈ t+ \ Ft } if t+ \ Ft = ∅ δt = ⎩ 1 if t+ \ Ft = ∅, or t+ = {∞} . For x ∈ C0 (Υ) and t ∈ Υ we set δt T x(t) = 1 + #Ft

x(t) −

x(s)

,

(2.5)

s∈Ft

where Ft = ∅ if t+ = {∞}. Clearly T : C0 (Υ) → ∞ (Υ) is a bounded linear operator. We prove Lemma 2.55. (Haydon [Hay99]) Let T be the bounded linear operator deﬁned in (2.5). Then T C0 (Υ) ⊂ c0 (Υ) and for every x ∈ C0 (Υ) and every t ∈ supp x we have [t, ∞) ∩ supp T x = ∅. Proof. In order to prove that T C0 (Υ) ⊂ c0 (Υ) it is enough to see that for every v ∈ Υ we have T 1l(0,v] ∈ c0 (Υ). Pick t ∈ (0, v) and set t = t+ ∩ (0, v]. Then T 1l(0,v] (t) =

δt (t ) − (t) 1lt+ \Ft (t ) ≤ ≤ (t ) − (t) . 1 + #Ft 1 + #Ft

This implies T 1l(0,v] (t) ≤ t∈Υ

δv + ( (t ) − (t)) ≤ δv + (v) . 1 + #Fv t≺v

Hence T 1l(0,v] ∈ 1 (Υ). We will show now that for every t ∈ supp x we have [t, ∞)∩ supp T x = ∅. Assume the contrary, i.e. there exists t ∈ supp x such that [t, ∞) ∩ supp T x = ∅ .

(2.6)

Set Λ = {v ∈ [t, ∞) : (v) = (t), v ∈ supp x} . Since t ∈ Λ we get that Λ = ∅. Taking into account that Λ is a non everbranching set we can ﬁnd u ∈ Λ such that [u, ∞) ∩ Λ is totally ordered. Pick v ∈ [u, ∞) ∩ Λ. From the deﬁnition of Fv , Λ, T and (2.6) we get

2.5 Co-σ-continuity of the Operator of Haydon in C(Υ), Υ a Tree

Fv ∩ supp x ⊂ [u, ∞) ∩ Λ ; x(v) =

x(s) .

37

(2.7) (2.8)

s∈Fv

Therefore there exists s ∈ Fv with x(s) = 0. From (2.7) we get that s ∈ [u, ∞) ∩ Λ. On the other hand # (v + ∩ [u, ∞) ∩ Λ) ≤ 1. Then Fv ∩ [u, ∞) ∩ Λ = {s}. This and (2.7) imply Fv ∩ supp x = {s}. Hence x(v) = x(s). So x is constant in [u, ∞) ∩ Λ. Therefore we can write [u, ∞) ∩ Λ = {v u : (v) = (u), x(v) = x(u)} . Since x and are continuous on Υ the set [u, ∞) ∩ Λ is compact. Let v0 be the maximal element of [u, ∞) ∩ Λ, we will show that x (v0 ) = 0 which is a contradiction. Indeed from (2.7) and the maximality of v0 we get Fv0 ∩

supp x = ∅. Hence from (2.8) we get x (v0 ) = 0. Proposition 2.56. The bounded linear operator T deﬁned by (2.5) is co-σ-continuous. Proof. In order to apply Proposition 2.33 let us deﬁne for every x ∈ C0 (Υ) the sets Ux = 1l(0,u] : ∃v, w ∈ supp T x, (0, u] = (0, v] ∩ (0, w] Vx = 1l(0,u] : ∃v ∈ supp T x, u ∈ (0, v], Fu ∩ (0, v] = ∅ Zx = Ux ∪ Vx . Since T x ∈ c0 (Υ) we have #supp T x ≤ ℵ0 . Hence #Ux ≤ ℵ0 . From (2.4) we get #Vx ≤ ℵ0 . So (2.9) #Zx ≤ ℵ0 .

∞ ∞ Let x, xn ∈ C0 (Υ) and T xn − T x∞ → 0. Then supp T x ⊂ m=1 n=m supp T xn . From the deﬁnition of Zx we have Zx ⊂

∞

Zxn .

(2.10)

n=1

Having in mind (2.9) and (2.10) the statement follows from Proposition 2.33 as soon as we show that ·∞ . (2.11) x ∈ spanZx Fix x ∈ C0 (Υ). For ε > 0 set Lε (|x|) = {t ∈ Υ : |x(t)| ≥ ε}, we show that there exists Dx,ε ⊂ Υ such that (0, v] . (2.12) #Dx,ε < ∞ , Dx,ε ⊂ supp T x , Lε (|x|) ⊂ v∈Dx,ε

38

2 σ-Continuous and Co-σ-continuous Maps

Since Lε (|x|) is compact and {(0, v] : v ∈ Lε (|x|)} is an open cover of Lε (|x|)

k we can ﬁnd u1 , u2 , . . . uk ∈ Lε (|x|) such that Lε (|x|) ⊂ i=1 (0, ui ]. According to Lemma 2.55 there exist vi ∈ supp T x, ui ∈ (0, vi ], i = 1, 2, . . . , k. In order k to prove (2.12) it is enough to set Dx,ε = {vi }1 . Having in mind that (0, v] is a well ordered set we have x1l(0,v] ∈ span

1l(0,t] : t ∈ {v} ∪ Λv,x

·∞

,

(2.13)

where Λv,x = t ∈ Υ : t ∈ (0, v[, x(t) = x (t ) where {t } = t+ ∩ (0, v] . Set

Wx = Ux ∪ 1l(0,t] : t ∈ {Λv,x : v ∈ supp T x} .

Taking into account that 1l {(0,v]:

v∈Dx,ε }

is a ﬁnite linear combination of

1l(0,u] ∈ Ux from (2.12) and (2.13) we get x1l {(0,v]: ·∞

v∈Dx,ε }

∈ span Wx

·∞

,

and dist (x, span Wx ) ≤ ε. Hence x ∈ span Wx . So in order to prove (2.11) it is enough to show that Wx ⊂ Zx . Since 1l(0,t] ∈ Ux ⊂ Zx for t ∈ supp T x it is enough to see that (2.14) 1l(0,t] ∈ Zx whenever T x(t) = 0 and t ∈ Λv,x for some v ∈ supp T x. For such t we have x(t) = x(s) , t ∈ (0, v] , x(t) = x (t ) where {t } = t+ ∩ (0, v] . (2.15) s∈Ft

If t ∈ / Ft we have Ft ∩ (0, v] = ∅ so 1l(0,t] ∈ Vx ⊂ Zx . If t ∈ Ft then from (2.15) it follows that there s ∈ Ft , s = t with x(s) = 0. According to Lemma 2.55 we can ﬁnd w ∈ supp T x with s ∈ (0, w]. Hence

(0, t] = (0, v] ∩ (0, w] so 1l(0,t] ∈ Ux therefore (2.14) is proved. The proof of Theorem 2.53 follows directly from Lemma 2.55, Proposition 2.56 and Theorem 1.6.

2.6 Co-σ-continuity in Weakly Countable Determined Banach Spaces The class of weakly countably determined Banach spaces was introduced and studied by M. Talagrand [Tal79] and L. Vaˇsak [Vas81]. The topological notion corresponds to an extension of K-analyticity called Lindel¨ of Σ-space in the Russian school of A. V. Arhangel’ski˘ı [Arca92]. To get an idea of the deep

2.6 Co-σ-continuity in Weakly Countable Determined Banach Spaces

39

interplay of this notion with geometric properties of Banach spaces see the papers of S. Negrepontis [Neg84], S. Mercourakis and S. Negrepontis [MN92] and the paper of V. Zizler in the Handbook of the Geometry of Banach spaces vol. II [Ziz03], as well as the monographs by R. Deville, G. Godefroy and V. Zizler [DGZ93] and by M. Fabian [Fab97]. Several deﬁnitions can be given for this notion (see [Fab97, Chap. 7]), we shall use the following: Deﬁnition 2.57. A topological space (X, T ) is called K-countably determined if it embeds in a compact space K in such a way that there is a sequence Kn of compact subsets of K such that for every x ∈ X and y ∈ K \ X there exists n0 ∈ N such that x ∈ Kn0 and y ∈ / Kn0 . It turns out that K-countably determined spaces have the above embedding in any compactiﬁcation of it. For a Banach spaces X to say that X is weakly countably determined means that (X, weak) is K-countably determined and it is so if, and only if, there exists a sequence {Kn : n ∈ N} of weak∗ compact subsets of X ∗∗ such that for every x ∈ X and y ∈ X ∗∗ \ X there is n0 ∈ N with x ∈ Kn0 and y ∈ / Kn0 . When (X, T ) is K-countably determined in the compact space K with the sequence {Kn : n ∈ N} we see that for some subset Σ ⊂ NN we will have X=

∞

Kσ(i)

σ∈Σ i=1

∞ and every set Aσ := i=1 Kσ(i) is a non-void compact subset of X for each σ ∈ Σ. Moreover if we ﬁx σ ∈ Σ and a sequence xn ∈ Kσ(1) ∩Kσ(2) ∩. . .∩Kσ(n) ∩X, ∞ ∞ K n ∈ N, then m=1 {xn : n ≥ m} ⊂ i=1 Kσ(i) ⊂ X and the sequence {xn } should have all its cluster points in (X, T ). Consequently {xn : n ∈ N} compact subset of (X, T ) since ∞

T

{xn : n ∈ N} = {xn : n ∈ N} ∪

T

is a

T

{xn : n ≥ m} =

m=1

= {xn : n ∈ N} ∪ = {xn : n ∈ N} ∪

∞ m=1 ∞

K

{xn : n ≥ m} ∩ X = {xn : n ≥ m}

K

K

= {xn : n ∈ N} .

m=1

Now we can formulate the following result providing automatic co-σcontinuity very often. Theorem 2.58. Let (X, T ) be a K-countably determined topological space and let be a metric on X which is ﬁner than T on every compact separable subset of (X, T ). Then there exists another metric d on X generating a topology ﬁner than the -topology and the T -topology and such that the identity map from (X, ) into (X, d) is σ-continuous.

40

2 σ-Continuous and Co-σ-continuous Maps

Proof. Let K be a compact space where X embeds and it is countably determined with a sequence {Kn : n ∈ N} of closed subsets of K. The family of sets Kn1 ∩ Kn2 ∩ . . . ∩ Knp ∩ V : V is -open, n1 , n2 , . . . , np ∈ N, p ∈ N is a basis of a metrizable topology on X, let say with a metric d, and the identity map from (X, ) into (X, d) is σ-continuous by Proposition 2.38. X, x ∈ X and Moreover d is ﬁner than and if {xn } is a sequence in ∞ limn→∞ d (x, xn ) = 0, we will have for σ ∈ Σ with x ∈ i=1 Kσ(i) ⊂ X and every positive integer p the existence of np ∈ N such that for n ≥ np we T

have xn ∈ Kσ(1) ∩ Kσ(2) ∩ . . . ∩ Kσ(p) , thus {xn : n ∈ N} is a compact subset T

of X and {xn : n ∈ N} ∪ {x} will be compact too. Since (x, xn ) → 0 and is ﬁner than T on separable compact subsets of X we should have that (xn ) converges to x in the T -topology too from where the conclusion follows.

Corollary 2.59. Let X be a weakly countably determined Banach space, let (Y, ) be a metric space and let Ψ : X → Y be a one-to-one map such that for every separable weakly compact subset K of X we have that Ψ−1 : ΨK → (K, weak) is continuous then Ψ is co-σ-continuous. Proof. According to Theorem 2.58 a metric d can be deﬁned on X such that Ψ−1 : (ΨX, ) → (X, d) is σ-continuous and the d-topology is ﬁner than the weak topology on X, then the identity (X, d) → (X, · ) is σ-continuous by Corollary 2.20 and ﬁnally

Ψ−1 : (ΨX, ) → (X, · ) is σ-continuous too by Corollary 2.41. In the linear case we have the following corollary of M. Raja [Raj04] where it is used to study Borel measurability of the inverse operator T −1 . Corollary 2.60. Let X be a weakly countably determined Banach space and let Y be a normed space with a continuous, one-to-one linear operator T : X → Y . Then T is co-σ-continuous. Proof. Since the metric (x, y) := T x − T yY , x, y ∈ X is weakly lower semicontinuous on every weakly compact subset of X the -topology should be ﬁner than the weak topology and the conclusion follows from the former result.

2.7 Co-σ-continuous Maps in C(K). The Case of the Compact of Helly. Let us recall that every compact topological space K can be considered, without loss of generality, embedded in a cube, K ⊂ [0, 1]Γ .

2.7 Co-σ-continuous Maps in C(K)

41

Deﬁnition 2.61. Given x ∈ C(K), K ⊂ [0, 1]Γ and Λ ⊂ Γ we will say that Λ controls x if x(s) = x(t) whenever s, t ∈ K and s Λ = t Λ .

(2.16)

Given x ∈ C(K) it is an easy consequence of the uniform continuity of x in K that there exists a countable subset Λ ⊂ Γ which controls x (This is just the theorem of Mibu [Eng66]). The space C(K) has structure of Banach algebra and we are going to take advantage of this structure in the next characterization. Theorem 2.62. Let K be a compact subset of [0, 1]Γ for some Γ. Given (Y, ) a metric space and a map Φ : C(K) → Y the following are equivalent: i) The map Φ is co-σ-continuous; ii) for every x ∈ C(K) there exists a countable set Λx ⊂ Γ such that n Λxn controls x whenever limn→∞ Φxn = Φx. Before proving this theorem we will introduce some notation and show some simple properties of the set of functions which are controlled by a ﬁxed set. Given Λ ⊂ Γ let PΛ : [0, 1]Γ → [0, 1]Λ the map deﬁned by PΛ s = s Λ for any s ∈ [0, 1]Γ . Lemma 2.63. Let K, Λ and x as in Deﬁnition 2.61 then Λ controls x if, and only if, for every ε > 0 there exist δ > 0 and F ⊂ Λ, F ﬁnite, such that |x(s) − x(t)| < ε whenever |s(γ) − t(γ)| < δ for every γ ∈ F . Proof. It is clear that any set Λ which fulﬁlls the last condition must control x. Conversely if Λ controls x let us assume there exists ε0 > 0 such that for any ﬁnite subset F ⊂ Λ and any n ∈ N we can ﬁnd sn,F , tn,F ∈ K with |x (sn,F ) − x (tn,F )| ≥ ε0 and |sn,F (γ) − tn,F (γ)| < 1/n for any γ ∈ F . The compactness of K gives cluster points s and t to the nets (sn,F ) and (tn,F ). Then we get |x (s) − x (t)| ≥ ε0 and s(γ) = t(γ) for any γ ∈ Λ. A contradiction.

Lemma 2.64. Let K a compact subset of [0, 1]Γ and Λ ⊂ Γ then the set of all functions x ∈ C(K) such that Λ controls x is {y ◦ (PΛ K ) : y ∈ C (PΛ K)}

(2.17)

Proof. It is clear that Λ controls any function of the set (2.17). Conversely let x ∈ C(K) such that Λ controls x. According to (2.16) given t ∈ PΛ K the value of x(s) is the same for any s ∈ K such that PΛ (s) = t. Then it makes sense to deﬁne y : PΛ K → R by the identity y(t) = x(s) where PΛ (s) = t for t ∈ PΛ K . The continuity of y follows from Lemma 2.63. Then x = y ◦ (PΛ K ).

(2.18)

42

2 σ-Continuous and Co-σ-continuous Maps

Lemma 2.65. Let K and Λ as in Deﬁnition 2.61 and let A be a subalgebra of {y ◦ (PΛ K ) : y ∈ C (PΛ K)} which contains the constant functions. Let us suppose that for any pair s, t ∈ K for which there exists γ ∈ Λ with s(γ) = t(γ) we can ﬁnd x ∈ A with x(s) = x(t). Then A is uniformly dense in {y ◦ (PΛ K ) : y ∈ C (PΛ K)}.

∞ In particular, if Λn ⊂ Γ and Λ = n=1 Λn then the algebra generated by {y ◦ (PΛn K ) : y ∈ C (PΛn K) , n ∈ N} is uniformly dense in {y ◦ (PΛ K ) : y ∈ C (PΛ K)}. Proof. To see the ﬁrst assertion let us observe that from (2.18) it follows that there exists an algebra B, B ⊂ C (PΛ K), which contains the constant functions such that A = {y ◦ (PΛ K ) : y ∈ B} . Since B separates points in PΛ K the Stone-Weierstrass theorem gives us that B is uniformly dense in C (PΛ K) and the ﬁrst part of the statement is proved. The second assertion follows in a straightforward way from the ﬁrst one.

Proof of Theorem 2.62. i)⇒ii). According to the theorem of Mibu for any x ∈ C(K) we can and do select a countable subset ∆x ⊂ Γ which controls x. Let Zx , Zx ⊂ C(K), separable subsets fulﬁlling the assertion of Theorem 2.32. For each x let Mx a ﬁxed countable dense subset of Zx . Set {∆y : y ∈ Mx } . Λx = From the choice of the sets Zx ’s we must have x ∈ Zx so Λx controls x. Moreover let xn ∈ C(K) be a sequence such that Φxn → Φx then x∈ {Zxn : n ∈ N} = {Mxn : n ∈ N} . Then

∞ n=1

Λxn controls x.

ii)⇒i). For x ∈ C(K) let Zx := {y ◦ (PΛx K ) : y ∈ C (PΛx K)}. Since PΛx K ⊂ [0, 1]Λx and Λx is countable we have that the space PΛx K is metric so Zx is separable. In order to check that the sets Zx verify the requirements of Lemma 2.33 let us consider a sequence x, xn ∈ C(K) such that

lim Φxn = Φx. From ii) it follows that Λ := Λxn controls x so there must exist y ∈ C(PΛ K) such that x = y ◦ (PΛ K ). Then according to Lemma 2.65 we have x ∈ alg {Zxn : n ∈ N} apply Proposition 2.33.

·

. Then to ﬁnish the proof it is enough to

In C(K) spaces we have a canonical map to c0 (Γ). Indeed let K ⊂ [0, 1]Γ the uniform continuity of every x ∈ C(K) allows us to deﬁne the oscillation map Ωx ∈ c0 (Γ) [Eng66], where

2.7 Co-σ-continuous Maps in C(K)

Ωx(γ) = sup x(t) − x(s) : t, s ∈ K , (t − s)Γ\{γ}

=0 .

43

(2.19)

In [Eng66] Ω was introduced looking for countable sets of coordinates which control a continuous function to obtain extensions of the theorem of Mibu. As a straightforward consequence of Corollary 1.21 we get the σ-slicely continuity of the oscillation map Ω. Moreover from Corollary 4.34.ii) we have Corollary 2.66. Let K be a compact subset of [0, 1]Γ for some Γ. The oscillation map Ω : C(K) → c0 (Γ) is pointwise-σ-slicely continuous. Therefore according to Theorem 1.15 the oscillation map transfers a LUR renorming from c0 (Γ) to C(K) whenever Ω is co-σ-continuous. Roughly speaking Theorem 2.62 says that this is so whenever it is possible to associate in a nice way to each x ∈ C(K) a set Λ which controls x. Of course if Λ ⊂ Γ controls x then Λ must contain supp Ωx = {γ ∈ Γ : Ωx(γ) > 0}. Proposition 2.67. Let K be a compact subset of [0, 1]Γ for some Γ, and let Q be a countable subset of Γ such that for every x ∈ C(K) the set Q ∪ supp Ωx controls x. Then Ω is co-σ-continuous. ∞

Proof. If {xn }n=1 is

∞a sequence in C(K) such that Ωx − Ωxn ∞ → 0 then supp Ωx ⊂ n=1 supp Ωxn . According to Theorem 2.62 the map

Ω : C(K) → c0 Γ) is co-σ-continuous. Given a set Γ and H ⊂ Γ the symbol 1lH denotes the characteristic function of H, i.e. 1lH (γ) = 1 if γ ∈ H and 1lH (γ) = 0 otherwise. Corollary 2.68. Let K be a compact subset of [0, 1]Γ such that for any s ∈ K and any H ⊂ Γ we have that s1lH ∈ K. Then given x ∈ C(K) the set supp Ωx controls x, Ω is co-σ-continuous and C(K) admits a pointwise lower semicontinuous LUR equivalent norm. In particular we obtain the well known fact that C [0, 1]Γ is LUR renormable for any set Γ, [Val90]. Proof of Corollary 2.68 Let x ∈ C(K), since x is uniformly continuous in K for every ε > 0 there exists a ﬁnite subset N ⊂ Γ and δ > 0 such that for every s, t ∈ K we have |x(s) − x(t)| < ε whenever |s(γ) − t(γ)| < δ for γ ∈ N .

(2.20)

Set N0 := N ∩ supp Ωx . According to Proposition 2.67 we shall have established the corollary if we prove that for every s, t ∈ [0, 1]Γ we have |x(s) − x(t)| < ε whenever |s(γ) − t(γ)| < δ for γ ∈ N0 .

(2.21)

44

2 σ-Continuous and Co-σ-continuous Maps

Indeed, put N \ N0 = {γ1 , γ2 , . . . , γn } . Now ﬁx s, t ∈ [0, 1]Γ such that |s(γ) − t(γ)| < δ, for all γ ∈ N0 . Take s0 := s, t0 := t and sk , tk , 1 ≤ k ≤ n, deﬁned by sk−1 (γ), γ ∈ Γ \ {γk } , tk−1 (γ), γ ∈ Γ \ {γk } , and tk (γ) = sk (γ) = 0, γ = γk ; 0, γ = γk . Our hypothesis on K gives that sk , tk ∈ K, 1 ≤ k ≤ n. According to the choice of N0 and (2.19) we have x(s) = x (s0 ) = x (s1 ) = . . . = x (sn ) ,

(2.22)

x(t) = x (t0 ) = x (t1 ) = . . . = x (tn ) .

(2.23)

On the other hand, according to the choice of sn we have |sn (γ) − tn (γ)| < δ, for all γ ∈ N . Thus from (2.20) it follows that |x (sn ) − x (tn )| < ε. The last inequality, (2.22) and (2.23) prove (2.21).

Corollary 2.69. [Alex82] Let G be a compact topological group. Then C(G) is LUR renormable. Proof. According to the Ivanovski˘ı-Vilenkin-Kuzminov theorem (see e.g. [Tod97, p. 81]) every compact topological group G is the continuous image of a Cantor cube [0, 1]Γ . So C(G) is a linear subspace of C [0, 1]Γ and the assertion follows from Corollary 2.68.

Despite of Corollary 2.68 there are compacta K for which C(K) admits a LUR renorming and we can ﬁnd some x ∈ C(K), x = 0, such that supp Ωx = ∅ so supp Ωx does not control x. As we are going to see in some these compacta the role of the set Q in Proposition 2.67 is essential to get a LUR renorming. Let us recall that the Helly space is the subspace H of [0, 1][0,1] consisting of all nondecreasing functions t : [0, 1] → [0, 1] endowed with the pointwise topology see e.g. [Kel55, p 164]. Then for H we can take [0, 1] as the set Γ so it makes sense to write Ωx(γ) for γ ∈ [0, 1]. There exist functions x ∈ C(H) which are not controlled by supp Ωx, as it is shown in the following Example 2.70. Let x : H → R be the real function deﬁned by the formula 1 t(γ)dγ . x(t) = 0

It is obvious that x ∈ C(H) and Ωx(γ) = 0 for every γ ∈ [0, 1]. Therefore supp Ωx does not control x.

2.7 Co-σ-continuous Maps in C(K)

45

The next lemma illustrates how the set supp Ωx can be enlarged to control x. Given x ∈ C(H) and η > 0 will denote by O(x, η) = {γ ∈ [0, 1] : Ωx(γ) > η} . Lemma 2.71. Given ε > 0 and x ∈ C(H) there exist η > 0 and L1 , . . . , Lk , pairwise disjoint open subintervals of [0, 1] such that j Lj ∩O(x, η) = ∅ and for any choice of points γj ∈ Lj , 1 ≤ j ≤ k, the set O(x, η) ∪ {γj : 1 ≤ j ≤ k} ε-controls x, i.e. there exists δ > 0 such that |x(s) − x(t)| < ε whenever |s(γ) − t(γ)| < δ for any γ ∈ O(x, η) ∪ {γj : 1 ≤ j ≤ k}. Before proving it let us see some consequences: As a result of the above lemma we have that if x is as in Example 2.70 everyone of the sets Q∩[0, 1] and (R\Q)∩[0, 1] controls x therefore, in general, we cannot associate to each x ∈ C(K) a canonical set which controls x even when C(K) is LUR-renormable. From Lemma 2.71 and Proposition 2.67 we get that Ω : C(H) → c0 ([0, 1]) is co-σ-continuous. This, Corollary 2.66 and Theorem 1.15 imply Theorem 2.72. If H is the Helly space then C(H) admits a pointwise lower semicontinuous LUR equivalent norm. Let us mention that the problem of the LUR-renormability of C(H) was imposed by M. I. Kadec in the middle of the 70’s and after that it was repeated many times. Starting at this time it was eventually proved that C(K) is LUR renormable if K is either a section of the ordinals, or a compact group, or the cube [0, 1]Γ , the two-arrows space, some scattered compacts, Eberlein compacts and its generalizations. In these cases this was a consequence of either the projective resolution of identity or of the three space property for LUR renorming. Most of these cases can be found in [DGZ93]. In the last few years the LUR renormability of C(K) when K is a tree or a totally ordered compact has been characterized [Hay99], [HJNR00]. Recently using topological games Kortezov [Kor00] proved that C(H) is σ-fragmentable (see [Bou93] too). Unfortunately his method of topological games gives only σ-fragmentability and will not give even that the identity map Id : (C(H), weak) → (C(H), · ∞ ) is σ-continuous. For γ ∈ [0, 1] and x ∈ C(H) set ˜ Ωx(γ) := inf sup x(t) − x(s) : (t − s)1l[0,1]\[γ−µ,γ+µ] ≡ 0, t, s ∈ H . µ>0

˜ Lemma 2.73. For all x ∈ C(H) and γ ∈ [0, 1] we have Ωx(γ) = Ωx(γ). ∞ ∞ ˜ Proof. Clearly Ωx(γ) ≥ Ωx(γ). On the other hand let {tn }n=1 , {sn }n=1 , ∞ {µn }n=1 be such that

46

2 σ-Continuous and Co-σ-continuous Maps

µn 0 , (tn − sn ) 1l[0,1]\[γ−µn ,γ+µn ] ≡ 0 ,

(2.24)

˜ and x (tn ) − x (sn ) → Ωx(γ) . ∞

∞

Without loss of generality we can assume that {tn }n=1 and {sn }n=1 are pointwise convergent. Let t and s be its limits. From (2.24) it follows ˜ ˜ consequently Ωx(γ) ≤ Ωx(γ).

(t−s)1l[0,1]\{γ} ≡ 0, and x(t)−x(s) = Ωx(γ) Proof of Lemma 2.71 Given ε > 0 let n ∈ N and a ﬁnite set R0 ⊂ [0, 1] such that for every s, t ∈ H we have |x(t) − x(s)| < ε/2 whenever sup |t(γ) − s(γ)| < 1/n .

(2.25)

γ∈R0

Let R1 := R0 ∩

{O(x, 1/j) : j ∈ N} . Since R1 is ﬁnite there exists η > 0

such that

Let R2 := R0 \

R1 ⊂ O(x, η) .

(2.26)

{O(x, 1/j) : j ∈ N} . Set := #R2 ∈ N. Now it will be

useful the following Claim. For any γ0 ∈ R2 there exists an open interval I0 containing γ0 such that for any β ∈ I0 we have O(x, η) ∪ (R2 \ {γ0 }) ∩ I0 = ∅ and |x(t) − x(s)| <

ε ε + 2 2

whenever

sup γ∈O(x,η)∪R2 \{γ0 }

|t(γ) − s(γ)| <

1 , n

|t(β) − s(β)| <

1 . n

Proof. From Lemma 2.73 it follows that there exists a µ > 0 such that ε sup |x(t) − x(s)| : (t − s)1l[0,1]\[γ0 −µ,γ0 +µ] ≡ 0 , t, s ∈ H < . (2.27) 4 Let I0 , γ0 ∈ I0 ⊂ [γ0 − µ, γ0 + µ], be open an interval with rational end-points such that I0 ∩ (O(x, η) ∪ R2 \ {γ0 }) = ∅. Now ﬁx a point β ∈ I0 and let t, s ∈ H for which sup γ∈O(x,η)∪R2 \{γ0 }

|t(γ) − s(γ)| <

1 1 , and |t(β) − s(β)| < . n n

Take t0 := t1l[0,1]\I0 + t(β)1lI0 , s0 := s1l[0,1]\I0 + s(β)1lI0 . Let us note that t0 , s0 ∈ H. Moreover from (2.27) it follows that

2.7 Co-σ-continuous Maps in C(K)

|x(t) − x(t0 )| <

ε ε and |x(s) − x(s0 )| < . 4 4

47

(2.28)

On the other hand from the choice of x0 and y0 we get sup γ∈O(x,η)∪R2 \{γ0 }

|t0 (γ) − s0 (γ)| <

1 , n

(2.29)

|t0 (γ0 ) − s0 (γ0 )| = |t(β) − s(β)| <

1 . n

Now from (2.25), (2.26) and (2.29) we have |x (t0 ) − x (s0 )| <

ε . 2

(2.30)

Then from (2.28) and (2.30) |x(t) − x(s)| ≤ |x(t) − x(t0 )| + |x(t0 ) − x(s0 )| + |x(s0 ) − x(s)| < This proves the claim.

ε ε + . 2 2

To ﬁnish the proof of Lemma 2.71 it is enough to repeat the argument of the above claim.

3 Generalized Metric Spaces and Locally Uniformly Rotund Renormings

3.1 Discreteness and Network Conditions A class of generalized metric spaces is a class of spaces deﬁned by a property shared by all metric spaces which is close to metrizability in some sense [Gru84]. The σ-spaces are deﬁned by replacing the base by network in the Bing-Nagata-Smirnov metrization theorem; i.e. a topological space is a σ-space if it has a σ-discrete network. Here we shall deal with a further reﬁnement replacing discrete by isolated or slicely isolated. Indeed we will see that the identity map from a subset A of a normed space is σ-slicely continuous if, and only if, the weak topology relative to A has a σ-slicely isolated network. If A is also a radial set then we have that the identity map Id : (X, weak) → (X, · ) is σ-slicely continuous and Theorem 1.1 in the Introduction says that this is the case if, and only if, X has an equivalent LUR norm. After our study of this class of maps we can now formulate the following theorem and its corollaries summarizing diﬀerent characterizations of LUR renormability for a Banach space. Theorem 3.1. Let X be a normed space and let F be a norming subspace of its dual. The following assertions are equivalent: i) X admits an equivalent σ(X, F )-lower semicontinuous LUR norm. ii) The identity map Id : (X, σ(X, F )) → (X, · ) is σ-slicely continuous. iii) The identity map above restricted to some radial set A in X is σ-slicely continuous. iv) There are a radial set A in X, a metric space Y and a map Φ : (A, σ(X, F )) → Y which is σ-slicely continuous and co-σ-continuous. v) Every discrete family of subsets of X for the norm topology is σ-slicely isolatedly decomposable in σ(X, F ).

∞ vi) The norm topology of X admits a network N = n=1 Nn where every Nn veriﬁes σ(X,F )

A ∩ conv ({B : B ∈ Nn , B = A})

= ∅ for every A ∈ Nn .

A. Molt´ o et al., A Nonlinear Transfer Technique for Renorming. Lecture Notes in Mathematics 1951, c Springer-Verlag Berlin Heidelberg 2009

49

50

3 Generalized Metric Spaces and Locally Uniformly Rotund Renormings

vii)There is a sequence {An : n ∈ N} of subsets of X such that the family {An ∩ H : H is a σ(X, F )-open half space, n ∈ N} is a network for the norm topology viii)The identity map Id : (X, σ(X, F )) → (X, · ) is σ-continuous and there is a sequence {An : n ∈ N} of subsets of X such that the family {An ∩ H : H is a σ(X, F )-open half space, n ∈ N} is a network for the σ(X, F )-topology Proof. i)⇐⇒ii) It is just Theorem 1.1 in the introduction. ii)⇐⇒iii) It follows from Proposition 2.46. iii)⇐⇒iv) This is because the proof of Theorem 1.15 when X is replaced by A works. ii)⇐⇒v) It follows from Proposition 2.22. ii)⇐⇒vi) From Proposition 2.24 we know that (ii) is equivalent to the existence of a network for the norm topology which is σ-slicely isolated in σ(X, F ). The theorem of Hahn-Banach asserts that a family of subsets {Dγ : γ ∈ Γ} in X is σ(X, F )-slicely isolated if, and only if, σ(X,F )

Dγ ∩ conv ({Dβ : β ∈ Γ, β = γ})

= ∅ , for all γ ∈ Γ

and the equivalence follows. ii)⇐⇒vii) follows from Proposition 2.39. viii)⇐⇒vii) follows from Corollary 2.43 since the norm topology is ﬁner than σ(X, F ).

When F = X ∗ we can state the following Corollary 3.2. A normed space X admits an equivalent LUR norm if, and only if, the weak topology has a σ-slicely isolated network. Proof. According to Corollary 2.18 if the weak topology has a σ-isolated network

then the identity map Id : (X, weak) → (X, · ) is σ-continuous. If ∞ N = =1 Nn is such a network where every Nn is even slicely isolated then we can take the sets An := {N : N ∈ Nn } , n ∈ N and it follows that the family of sets {An ∩ H : H is a weak-open half space, n ∈ N} is a network for the weak topology too. To ﬁnish the proof it is enough to remember the equivalence (i)⇐⇒(viii) in the former theorem.

3.1 Discreteness and Network Conditions

51

The same proof works for a radial subset A of X due to the equivalence (iii) in the previous theorem, so we have Corollary 3.3. A normed space X admits an equivalent LUR norm if, and only if, the weak topology restricted to some radial subset of X has a σ-slicely isolated network. Remark 3.4. When we have a weakly LUR norm on a normed space X; i.e. a norm such that whenever x and xn are elements of the unit sphere SX such that (xn + x) /2 → 1 implies that (xn ) converges weakly to x, we proved [MOTV99] that on the unit sphere there exists a σ-slicely isolated network for the weak topology and so an equivalent LUR norm on X. See Sect. 3.3 for more information on this problem. If every point of the unit sphere of a normed space is a denting point for the unit ball then the weak and the norm topologies coincide on the unit sphere and so the weak topology has a σ-discrete basis in the unit sphere which is going to be a σ-slicely discrete network too. The conclusion that such normed space should be LUR renormable is a result of the third named author in [Tro85]. A situation in which the condition {An ∩ H : H is a weak open half space, n ∈ N} is a network of the weak topology restricted to the unit sphere holds for An = SX , n ∈ N, is when the slices containing a given point x ∈ SX are a basis of neighbourhoods for the relative weak topology. Choquet’s lemma [Cho69] asserts that this happens when every x ∈ SX is an extreme point of the bidual unit ball , which turns out to be equivalent to the so called weakly MLUR property for the given norm [KR82]. Therefore we have (see [MOT97]): Corollary 3.5. A normed space admits an equivalent LUR norm if, and only if, the identity map from (X, weak) into X (or from (SX , weak) into SX ) is σ-continuous and it admits an equivalent weakly MLUR norm. Another interesting case is the next one, where given a norming subspace F ⊂ X ∗ we look for new σ(X, F )-lower semicontinuous LUR renormings. Corollary 3.6. Let X be a normed space and let F be a norming subspace in X ∗ . If X is LUR renormable and for a σ(X, F )-lower semicontinuous norm | · | in X we have that the topologies weak and σ(X, F ) coincide in the unit sphere SX = {x ∈ X : |x| = 1} then X admits an equivalent σ(X, F )-lower semicontinuous LUR norm. Proof. The identity map Id : (SX , weak) → SX is σ-slicely continuous, let us see that it is also σ-slicely continuous for σ(X, F ). Indeed take L := {x ∈ X : g(x) > µ} for g ∈ X ∗ , µ ∈ R and x0 ∈ SX ∩ L. We claim that σ(X,F )

x0 ∈ / BX \ L

. If this is not the case there must exist a net {xβ }β with

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3 Generalized Metric Spaces and Locally Uniformly Rotund Renormings

|xβ | ≤ 1 = |x0 | and g (xβ ) ≤ µ for every β such that σ(X, F ) − limβ xβ = x0 . Since | · | is σ(X, F )-lower semicontinuous we have 1 = |x0 | ≥ lim sup |xβ | ≥ lim inf |xβ | ≥ |x0 | β

β

so limβ |xβ | = |x0 |. Our assumption on SX gives that weak − lim xβ = x0 β

which contradicts that x0 ∈ L and xβ ∈ X \ L for all β. The theorem of HahnBanach in the duality < X, F > tells us that there exists f ∈ F separating σ(X,F )

BX \ L from x0 so x0 ∈ {x ∈ X : f (x) > ξ} ∩ BX ⊂ L. So for every weak open half space L with x0 ∈ SX ∩ L we ﬁnd a σ(X, F )-open half space H with x0 ∈ H ∩ BX ⊂ L and consequently the identity map Id : (SX , σ(X, F )) → X will be σ-slicely continuous too and the conclusion follows from Theorem 3.1-(iii).

For instance we have the following result of M. Raja [Raj02]. Corollary 3.7. If X is a Banach space such that the weak∗ and the weak topologies coincide on the dual unit sphere then X ∗ has an equivalent dual LUR norm. Proof. By [NP75] X ∗ has the Radon-Nikodym property and by the result of

M. Fabian and G. Godefroy [FG88] X ∗ has a LUR equivalent norm. Another corollary is obtained playing with homogeneity; to formulate it we need the following Deﬁnition 3.8. A function Φ from a locally convex linear topological space X into a metric space (Y, ) is said

∞ to be ε-σ-slicely continuous for a ﬁxed ε > 0 if we can decompose X = n=1 Xn in such a way that for every n ∈ N and every x ∈ Xn there is an open half space H in X with x ∈ H and such that osc (ΦH∩Xn ) < ε. Then we have Corollary 3.9. Given a normed space X and a norming subspace F in X ∗ we have that X admits an equivalent σ(X, F )-lower semicontinuous LUR norm if, and only if, the identity map Id : (X, σ(X, F )) → (X, · ) is ε-σ-slicely continuous for some ε > 0. Proof. We shall prove that if the identity map is ε-σ-slicely continuous in X for a ﬁxed ε > 0 then it is σ-slicely continuous restricted to the unit sphere of X so the conclusion will follow from Theorem 3.1–iii).

∞Indeed let Xn be as in the deﬁnition above, for η > 0 we can split SX = n=1 Sn in such a way that x ∈ Sn if, and only if, (ε/η)x ∈ Xn . Then for such x ∈ Sn there exists fx ∈ F and δx > 0 such that (ε/η)x − (ε/η)y < ε whenever y ∈ Sn and fx (x − y) < δx η/ε.

3.2 Fragmentability Conditions

53

Corollary 3.10. For a normed space X and a norming subspace F in X ∗ we have that X admits an equivalent σ(X, F )-lower semicontinuous LUR norm if, and only if, there exists ε > 0 and Φ : X → X ε-σ-slicely continuous for σ(X, F ) such that x − Φx < ε for all x ∈ X. Proof. The identity map Id : (X, σ(X, F )) → (X, · ) is 3ε-σ-slicely continuous.

Corollary 3.11. A normed space X is LUR renormable if, and only if, there exists a metric d on X generating a topology ﬁner than the weak topology such that the identity map Id : (X, weak ) → (X, d) is σ-slicely continuous. Proof. It follows from Theorem 1.15 and Corollary 2.36.

We can also give an alternative proof for our transfer result in terms of maps. Another proof of Theorem 1.15. Let Φ : X → (Y, ) be co-σ-continuous and σ-slicely continuous for σ(X, F ). According to Proposition 2.30 the ﬁbers of Φ are separable and every selector for Φ−1 is σ-continuous. For every x ∈ X we choose a sequence {ynx : n ∈ N} dense in Φ−1 (Φx) for the norm topology. Let Ξn : ΦX → X be the selector choosing ynx in the ﬁber of x. Then Ξn is σ-continuous for every n ∈ N. If we consider the composition maps {Ξn ◦ Φ : X → X : n ∈ N} they are σ-slicely continuous for σ(X, F ) according to Propositions 2.39 and 2.40. For every x ∈ X we have ·

x ∈ {Ξn ◦ Φx : n ∈ N} . Then according to Proposition 2.23 the identity map Id : (X, σ(X, F )) → X is σ-slicely continuous.

3.2 Fragmentability Conditions Following J. Jayne, I. Namioka and C. A. Rogers [JNR93] we say that a topological space (X, T ) is σ-fragmented by a metric d on the set X if, for each

∞ ε > 0 we can write X = n=1 Xn,ε in such a way that for each n an each non empty subset C ⊂ Xn,ε there exists a T -open set V with V ∩ C = ∅ and d-diam(V ∩ C) < ε. Using our terminology for maps that means that the identity map from (X, T ) into (X, d) is σ-fragmented (see Deﬁnition 2.11). On the other hand if every non empty subset of X contains non empty relatively T -open subsets of arbitrarily small d-diameter then (X, d) is said to be fragmented by the metric d; i.e. for every ε > 0 and every non empty subset C of X there exists a T -open set V with V ∩ C = ∅ and d-diam(V ∩ C) < ε. This term is due to J. Jayne and C. A. Rogers in connection with their work on Borel selectors for set valued maps taking values in subsets of certain Banach spaces [JR85]. Let us remark that for hereditarily Baire spaces (X, T ), it is

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3 Generalized Metric Spaces and Locally Uniformly Rotund Renormings

equivalent that (X, T ) is fragmented by the metric d to say that the identity map from (C, T ) into (C, d) has a point of continuity for every closed subset C of X; i.e. the identity map from (X, T ) into (X, d) is barely continuous (see Deﬁnition 2.9) [Nam87]. A very useful internal characterization of the topological spaces (X, T ) for which there exists a metric d on it fragmenting (X, T ) was given by N. Ribarska [Rib87] and by P. Kenderov and W. Moors [KM96], see also [KM99]. The σ-fragmented spaces are related with ˇ descriptive topology through Cech-analytic spaces. Following D. Fremlin (see ˇ [JNR93]) a completely regular topological space (X, T ) is said to be Cechanalytic if it is a Souslin set in some (equivalently every) compactiﬁcation of X, i.e. it is obtained through the Souslin operation on Borel sets in some comˇ pactiﬁcation of X. An important result in [JNR93] is that for a Cech-analytic space (X, T ) and a lower semicontinuous metric d on it, the space (X, T ) is σ-fragmented by d whenever every compact subset of (X, T ) is fragmented by d. They mainly apply their results to Banach spaces with the weak topology and the norm metric calling them σ-fragmentable Banach spaces. As weakly compact subsets of Banach spaces are fragmented by the norm metric [Nam74] ˇ they show that every weakly Cech-analytic Banach space is σ-fragmentable. In particular, Banach spaces which are Borel subsets of the bidual with the weak∗ topology are σ-fragmentable. Spaces of this kind are the Banach spaces X which have an equivalent Kadec norm, i.e. a norm such that the weak and the norm topologies coincide on the unit sphere according to a previous result of G. A. Edgard and W. Schachermayer [Edg77&79]. Summarizing we have ˇ LUR norm =⇒ Kadec norm =⇒ weakly Cech-analytic =⇒ σ-fragmentable Let us remark that no example of σ-fragmentable Banach space without an equivalent Kadec norm is known at the present time and so in the last years a lot of research has been done along these classes of spaces penetrating more on the connections between the weak and the norm topologies [DJP97], [DJP], [JNR92] and [Han01]. Indeed the ﬁrst example of a Banach space with a Kadec norm and without en equivalent LUR norm is due to R. Haydon in the remarkable paper [Hay90] where it is proved that for trees Υ, the Banach space C0 (Υ) has a Kadec norm if, and only if, it is σ-fragmentable, and moreover it is shown that there are trees Υ such that C0 (Υ) has no equivalent LUR norm despite C0 (Υ) is σ-fragmentable. More examples recently appear in [HJNR00]. In our paper [MOTV00] we prove that in spaces with RNP to have and equivalent Kadec norm is equivalent to have an equivalent LUR norm. Two equivalent approaches have been considered dealing with the problem of the reverse implications between σ-fragmentability and Kadec renormings. Jayne, Namioka and Rogers say that a Banach space X has a countable cover by

∞sets of small local diameter whenever for every ε > 0 we can write X = n=1 Xn,ε and for every n ∈ N and every x ∈ Xn,ε a weak-open set U can be found such that · -diam(U ∩ Xn,ε ) < ε, [JNR92]. In our terminology that means that the identity map from (X, weak) into X is σ-continuous. On the other hand according to the R. Hansell concepts [Han01], a Banach space

3.2 Fragmentability Conditions

55

is descriptive when there is a complete metric space W and a continuous surjection Ξ : W → (X, weak) sending each discrete family {Dγ : γ ∈ Γ} in W into {ΞDγ : γ ∈ Γ} a σ-isolatedly decomposable family in (X, weak). Hansell showed that W can be chosen always equal to the Banach space itself with the norm-topology. As we pointed out in [MOTV99] a Banach space is descriptive if, and only if, it has a countable cover by sets of small local diameter (see the equivalence between (i) and (ii) in Proposition 2.2). It is proved in [Han01] that ˇ Kadec norm =⇒ descriptive =⇒ Cech-analytic and that descriptive ⇐⇒ σ − fragmentable and hereditarily weakly θ-reﬁnable for the weak topology (see Proposition 2.12). M. Raja has showed how the descriptive Banach spaces are characterized by the existence of a function ϕ : X → R which is positively homogeneous and lower semicontinuous for the weak topology with · ≤ ϕ(·) ≤ (1 + ε) · , and such that the weak and the norm topologies coincide on the pseudo-sphere {x ∈ X : ϕ(x) = 1} [Raj99]. In the dual case the σ-continuity of the identity map Id : (X ∗ , weak∗ ) → X ∗ is equivalent to the existence of an equivalent dual LUR norm on X ∗ [Raj02]. The ﬁrst examples of Banach spaces without the weakly θ-reﬁnability property for the weak topology has been given in [DJP97], [DJP] but these spaces are not σ-fragmentable. Our main transfer result also works for descriptive and σ-fragmentable Banach spaces. Indeed with the same proof as in Theorem 1.15 above we have the following results stated for one-to-one maps by L. Oncina [Onc00] taking (ii) of Theorem 2.28 as the deﬁnition of co-σ-continuity, and for mappings into c0 (Γ) by R. Hansell to transfer network conditions, [Han01]. Theorem 3.12. Let X be a normed space and let F be a subspace of its dual which separates points of X. The identity map Id : (X, σ(X, F )) → X is σ-continuous if, and only if, there exists a metric space (Y, ) and a map Φ : X → Y which is co-σ-continuous for the norm and σ-continuous for σ(X, F ). and Theorem 3.13. Let X be a normed space and let F be a subspace of its dual which separates points of X. The identity map Id : (X, σ(X, F )) → X is σ-fragmented if, and only if, there exists a metric space (Y, ) and a map Φ : X → Y which is co-σ-continuous and σ-fragmented for σ(X, F ).

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3 Generalized Metric Spaces and Locally Uniformly Rotund Renormings

Proof. Following the proof

of Theorem 1.15 if Φ : X → Y is co-σ-continuous ∞ apply the σ-fragmentability we have the splitting X = n=1 Xn,p,ε . Now we

∞ p so that for every of the map Φ for ﬁxed 1/p and we have X = m=1 Xm p there exists a σ(X, F )-open set V , m and every non empty subset C ⊂ Xm V ∩ C = ∅ and osc (ΦV ∩C ) < 1/p. Consequently for every non empty subset p we have that C ⊂ Xn,p,ε ∩ Xm · − diam(C ∩ V ) < ε since for x, y ∈ C∩V , (Φx, Φy) < 1/p < δ(x, n,

ε) and consequently x−y < p ε since x, y ∈ Xn,p,ε . So the decomposition X = {Xn,p,ε ∩ Xm : m, n, p ∈ N} works again in that case.

Corollary 3.14. A normed space X is σ-fragmentable if, and only if, there exists a metric space (Y, ) and a map Φ : X → Y which is co-σ-continuous and σ-fragmented for the weak topology. As we have seen in Corollary 2.36 a simple way to have a metric d on a normed space X for which the identity map from X into (X, d) is co-σ-continuous is to assume that the d-topology is ﬁner than the weak topology. Consequently we have Corollary 3.15. If X is a normed space with a metric d whose topology is ﬁner than the weak topology we have i) The identity map Id : (X, weak) → X is σ-continuous for the norm topology whenever it is for the d-topology. ii) The identity map Id : (X, weak) → X is σ-fragmented for the norm topology whenever it is for the d-topology. iii) X is σ-fragmentable whenever (X, weak) is fragmented by d. Remark 3.16. Fragmenting metrics ﬁner than the weak topology were considered by R. Hansell [Han01, Theorem 6.4.] and ii) above follows also from there. In [KM99] it is proved that a Banach space X is σ-fragmentable if, and only if, there is a fragmenting metric d on X for the weak topology such that the d-topology is ﬁner than the weak topology. P. Kenderov and W. Moors arrived to that with very interesting examples studied before. For instance they showed this fact for a Banach space X such that the dual norm in X ∗ has a sphere where the weak and the weak∗ topologies coincide, we know that this is equivalent to have an equivalent dual LUR norm on X ∗ [Raj02], then X itself is fragmented with a metric d ﬁner than the weak topology and X must be σ-fragmentable too. It has been an important question that remained open for a long time whether the metric they obtain can be reﬁned in such a way that the identity map Id : (X, weak) → (X, d) is σ-continuous (or σslicely continuous). Let us observe that if it is so the Banach space X would be descriptive (or LUR renormable) and if it is not it should be possible to have examples of σ-fragmentable non descriptive Banach spaces. A surprising

3.2 Fragmentability Conditions

57

recent result of R. Haydon asserts that the answer is yes for Banach spaces C(K) [Hay]. Indeed for a σ-discrete compact space K he has shown that C(K) has an equivalent LUR norm. The equivalence between the existence of a dual LUR renorming in C(K)∗ and the σ-discreteness of K was proved by M. Raja in [Raj02] and it goes back to R. Hansell [Han01] who showed that K is σ-discrete if, and only if, the dual C(K)∗ is dual descriptive (see Sect. 3.3). Recently Haydon has made an impressive progress on the matter showing that X is LUR renormable if X ∗ has a dual LUR norm. Even more, if K is a compact space of the Namioka-Phelps class, i.e. K is homeomorphic to a weak∗ compact subset of a dual space X ∗ with a dual LUR norm, or equivalently K has a lower semicontinuous metric such that the identity map from K into (K, ) is σ-continuous (see [Raj02] and [OR04]), then the Banach space C(K) is LUR renormable. Another interesting case is when the Banach space X has a Gˆ ateaux differentiable norm. As D. Preiss, I. Namioka and R. Phelps showed in [PNP90], solving a long standing open problem, the Banach space X is a weak Asplund too in that case. The analysis of the game played in the proof made by N. Ribarska lead her to see that the dual ball BX ∗ endowed with the weak∗ topology is fragmented by a metric d which generates a topology ﬁner than the weak∗ topology by compactness ([Rib92]). P. Kenderov and W. Moors then observed ateaux diﬀerentiable norm then the bidual that when a dual space X ∗ has a Gˆ unit ball is fragmented with a metric d which fragments (BX , weak) and the d-topology is again ﬁner than the weak topology so X itself is σ-fragmentable too [KM96]. It remains unknown if X has an equivalent LUR norm in that case too. Let us observe that once more it will be enough to construct the former metric d in such a way that the identity map from (SX , weak) into (SX , d) is σ-continuous instead of fragmented. In that case the open half spaces give a basis of neighbourhoods for the weak topology in the unit sphere due to the Gˆ ateaux diﬀerentiability of the dual space X ∗ .

∞ Example 3.17. Let K be a compact space with K = n=1 Kn where every Kn is closed and such that C (Kn ) admits an equivalent pointwise-lower semicontinuous LUR norm (resp. the identity map from (C (Kn ) , pointwise) into C (Kn ) is σ-continuous or σ-fragmented) for every n ∈ N. Then C(K) admits an equivalent pointwise-lower semicontinuous LUR norm (resp. the identity map from (C (K) , pointwise) into C(K) is σ-continuous or σ-fragmented). This example completely solves a question of Haydon in [Hay94] and it was studied before for σ-fragmentability of (C (K) , pointwise) by P. Kenderov and W. Moors [KM99] and for descriptiveness by L. Oncina [Onc00]. The same scheme works for all cases if we apply our transfer theorems 1.15, 3.12 and 3.13 for every one of the cases. Indeed if Y := c0 (N) C (Kn ) and ∞ T : C(K) → Y is deﬁned by T f = n−1 f Kn n=1 it is clear that T is a linear continuous map which is co-σ-continuous too by Corollary 3.30. If we consider in Y the pointwise convergence topology T in every one of the

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3 Generalized Metric Spaces and Locally Uniformly Rotund Renormings

factors it follows that Y admits a T -lower semicontinuous LUR norm (resp. identity from (Y, T ) into Y is σ-continuous or σ-fragmented) and since T is linear and pointwise to T -continuous it now follows that T is σ-slicely continuous from (C (K) , pointwise) into Y by Corollary 2.42 (resp. σ-continuous o σ-fragmented). The conclusion now follows from Theorem 1.15 (resp. 3.12 or 3.13).

3.3 LUR Conditions for Pointwise Convergence Topologies The relationship between weakly LUR and LUR renormability of a normed space X was completely solved with our Theorem 1.11 [MOTV99]. The main idea for proving it is to construct in the unit sphere of a weakly LUR norm a σ-slicely discrete network for the weak topology which turns out to be enough for the LUR renormability of X itself (see Corollary 3.3). We shall work here with topologies weaker than the weak topology, i.e. topologies of the form σ(X, F ) for some norming subspace F ⊂ X ∗ and we will see that given a σ(X, F ) LUR norm (see Deﬁnition 1.10) it is always possible to construct a metric on X such that the identity from (X, σ(X, F )) into (X, ) should be σ-slicely continuous. If we can connect the metric and the norm of X with co-σ-continuity we should have an equivalent σ(X, F )-lower semicontinuous LUR norm on X by our transfer theorem 1.15. Since the metric is always ﬁner than the σ(X, F )-topology, this should be the case for F = X ∗ by Corollary 2.36, and for σ(X, F ) when (X, σ(X, F )) is σ-fragmented by the norm by Theorem 2.14. In particular if a dual space X ∗ has the Radon-Nikodym property and a dual weak∗ LUR norm then X ∗ admits an equivalent dual LUR norm, see Corollary 3.24. Since our intention is to construct metrics from σ(X, F ) LUR norms it is not a surprise that we shall need arguments related with paracompactness. To formulate our main tool let us remember the following Deﬁnition 3.18. A function s : X × X → R+ is a symmetric on the set X if for each x, y ∈ X we have i) s(x, y) = 0 ⇒ x = y ii) s(x, y) = s(y, x) Since we do not have the triangle inequality the usual ε-balls Bs (x, ε) := {y ∈ X : s(x, y) < ε} do not necessarily form a base for a topology. They form a network for a topology T deﬁned in the following way: A subset U ⊂ X is T -open if, and only if, for each x ∈ U there exists ε > 0 with Bs (x, ε) ⊂ U . A topological space (X, T ) is said to be symmetrizable (see [Gru84, Sect. 9]) if there exists

3.3 LUR Conditions for Pointwise Convergence Topologies

59

a symmetric s on X satisfying the following condition: for U ⊂ X we have U ∈ T , if, and only if, for each x ∈ U there exists ε > 0 with Bs (x, ε) ⊂ U . (X, T ) is symmetrizable and ﬁrst countable if, and only if, {Bs (x, ε) : ε > 0} forms a (not necessarily open) neighbourhood base at x; in that case it is said that (X, T ) is semi-metrizable ([Gru84, Theorem 9.6]). We have the following Main Lemma 3.19. Let A be a symmetrizable subset of a locally convex linear topological space X and let s : A × A → R+ a symmetric for the topology induced on A by the weak topology. If for every x ∈ A and every ε > 0 there exists an open half-space H in X with x ∈ H and s-diam (H ∩ A) < ε then any cover of A with open half spaces has a σ-slicely-discrete reﬁnement. Proof. Let {Hγ : γ < µ} be a family of open half-spaces covering A, i.e. Hγ = {x ∈ X : fγ (x) > λγ }, γ < µ, for fγ continuous linear form on X and λγ ∈ R. Set Dγ = A ∩ Hγ and for every positive integer n let us consider the sets 1 Dγn = x ∈ Dγ : fγ (x) ≥ λγ + . n We have Dγ =

∞

Dγn for every γ < µ. Let us deﬁne for every γ < µ the sets

n=1

{Dβ : β < γ} and Mγn = Dγn \ {Dβ : β < γ} ,

then we also have Mγ = n Mγn for every γ < µ. We are going to show n that every family Mγ : γ < µ is σ-slicely-discretely decomposable and the assertion follows. We begin by proving that if x ∈ Mγn , y ∈ Mβn with γ = β then we have either |fγ (x − y)| ≥ 1/n or |fβ (x − y)| ≥ 1/n. Indeed, assume for instance that β < γ, since x ∈ / Dβ and y ∈ Mβn it follows that fβ (x) ≤ λβ and fβ (y) ≥ λβ + 1/n. So fβ (y) − fβ (x) ≥ 1/n. Mγ = Dγ \

Now for every x ∈ A let γ(x) be the ordinal for which x ∈ Mγ(x) . Since s is a symmetric for the topology induced on A by the weak topology we should have for every x ∈ A and every η > 0, a number δ(x, η) > 0 such that fγ(x) (x − y) < η whenever y ∈ A and s(x, y) < δ(x, η) . Let us deﬁne the sets 1 Sp (η) = x ∈ A : δ(x, η) > where p ∈ N , and η > 0 . p Thus for every η > 0 we have A =

∞

Sp (η), and for every p ∈ N we get

p=1

fγ(x) (x − y) < η and fγ(y) (x − y) < η whenever s(x, y) <

1 and x, y ∈ Sp (η) . p

(3.1)

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3 Generalized Metric Spaces and Locally Uniformly Rotund Renormings

We claim that for each n, p ∈ N the family n Mγ ∩ Sp (1/n) : γ < µ

(3.2)

is slicely-discrete in A. Indeed, from the choice of the sets Sp (1/n) it follows that γ(x) = γ(y) whenever s(x, y) < 1/p and x, y ∈

(3.3)

Mγn ∩ Sp (1/n) : γ < µ

since otherwise we would have fγ(x) (x − y) ≥ 1/n or fγ(y) (x − y) ≥ 1/n which contradicts (3.1) above. According to our hypothesis we can choose for every x ∈ A an open half-space H = {y ∈ X : f (y) > α} with x ∈ H and s-diam (H ∩ A) < 1/p. Let us see that H meets at most one set of the family (3.2). Indeed, suppose we have y ∈ H ∩ Mαn ∩ Sp (1/n) for some α < µ and z ∈ H ∩ Mβn ∩ Sp (1/n) for some β < µ . Then y, z ∈ H ∩ A and according to the choice of H we have s(y, z) < 1/p so from (3.3) it follows that β = γ(z) = γ(y) = α. Thus the family (3.2) is slicely-discrete. Since for every γ < µ we have Mγn ∩ Sp (1/n) Mγ = n,p

we conclude that the family {Mγ : γ < µ} is σ-slicely discretely decomposable and the lemma follows.

A natural situation in which the hypotheses of Lemma 3.19 are satisﬁed is given by the following Example 3.20. Let X be a normed space and let F be a 1-norming subspace of X ∗ such that X is a σ(X, F ) LUR space. If we deﬁne the function

x + y

x, y ∈ SX s(x, y) = 1 − 2 then we have a symmetric for the topology induced on SX by σ(X, F ), and for every ε > 0 and every x ∈ SX there exists a σ(X, F )-open half space H with x ∈ H and s-diam (H ∩ SX ) < ε. Proof. It is a simple consequence of the deﬁnition of σ(X, F ) LUR norm that s induces the topology σ(X, F ) in SX . Indeed since F is norming we get that · is σ(X, F )-lower semicontinuous, then given x ∈ SX the set

3.3 LUR Conditions for Pointwise Convergence Topologies

{y ∈ SX

: s(x, y) < ε} = y ∈ SX :

61

x + y

>1−ε

2

is open for the topology induced in SX by σ(X, F ). Moreover for any se∞ quence (xn )1 and x in SX we have that s (xn , x) → 0 if, and only if, (xn + x) /2 → 1 which implies xn → x in σ(X, F ). Thus the ε-balls for s are a base of neighborhoods at every point x ∈ SX for the σ(X, F )-topology and s symmetrizes the topology induced in SX by σ(X, F ), even it is semi-metrizable with s [Gru84, Sect. 9]. Moreover for x ∈ SX and ε > 0 if we choose f ∈ F ∩BX ∗ with f (x) > 1−ε then we ﬁx H = {y ∈ X : f (y) > 1 − ε}, which is σ(X, F )-open half space, and for every y, z ∈ H ∩ SX we have

y + z

≥f y+z >1−ε

2 2 and so s(y, z) = 1 − (y + z)/2 < ε and s-diam (H ∩ SX ) ≤ ε.

We are now ready to prove the main result of this section: Theorem 3.21. Let X be a normed space and let F be a 1-norming subspace of X ∗ such that X is a σ(X, F ) LUR space. Then there exists a metric on X, generating a topology which is ﬁner than σ(X, F ), such that the identity map Id : (X, σ(X, F )) → (X, ) is σ-slicely continuous. Proof. Example 3.20 shows that we can apply Lemma 3.19 to the unit sphere SX endowed with the topology σ(X, F ). So ﬁx ε ∈ (0, 1) and Dε , a family of slices of SX covering it, with Dε = {S (fγ , ε) : γ < µε } where fγ ∈ F ∩ SX ∗ , µε is an ordinal and S (fγ , ε) is denoting the slice of the unit sphere generated by f with width ε, i.e. S (fγ , ε) = {x ∈ SX : f (x) > 1 − ε}. There must exist a σ-slicely ∞ discrete reﬁnement Mε of Dε . If we take M1/n we obtain a σ-slicely n=2

discrete family which is also a network for the topology induced in SX by σ(X, F ). Indeed it is enough to observe the fact that the family {S (f, ε) : x ∈ S (f, ε) , f ∈ F ∩ SX ∗ , 0 < ε < 1} is a base of neighborhoods of x for the σ(X, F )-topology

induced in SX by the σ(X, F ) LUR condition on the norm and therefore 0<ε<1 Mε is a network too. It is not a restriction to assume that all the sets in our σ-slicely discrete network for σ(X, F ) are σ(X, F )-closed since otherwise we could take their σ(X, F )-closures and we still have a σ-slicely discrete network for σ(X, F ). Let

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3 Generalized Metric Spaces and Locally Uniformly Rotund Renormings

N = Nn such a network where every Nn is slicely-discrete family of σ(X, F )closed sets for σ(X, F ). Now for every n let us deﬁne for (x, y) ∈ SX × SX 0 if x and y are in the same element of Nn dn (x, y) := 1 otherwise. dn is a σ(X, F )-lower semicontinuous semi-metric on SE . We now deﬁne d(x, y) :=

∞ 1 d (x, y) , n n 2 n=1

∀(x, y) ∈ SX × SX

providing us a σ(X, F )-lower semicontinuous metric on the sphere SX generating a topology ﬁner than σ(X, F ) and such that the identity map from (SX , σ(X, F )) → (SX , d) is σ-slicely continuous. Indeed the family of ﬁnite intersections of sets in N is a basis for the d-topology which is σ-slicely discrete too and Proposition 2.24 ensures the conclusion. Let us consider now the metric x y , (x, y) = max d , x − y , for all x, y ∈ X \ {0} . x y Let us show that the identity map from (X \ {0}, σ(X, F )) into (X \ {0}, ) is σ-slicely continuous too. It follows using the same arguments as in Proposition 2.46. Indeed for every x ∈ X \ {0} set x/x ∈ SX and denote the map Φ from X \ {0} onto SX deﬁned in that way, i.e. Φx = x/x. If H denotes the family of all the σ(X, F )-open half spaces of X according to Proposition 2.44, Φ must have the property P (H, H). Since the identity map above in the unit sphere SX is σ-slicely continuous from Propositions 2.39 and 2.40 we have that Φ : (X \ {0}, σ(X, F )) → (SX , d) is σ-slicely continuous too. For every positive rational number q the map Φq x := qΦx is σ-slicely continuous from (X \ {0}, σ(X, F )) into (X \ {0}, ) since (Φq x, Φq y) = d (Φx, Φy) for every x, y ∈ X \ {0}. Finally let us observe that x = limq→x Φq x in the -distance for every x ∈ X \ {0} since (x, Φq x) = |x − q|. Consequently

x ∈ {Φq x : q ∈ Q+ } for every x ∈ X \ {0} then from Proposition 2.23 it follows that the identity map from (X \ {0}, σ(X, F )) into (X \ {0}, ) is σ-slicely continuous. To extend the result above to the whole space X we can assume without loss of generality that is bounded, ≤ 1, and deﬁne (0, x) = 1 for all x = 0, x ∈ X. Then the identity map from (X, σ(X, F )) into (X, ) is σ-slicely continuous too and the proof is complete.

Corollary 3.22. Let X be a normed space and let F be a 1-norming subspace of X ∗ such that X is a σ(X, F ) LUR space. Then there exists a σ-slicely continuous map Φ : X → c0 (Γ) which is one-to-one and such that Φ−1 is pointwise to σ(X, F ) continuous.

3.3 LUR Conditions for Pointwise Convergence Topologies

63

Proof. It is enough to remember that every metric space is homeomorphic to some subset of a c0 (Γ) space, for an adequate set Γ, endowed with the pointwise topology.

Corollary 3.23. If a normed space X has a weakly LUR norm then the identity map Id : (X, weak) → (X, ·) is σ-slicely continuous and consequently X is LUR renormable. Proof. Let be a metric generating a topology ﬁner than the weak topology and such that the identity map (X, weak) → (X, ) is σ-slicely continuous. Since the -topology is ﬁner than the weak topology we can apply Corollary 2.36 to get that the identity map from (X, ) to (X, · ) is σ-continuous. Then according to Propositions 2.39 and 2.40 the identity from (X, weak) to (X, · ) is σ-slicely continuous.

Corollary 3.24. If a dual Banach space X ∗ has a dual weak∗ LUR norm and the RNP then the identity map Id : (X ∗ , weak∗ ) → (X ∗ , · ) is σ-slicely continuous and consequently X ∗ has a dual LUR equivalent norm. Proof. The RNP allows us to modify the construction of the proof of Theorem 3.21 and to obtain the family Dε made

up by slices S (fγ , εγ ) of the unit sphere SX with · -diam (S (fγ , εγ ) \ ( {S (fβ , εβ ) : β < γ})) < ε and fγ ∈ BX for all γ < µε (instead of ﬁxing ε as the width of them). If we do it the network we obtain is a network for the · -topology on SX ∗ which is σ-slicely discrete in (SX ∗ , weak∗ ) so the identity map Id : (SX ∗ , weak∗ ) → (SX ∗ , · ) is σ-slicely continuous. The conclusion follows now applying Proposition 2.46.

Remark 3.25. i) M. Raja [Raj03] has recently obtained a converse of this result, showing that if X ∗ is a dual Banach space with a metric d such that the identity map Id : (X ∗ , weak∗ ) → (X, d) is σ-continuous, then there exists an equivalent norm in X ∗ which is weak∗ LUR. ii) We can obtain another proof of Corollary 3.24 if we argue as in Corollary 3.23 since in that case we also know that the identity map from (X ∗ , ) into (X ∗ , · ) is σ-continuous. The RNP property tells us that the identity from the weak∗ topology to · is σ-fragmented, so it is from to · since is ﬁner than the weak∗ topology. It is enough to apply now Theorem 2.14 to complete the proof. Indeed we can formulate the following Corollary 3.26. Let X be a normed space and F ⊂ X ∗ be a 1-norming subspace such that X is a σ(X, F ) LUR space. If (X, σ(X, F )) is σ-fragmented by the norm of X then the identity map Id : (X, σ(X, F )) → X is σ-slicely continuous. Consequently X has an equivalent σ(X, F )-lower semicontinuous LUR norm.

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3 Generalized Metric Spaces and Locally Uniformly Rotund Renormings

Proof. According to theorem 3.21 there is a metric on X generating a topology ﬁner than σ(X, F ) such that the identity map from (X, σ(X, F )) into (X, ) is σ-slicely continuous. Since (X, σ(X, F )) is σ-fragmented by the norm of X the identity map from (X, ) into (X, · ) is σ-continuous since (X, ) is also σ-fragmented by the norm of X (Theorem 2.14). Then the composition map from σ(X, F ) to · must be σ-slicely continuous by Corollary 2.42.

Remark 3.27. If X is a normed space, F ⊂ X ∗ is a norming subspace and K ⊂ X is a σ(X, F )-compact subset which is fragmented with the norm · of X ·

then spanK is σ(X, F )-fragmentable and it always has an equivalent LUR norm [CNO03]. These class of spaces has been recently studied in [CNO03] in connection with an open problem stated by M. Talagrand. In this space the existence of an equivalent σ(X, F )-lower semicontinuous LUR norm it is equivalent to the existence of a σ(X, F ) LUR norm.

3.4 From the Linear to the Nonlinear Transfer Technique Theorem 1.6 in the introduction can be formulated for topologies weaker than the weak topology as follows. Theorem 3.28. Let Y be a normed space with a σ(Y, G)-lower semicontinuous LUR equivalent norm for some norming subspace G ⊂ Y ∗ . Let X be a normed space and let F be a norming subspace of X ∗ . If T : (X, σ(X, F )) → (Y, σ(Y, G)) is a continuous linear map which is co-σ-continuous for the norm topology then X admits an equivalent σ(X, F )-lower semicontinuous LUR norm too. Proof. Since preimages of σ(Y, G)-open half spaces are σ(X, F )-open half spaces and the identity map Id : (Y, σ(Y, G)) → (Y, · ) is σ-slicely continuous we have that T is σ-slicely continuous from (X, σ(X, F )) into (Y, · ). The conclusion follows Theorem 1.15 in the introduction.

Applying Proposition 2.33 we obtain the following consequences. Corollary 3.29. Let Y be a LUR space and let T be a bounded linear operator ∞ from the normed space X into Y such that for every bounded sequence {xn }1 in X with T xn − T x → 0 we have that ∞ ·

x ∈ span {xn }1

∞ ·

(in particular, whenever x ∈ conv {xn }1 is LUR renormable.

, or weak-lim xn = x). Then X

3.4 From the Linear to the Nonlinear Transfer Technique

65

The conditions imposed in this corollary are very natural when we are dealing with C(K) spaces, K compact, since Grothendieck’s theorem [Die75, p.156] asserts that a bounded set L of C(K) is weakly compact if and only if L is compact in the topology of pointwise convergence on K. So from the previous corollary we obtain Corollary 3.30. Let Y be a LUR space and let T be a bounded linear operator ∞ from C(K) into Y such that for every bounded sequence {xn }1 in C(K) with ∞ T xn − T x → 0 the sequence {xn }1 pointwise converges to x. Then C(K) is LUR renormable. Let us remember that given a Banach space X a subset B ⊂ BX ∗ is called a James boundary of X if for every x ∈ X there is g ∈ B such that g(x) = x. For instance the set of the extreme points of the dual unit ball is a James boundary ([FHHMPZ01, p. 79]). It follows that any bounded sequence (xn ) in X such that g (xn ) converges to g(x) for every g ∈ B a James boundary of X is also weakly convergent (Theorem 3.60 in [FHHMPZ01]), therefore we also have Corollary 3.31. Let Y be a LUR space and T : X → Y be a bounded linear operator such that for every bounded sequence (xn ) in X with T xn − T x → 0 the sequence (xn ) converges to x pointwise on a James boundary B ⊂ BX ∗ for X. Then X is LUR renormable. Another interesting case is given by the following Corollary 3.32. Let T be bounded linear operator from the Banach space X into the LUR space Y such that T ∗ Y ∗ is norm dense in X ∗ . Then X is LUR renormable. Proof. Since T ∗ Y ∗ is norm dense in X ∗ it is well known that on bounded sets of X the weak topology coincides with the topology of pointwise convergence on the elements of T ∗ Y ∗ . Thus we have that T −1 : T BX → X is continuous from · Y to the weak topology and by Corollary 3.29 we get the result.

Let us observe that in the linear case is not a restriction to assume that the operator is one-to-one. Indeed we have the following Proposition 3.33. If X is a normed space and E is a closed separable subspace of X then the canonical quotient map Q : X → X/E is co-σ-continuous. Proof. For every x in X we set Zx := x + E which is a separable subset of X. If {xn : n ∈ N} is a sequence in X with lim Qxn = Qx then x ∈

· {Zxn : n ∈ N} and we can apply Theorem 2.32 to get the conclusion. Indeed the norm in the quotient space is Qx = inf{z : z ∈ x + E} = · −dist(x, E).

Then we have

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3 Generalized Metric Spaces and Locally Uniformly Rotund Renormings

Proposition 3.34. For normed spaces X and Y and a bounded linear operator T : X → Y the following are equivalent (i) T is co-σ-continuous; (ii)Ker T is separable and T˜ : X/Ker T → Y is co-σ-continuous; where T˜ is the map deﬁned by T˜ ◦ Q = T and Q : X → X/Ker T is the canonical map. Proof. (i)⇒(ii) If follows from Proposition 2.30. Indeed since the ﬁbers of T are separable we have that Ker T is a closed separable subspace of X. Moreover any selector of T −1 is σ-continuous and so is the composition with Q by Corollary 2.41. It follows that T˜−1 : T X → X/Ker T is σ-continuous and so T˜ is co-σ-continuous. (ii)⇒(i) It follows form the fact that the composition of co-σ-continuous maps between metric spaces is co-σ-continuous too. (It is enough to apply, for instance, Theorem 2.28.(ii).)

When the range space Y is only σ(Y, G) LUR renormable we can require the gluing separability property characterizing co-σ-continuous maps (Theorem 2.32 and Proposition 2.33) for the weaker topology involved, and thanks to the metric constructed in Theorem 3.21 we can formulate the following Theorem 3.35. Let Y be a normed space with a σ(Y, G) LUR norm which is σ(Y, G)-lower semicontinuous where G ⊂ Y ∗ . Let X be a normed space and let F ⊂ X ∗ a norming subspace of it. If T : (X, σ(X, F )) → (Y, σ(Y, G)) is a linear continuous operator such that for every x ∈ X there exists a separable subspace Zx of X with ·

x ∈ span {Zxn : n ∈ N} ∞

whenever σ(Y, G) − lim T xn = T x and {xn }1 is a bounded sequence in X then X admits an equivalent σ(X, F )-lower semicontinuous LUR norm. Proof. Let be the metric on Y provided by Theorem 3.21 so that the identity map from (Y, σ(Y, G)) into (Y, ) is σ-slicely continuous. Since preimages of σ(Y, G)-open half spaces are σ(X, F )-open half spaces we have that T from (X, σ(X, F )) into (Y, ) is σ-slicely continuous too. If we remind that the metric gives a topology ﬁner than σ(Y, G), the conditions imposed on T implies that T is co-σ-continuous by Proposition 2.33 and the fact that X itself is σ-bounded. Our transfer theorem 1.15 ﬁnishes the proof.

Let us describe now an application of this theorem in weakly countably determined Banach spaces. We shall need some preparatory work on a space studied by S. Mercourakis. The next assertion is an useful tool for renorming.

3.4 From the Linear to the Nonlinear Transfer Technique

67

∞

Lemma 3.36 (The convex arguments). Let { · n }1 be a sequence of ∞ semi-norms in a linear space X such that the sequence {xn }1 is bounded for all x ∈ X. Then 1/2 ∞ −n 2 2 · n ·= 1

is also a semi-norm in X and for all x,y ∈ X and n ∈ N we have 2 (xn − yn ) ≤ 2 x2n +y2n −x+y2n ≤ 2n 2 x2 + y2 −x + y2 . Moreover we have lim 2 xk 2n + yk 2n − xk + yk 2n = 0 , lim (xk n − yk n ) = 0 (3.4) k

k

whenever xk , yk ∈ X and lim 2 xk 2 + yk 2 − xk + yk 2 = 0 . k

(3.5)

Proof. For all x, y ∈ X and n ∈ N we have 2 2 0 ≤ (xn − yn ) = 2 x2n + y2n − (xn + yn ) ≤ ∞ 2−k 2 x2k + y2k − x + y2k = ≤ 2 x2n + y2n − x + y2n ≤ 2n 1

= 2n 2 x2 + y2 − x + y2 . Now it is straightforward to deduce (3.4) from (3.5).

It is well known that the classical Day’s norm on c0 (Γ) is a lattice LUR norm (see e.g. [Die75, p. 94]). We give here a new proof of this fact which uses more convex arguments instead of combinatorial ones. Another lattice LUR in c0 (Γ) can be found in [DGZ93, p. 282] using Deville’s master lemma [DGZ93, p. 279] . The following notation will be used only until the end of the section. For x ∈ ∞ (Γ) and n ∈ N set ⎧⎛ ⎫ ⎞1/2 ⎪ ⎪ ⎨ ⎬ x2 (γ)⎠ : N ⊂ Γ , #N = n . xn = sup ⎝ ⎪ ⎪ ⎩ γ∈N ⎭ We introduce an equivalent lattice norm on ∞ (Γ) by the formula

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3 Generalized Metric Spaces and Locally Uniformly Rotund Renormings

· Day =

∞

1/2 −n

2

·

2n

.

1

It is easy to see that · Day is just the norm of Day, i.e. xDay = sup

∞

1/2 −n+1 2

2

x (γn )

,

(3.6)

1 ∞

where the supremum is taken over all the sequences {γn }1 of distinct elements of Γ. For x ∈ ∞ (Γ) and ε > 0 set Lε (|x|) = {γ ∈ Γ : |x(γ)| ≥ ε} , η(x, ε) = ε2 − sup x2 (γ) : γ ∈ / Lε (|x|) . ε Lemma 3.37. Let x ∈ ∞ (Γ) and ε > 0 be such that is a ﬁnite L = L (|x|) 2 % 2 ε − ε − η , η/2 . or empty set. Let η = η(x, ε) > 0, = #L, δ = min

Let y ∈ ∞ (Γ) and 2 x2Day + y2Day − x + y2Day < 2− −1 δ .

(3.7)

Then x − y∞ < 3ε. Proof. From the convex arguments (Lemma 3.36) and (3.7) we get for all n∈N 2 (xn − yn ) ≤ 2 x2n + y2n − x + y2n < 2n− −1 δ . (3.8) ε We consider two cases. First suppose L = ∅. We show % that L (|y|) √ = ∅ 2 too. Indeed otherwise we would have y1 − x1 ≥ ε − ε − η ≥ δ which contradicts (3.8). So Lε (|y|) = ∅ and in this case we have x − y∞ ≤ x∞ + y∞ < 2ε.

Assume now that L = ∅, i.e. ≥ 1. Pick M ⊂ Γ, #M = , µ ∈ Γ \ M such that 2 2 x+y2 ≤ (x(γ) + y(γ)) +δ, x+y +1 ≤ (x(γ) + y(γ)) +δ . γ∈M ∪{µ}

γ∈M

(3.9) First we show that M =L.

(3.10)

Assume the contrary. Since #L = #M = we can ﬁnd α ∈ L \ M and β ∈ M \ L. From the choice of δ we get x2 (α) − x2 (β) ≥ η ≥ 2δ .

(3.11)

3.4 From the Linear to the Nonlinear Transfer Technique

69

Set N = {α} ∪ M \ {β}. From (3.9) and (3.11) we get ⎛ ⎞ x2 (γ) + y 2 (γ) + δ ⎠ = x + y2 ≤ 2 ⎝ γ∈M

⎛ = 2⎝

⎞ x2 (γ) − x2 (α) − x2 (β) + y 2 (γ) + δ ⎠ ≤ 2 x2 + y2 − δ

γ∈N

γ∈M

which contradicts (3.8) so (3.10) is proved. Again from (3.9), (3.8) and (3.10) we get 2 2 x2 (γ) + y 2 (γ) − (x(γ) − y(γ)) = 2 (x(γ) + y(γ)) ≤ γ∈L

γ∈L

γ∈M

≤ 2 x2 + y2 − x + y2 + δ < 2δ ≤ η ≤ ε2 . Hence sup{|x(γ) − y(γ)| : γ ∈ L} < ε .

(3.12)

sup{|y(γ)| : γ ∈ / L} < 2ε .

(3.13)

Now we show that Assume the contrary. Since µ ∈ / L there exists λ ∈ / L such that |y(λ)| ≥ max{|y(µ)| , 2ε} .

(3.14)

Taking into account again that µ ∈ / L we have |y(λ)| − |x(µ)| > 2ε − ε = ε. This (3.9), (3.10) and (3.14) imply 2 x2 +1 + y2 +1 − x + y2 +1 ≥ ⎛

≥ 2⎝

x2 (γ) +

γ∈L∪{µ}

=

⎞ y 2 (γ)⎠ −

γ∈L∪{λ}

2

(x(γ) + y(γ)) − δ =

γ∈M ∪{µ}

(x(γ) − y(γ))2 + 2 x2 (µ) + y 2 (λ) − (x(µ) + y(µ))2 − δ ≥

γ∈L 2

≥ x2 (µ) + y 2 (λ) − 2|x(µ)y(λ)| − δ = (|y(λ)| − |x(µ)|) − δ > ε2 − δ > η − δ ≥ δ which contradicts (3.8) so (3.13) is proved. Since |x(γ)| < ε for γ ∈ / L from (3.12) and (3.13) we get x−y∞ < 3ε.

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3 Generalized Metric Spaces and Locally Uniformly Rotund Renormings

From the above lemma in a straightforward way it follows Proposition 3.38. There exists an equivalent lattice LUR norm on c0 (Γ). Deﬁnition 3.39. Let Λ be a set and (Z, ) a separable metric space. A function x ∈ ∞ (Z × Λ) belongs to c1 (Z × Λ) if for every compact subset K of Z the restriction of x to K × Λ belongs to c0 (K × Λ). The next lemma is from [DGZ93, p. 248]. Lemma 3.40. Let x ∈ c1 (Z × Λ), t0 = (z0 , λ0 ) ∈ Z × Λ, ε > 0. Then there exists an open neighbourhood V of z0 such that #Lε (|x1lV ×Λ |) < ∞. Proof. We show that there exists an open neighbourhood V such that x ((V \ {z0 }) × Λ) ⊂ (−ε, ε) .

(3.15)

Assume the contrary then there exists a sequence tn = (zn , λn ) ∈ Z×Λ, n ∈ N, with zm = zn for m = n, lim (zn , z0 ) = 0 and |x (tn )| ≥ ε. Since K = {zn }n≥0 is compact we get that xK×Λ ∈ c0 (K × Λ) which is a contradiction. From (3.15) we get Lε (|x1lV ×Λ |) = Lε x1l{z0 }×Λ . Since x1l{z0 }×Λ ∈

c0 ({z0 } × Λ) we get #Lε (|x1lV ×Λ |) < ∞. Corollary 3.41. If x ∈ c1 (Z × Λ) then supp x contains at most countable many points. Having in mind that every separable metric space is Lindel¨ of the proof follows directly from the former lemma. Proposition 3.42. [DGZ93, p. 287] The space c1 (Z ×Λ) admits an equivalent lattice pointwise LUR norm which is pointwise lower semicontinuous. ∞

Proof. Let {Vm }1 be a base of open sets of Z and let · Day be the norm in ∞ (Z × Λ) deﬁned in (3.6). For x ∈ ∞ (Z × Λ) we set |x| =

∞

1/2 −m

2

2 x1lVm ×Λ Day

.

(3.16)

1

Let x ∈ c1 (Z × Λ), xn ∈ ∞ (Z × Λ) and limk |(xk + x) /2| = |xk | = |x|. Then from the convex arguments (Lemma 3.36) we get for m ∈ N

(xk + x)

1lVm ×Λ = lim xk 1lVm ×Λ Day = x1lVm ×Λ Day . (3.17) lim

k k 2 Day We show that limk xk (t) = x(t) for all t ∈ Z ×Λ. Indeed pick t ∈ Z ×Λ and ε > 0. According to the previous lemma we can ﬁnd Vm such that t ∈ Vm × Λ and #Lε (|x1lVm ×Λ |) < ∞. Then #L2ε (|x1lVm ×Λ |) < ∞ and η (x1lVm ×Λ , 2ε) > 0. From Lemma 3.37 and (3.17) we get lim sup (xk − x) 1lVm ×Λ ∞ ≤ 6ε. Since k

t ∈ Vm × Λ we have lim supk |xk (t) − x(t)| ≤ 6ε.

3.4 From the Linear to the Nonlinear Transfer Technique

71

Theorem 3.43. Let Y be a weakly countably determined Banach space and let X be a subspace of Y ∗ such that for every x ∈ X there exists y ∈ BY with x(y) = x. Then X admits an equivalent σ(X, Y )-lower semicontinuous LUR norm. Proof. Since Y is weakly countably determined there exists a separable metric space Z, and index set Λ and a one-to-one bounded linear operator T : Y ∗ → c1 (Z × Λ) that is weak∗ to pointwise continuous (see [DGZ93, p. 251]). By Proposition 3.42 the space c1 (Z × Λ) has a pointwise lower semicontinuous and pointwise LUR equivalent norm and T X veriﬁes the conditions of Theorem 3.35 for (X, σ(X, F )) and c1 (Z × Λ) with the pointwise convergence topology. Indeed, given a sequence {xn } and x ∈ BX such that limn T xn (t) = T x(t) for all t ∈ Z × Λ, since T is an homeomorphism from (BY ∗ , weak∗ ) into T BY ∗ with the pointwise convergence topology by compactness, we will have that σ(X, Y )-limn xn = x, but BY is a James-boundary of X by assumption and we have σ (X, X ∗ )-limn xn = x too (Theorem 3.60 in [FHHMPZ01]), and so ·

x ∈ span {xn }

from where the conclusion follows.

A straightforward consequence is the result in [FT90]: Corollary 3.44. If a Banach space X has a dual space X ∗ weakly countably determined then X admits an equivalent LUR norm. Remark 3.45. In a dual space X ∗ which is weakly countably determined M. Fabian [Fab91] constructed a dual LUR norm. R. Haydon has recently proved that X is LUR renormable provided X ∗ is a dual LUR Banach space [Hay]. Let us ﬁnish this section with a nonlinear version of Theorem 1.12 in the introduction. We hope it motivates the reader for the characterizations we shall obtain in the next chapter for σ-slicely continuous maps in terms of subdiﬀerentiability properties. Theorem 3.46. Let Y be a subspace of ∞ c (Γ) with a pointwise lower semicontinuous and pointwise LUR norm on it, let X be a normed space with F ⊂ X ∗ a norming subspace for it and let Φ : X → Y a map between them. We assume that i) There is a sequence {An : n ∈ N} of subsets of X such that for every x ∈ X and every pointwise open half space L of Y with Φx ∈ L we have n ∈ N and a σ(X, F )-open half space H of X so that x ∈ H ∩ An ⊂ Φ−1 (L) .

72

3 Generalized Metric Spaces and Locally Uniformly Rotund Renormings

ii) There is a family {Xγ : γ ∈ Γ} of separable subspaces of X such that for every x ∈ X we have x ∈ span {Xγ : γ ∈ supp Φx}

·

.

Then X admits an equivalent σ(X, F )-lower semicontinuous LUR norm. Proof. According to Theorem 3.21 there exists a metric on Y generating a topology ﬁner than the pointwise convergence topology restricted to Y and such that the identity map Id : (Y, pointwise) → (Y, ) is σ-slicely continuous. The map Φ veriﬁes the property P (σ(X, F ) − open half space, pointwise-open half spaces) by our assumption and consequently Φ : (X, σ(X, F )) → (Y, ) is σ-slicely continuous by Corollary 2.42. Moreover for every x = 0 set ·

Zx := span {Xγ : γ ∈ supp Φx}

which is a separable subspace of X fulﬁlling the conditions of Proposition 2.33, then we conclude that Φ : X → (Y, ) is co-σ-continuous too. Indeed if ∞ ∞ (Φxn )n=1 is sequence converging to Φx in (Y, ) we have that (Φxn )n=1 converges to Φx in the pointwise topology too, and we have ∞

supp Φx ⊂

supp Φxn .

n=1

So {Xγ : γ ∈ supp Φx} ⊂

∞

{Xγ : γ ∈ supp Φxn }

n=1

and ﬁnally x ∈ Zx ⊂ span {Zxn : n ∈ N}

·

.

4 σ-Slicely Continuous Maps

4.1 Examples All examples of σ-slicely continuous maps are connected somehow with LUR Banach spaces. It is clear that if x is a denting point of a set D and Φ is a norm continuous map at x then Φ is slicely continuous at x. Hence if X is a LUR normed space then every norm continuous map Φ on BX is slicely continuous on SX . Toru´ nczyk [Tor81] proved that every Banach space X is norm homeomorphic to some Hilbert space H. When X has an equivalent LUR norm | · | and the previous homeomorphism h : X → H can be constructed so that h(x) = |x| for all x ∈ X we will have that both h B(X,|·|) and h−1 BH are slicely continuous on S(X,|·|) and SH respectively. That is the case previously obtained by Kadec [Kad66], [Kad67] and Anderson [And66] in the BessagaPe lczy´ nski form [BP75, Proposition 5.1] for the separable case as well as in other cases [Tro67], [BP75, Chap. VII, § 3, § 4]. From Toru´ nczyk, Theorem 1.1 and Corollary 2.42 we have the following consequence: Proposition 4.1. A Banach space X is LUR-renormable if and only if there exists a norm to norm homeomorphism Φ from X onto a Hilbert space such that Φ and Φ−1 are σ-slicely continuous. A relevant map fulﬁlling the conditions of the transfer technique (Theorem 1.15) is Day’s map, D : c0 (Γ) → 2 (Γ) deﬁned by Dx (γn ) = 2−n x (γn ) if supp x = {γn }n≥1 , |x (γ1 )| ≥ |x (γ2 )| ≥ . . . ≥ |x (γn )| ≥ . . . > 0, and Dx (γ) = 0 if γ∈ / supp x. Proposition 4.2. The Day map D : c0 (Γ) → 2 (Γ) is a σ-slicely continuous and co-σ-continuous map. In particular c0 (Γ) is LUR-renormable. Proof. In order to simplify the proof we assume that Γ is well ordered and A. Molt´ o et al., A Nonlinear Transfer Technique for Renorming. Lecture Notes in Mathematics 1951, c Springer-Verlag Berlin Heidelberg 2009

73

74

4 σ-Slicely Continuous Maps

i < j if α < β , x (α) = x (β) = 0 ,

(4.1)

Dx (α) = 2−i x (α) , Dx (β) = 2−j x (β) . Fix ε > 0, for every x ∈ c0 (Γ) set Lε (|x|) = {γ : |x(γ)| ≥ ε}. Let m, p ∈ N, m m m q ∈ Q, r, s ∈ Qm , n ∈ Nm , r = (ri )1 , s = (si )1 , n = (ni )1 , r1 > s1 > r2 > −1 s2 > . . . rm > sm , |ri |, |si | ≥ ε − p > 0, θ(r, s) = min1≤i≤m−1 (si − ri+1 ). m,p,q be the set of all x ∈ c0 (Γ) which satisfy the following conditions: Let Xn,r,s x (α) = x (β) if x (α) , x (β) ∈ [si , ri ] , # {γ : x(γ) ∈ [si , ri ]} = ni , {x(γ) : γ ∈ Lε (|x|)} ⊂

m

(si , ri ) ,

(4.2)

i=1

ε − sup {|x(γ)| : γ ∈ / Lε (|x|)} > p−1 ,

(4.3)

and

δ2 x2 (γ) : γ ∈ Lε (|x|) < 3 −1 where δ = min p , θ(r, s) . 0≤q−

(4.4)

&m

m,p,q . Let Lε (|x|) = {γi }1 , = 1 ni , Dx (γi ) = 2−i x (γi ), Pick x ∈ Xn,r,s i = 1, 2, . . . , , and let f be the linear functional on c0 (Γ) deﬁned by the formula

f (u) = x (γi ) u (γi ) , u ∈ c0 (Γ) . 1

Pick y ∈

m,p,q Xn,r,s

with f (x − y) < δ 2 /3. From (4.4) we get 2

(x (γi ) − y (γi )) = 2f (x − y) +

1

y 2 (γi ) − x2 (γi ) <

(4.5)

1

2f (x − y) +

2 x (γ) : γ ∈ Lε (|x|) < δ 2 . y 2 (γ) : γ ∈ Lε (|y|) −

From (4.1), (4.2) and (4.3) we get that Lε (|x|) = Lε (|y|)

and Dy (γi ) = 2−i y (γi ) , i = 1 , 2 , . . . , .

For γ ∈ / Lε (|x|) we have |x(γ)|, |y(γ)| ≤ ε and since δ ≤ p−1 < ε from (4.5) we get Dx − Dy2 ≤

1

2

(x (γi ) − y (γi )) +

∞

+1

−i

2 2ε

2

1/2 < 2ε .

4.1 Examples

75

m,p,q Now we prove that D is co-σ-continuous. Indeed let x, y ∈ Xn,r,s , −1 &m −

= 1 ni , δ = min p , θ(r, s) and Dx − Dy2 < 2 δ. Then |Dx(γ) − Dy(γ)| < 2− δ for γ ∈ Lε (|x|). This (4.1), (4.2) and (4.3) imply that Lε (|x|) = Lε (|y|) and |x(γ) − y(γ)| < δ for γ ∈ Lε (|x|). Since δ ≤ p−1 < ε we get x − y∞ ≤ 2ε. So D is co-σ-continuous. From Theorem 1.15 we get again

that c0 (Γ) is LUR renormable.

The next example gives us another proof of the LUR renormability of c0 (Γ) using a map which is diﬀerent of Day’s map. Proposition 4.3. Given ε > 0 the map Φε : c0 (Γ) → c0 (Γ) deﬁned by the formula Φε x = x1lLε (|x|) where Lε (|x|) = {γ ∈ Γ : |x(γ)| ≥ ε} is σ-slicely continuous and sup {Φε x − x∞ : x ∈ c0 (Γ)} ≤ ε. In particular c0 (Γ) is LUR renormable. p,q Proof. For η > 0 and , p ∈ N and q ∈ Q+ we denote by X ,η the set of all ε x ∈ c0 (Γ) for which #L (|x|) = ,

ε − sup {|x(γ)| : γ ∈ / Lε (|x|)} > p−1 , and 0≤q−

δ2 x2 (γ) : γ ∈ Lε (|x|) < , 3

(4.6)

where δ = min p−1 , η . Evidently p,q ∪ {x ∈ c0 (Γ) : Lε (|x|) = ∅} . c0 (Γ) = X ,η : p, q ∈ N, q ∈ Q+ p,q and deﬁne the linear function f by the formula Pick x ∈ X ,η

f (u) =

{x(γ)u(γ) : γ ∈ Lε (|x|)} ,

u ∈ c0 (Γ) .

p,q with Take y ∈ X ,η

δ2 . 3 From (4.6) we get as in the former proposition f (x − y) <

δ2 (x(γ) − y(γ))2 : γ ∈ Lε (|x|) ≤ 2f (x − y) + < δ2 . 3

This and (4.6) imply Lε (|x|) = Lε (|y|). Hence Φε x − Φε y < η. So Φε is σslicely continuous. From Corollary 3.10 we get that c0 (Γ) is LUR renormable.

Duality map gives us another example of σ-slicely continuous map.

76

4 σ-Slicely Continuous Maps

Proposition 4.4. Let X be a Banach space with Fr´echet diﬀerentiable norm and let ∂ be the duality mapping from SX into SX ∗ , i.e. ∂x(x) = 1 for all x ∈ ateaux SX . Then ∂ is σ-slicely continuous. If in addition the norm of X ∗ is Gˆ diﬀerentiable then ∂ is co-σ-continuous and hence X is LUR renormable. Proof. Pick ε > 0. According to Smulyan test (see e.g. [DGZ93, Theorem 1.4.]) for every x ∈ SX we can ﬁnd δx >0 such that ∂x − f < ε whenever f ∈ SX ∗ and f (x) > 1 − δx . Set Sn = x ∈ SX : δx > n−1 . Pick x ∈ Sn and let ∂x(y) > 1 − n−1 for some y ∈ Sn . Since y ∈ Snwe get ∂x(y) > 1 − δy . Hence ∂y − ∂x < ε. So osc ∂ Rn ≤ 2ε where Rn = y ∈ Sn : ∂x(y) > 1 − n−1 . Assume now that the norm of X ∗ is Gˆ ateaux diﬀerentiable then (see e.g. [DGZ93, Theorem 1.4.] we have that ∂ −1 is norm-to-weak continuous. ∞ ·

, whenever ∂x − ∂xn → 0. This and This implies that x ∈ conv {xn }1 Proposition 2.33 show that ∂ is co-σ-continuous. Now from Theorem 3.1.iv we get that X is LUR renormable.

The next proposition gives us another example of a slicely continuous map. Proposition 4.5. Let (X, · ) be a normed space, let F be a 1-norming subspace of X ∗ and let K be a σ(X, F )-compact, convex and balanced subset ·

of X, such that span K = X. Suppose that · K , the Minkowski functional of K, is a LUR norm in span K. Let Φ : X → K be the best approximation map i.e. x − Φx = dist(x, K) for all x ∈ X. Then Φ is norm continuous and ΦC is σ(X, F )-slicely continuous on D1 = {u : dist(u, K) = 1}, where C = BX + K. Before proving the above proposition we need two simple assertions. Lemma 4.6. Let f ,fn , n ∈ N, linear functionals on a linear space X, let vi ∈ Li ⊂ X, i = 1, 2, v = v1 + v2 , L = L1 + L2 , 0 ∈ L1 ∩ L2 , such that f (v) = sup f , L

sup fn ≤ f (v) , n ∈ N , L

lim fn (vi ) = f (vi ) , i = 1, 2 . n

Then sup f = f (vi ) ,

i = 1, 2 ;

(4.7)

i = 1, 2.

(4.8)

Li

lim sup fn = f (vi ) , n

Li

Proof. We have 0 ≤ sup f −f (v1 ) = sup f −f (v)− sup f − f (v2 ) = − sup f − f (v2 ) ≤ 0 L1

L

L2

L2

4.1 Examples

77

which implies (4.7). Since fn (v) ≤ supL fn ≤ f (v) and limn fn (v) = f (v) we get limn supL fn = f (v). We have for all n ∈ N fn (v1 ) ≤ sup fn = sup fn − sup fn ≤ sup fn − fn (v2 ) ≤ f (v) − fn (v2 ) . L1

L

L2

L

Letting n → ∞ we get f (v1 ) = lim fn (v1 ) ≤ lim inf sup fn ≤ lim sup sup fn ≤ n

L1

L1

≤ lim (f (v) − fn (v2 )) = f (v1 ) , n

which implies (4.8).

The following result which is a variant of the bipolar theorem gives the well known characterization of 1-norming subspaces, it can be found in the book [FHHMPZ01, p. 134]. Lemma 4.7. Let (X, · ) be a normed space, given a subspace F of X ∗ the following are equivalent: i) F is 1-norming; weak∗ = BX ∗ ; ii) BX ∗ ∩ F iii) BX is σ(X, F )-closed; iv) The norm · is σ(X, F )-lower semicontinuous. Proof of Proposition 4.5. We ﬁrst show that C = {u ∈ X : dist(u, K) ≤ 1}.

(4.9)

Indeed, let u such that dist(u, K) ≤ 1 then u − Φu ≤ 1 so u ∈ BX + K = C. Take u ∈ BX + K then there exist v ∈ K, w ∈ BX such that u = v + w. Then dist(u, K) ≤ u − v = w ≤ 1 and (4.9) is proved. Taking into account that K is σ(X, F )-compact and BX is σ(X, F )-closed we have that C is σ(X, F )-closed. Then | · | the Minkowski functional of C is an equivalent σ(X, F )-lower semicontinuous norm in X. From (4.9) it follows that S(X,|·|) = D1 . Set V = span K. For f ∈ V ∗ we denote f K = supK f . Pick x ∈ D1 . Then |x| = 1. We can ﬁnd f ∈ X ∗ supporting x with respect to | · |, i.e. f (x) = |f | = sup f > 0 .

(4.10)

C

Since the identity operator Id : V → X has dense range we get that the transpose operator Id∗ : X ∗ → V ∗ is one-to-one. We write g instead of Id∗ g for any g ∈ X ∗ and we assume that f K = 1 .

(4.11)

78

4 σ-Slicely Continuous Maps

Since x = (x − Φx) + Φx, x − Φx ∈ BX , Φx ∈ K we get from Lemma 4.6 and (4.11) f K = sup f = f (Φx) = 1 .

(4.12)

K

Since |·| is a σ(X, F )-lower semicontinuous norm according to Lemma 4.7 we get that F is 1-norming for (X, | · |) and therefore B(X,|·|)∗ ∩ F Hence there exist fn ∈ F , n ∈ N, such that |fn | ≤ |f | ,

lim fn (Φx) = f (Φx) , n

weak∗

= B(X,|·|)∗ .

lim fn (x − Φx) = f (x − Φx) . (4.13) n

Since |fn | = supC f , fn = supBX fn , fn K = supK fn , from Lemma 4.6, (4.10) and (4.12) we get lim fn K = f K = 1 , n

lim fn = f (x − Φx) . n

(4.14)

Pick an arbitrary y ∈ X since dist(y, K) = y − Φy we have fn (Φy) = fn (y) − fn (y − Φy) ≥ fn (y) − dist(y, K) fn =

(4.15)

= 1 − (dist(y, K) fn − fn (x − Φx)) − (1 − fn (Φx)) − fn (x − y) . Pick ε > 0. Since · K is a · LUR norm in V there exists δ > 0 such that Φx − v < ε whenever v ∈ K and Φx + vK > 2 − 8δ. From (4.12), (4.13) and (4.14) it follows that there exists m ∈ N such that fm − fm (x − Φx) < δ ,

1 − fm (Φx) < δ ,

fm K < 1 + δ . (4.16)

Pick y ∈ C with fm (x − y) < δ. From (4.15) and (4.16) fm (Φy) > 1 − 3δ .

(4.17)

Setting y = x in (4.17) we get fm (Φx) > 1 − 3δ .

(4.18)

From (4.16), (4.17) and (4.18) we have fm (Φx + Φy) 2 − 6δ > 2 − 8δ . > fm K 1+δ Hence Φx + ΦyK > 2 − 8δ. Since ΦyK ≤ 1 we get that Φx − Φy < ε. Then we have proved that ΦC is σ(X, F )-slicely continuous at x ∈ D1 .

4.1 Examples

79

We will show now that Φ is norm continuous at x. Set η = δ/2 fm and pick y ∈ X with x − y < η. Then dist(y, K) ≤ dist(x, K) + y − x < 1 + η. This, (4.15) and (4.16) imply fm (Φy) > 1 − ((1 + η) fm − fm (x − Φx)) − (1 − fm (Φx)) − fm x − y > 1 − 2δ − 2η fm > 1 − 3δ . Now as above we get Φx − Φy < ε so Φ is norm continuous at x. In this way we have proved that Φ is norm continuous at X \ K. Pick now x ∈ K and let x − y < ε. Since x ∈ K we have Φx = x and y − Φy = dist(y, K) ≤ y − x < ε. Hence Φx − Φy ≤ x − y + y − Φy < 2ε.

The former proposition is the key for the next Theorem 4.8. Let X be a normed space and let F be a 1-norming subspace of X ∗ . Let Y be a LUR Banach space and let T : Y → X be a bounded linear · operator such that K = T BY is σ(X, F )-compact and T Y = X. Given ε > 0 there exists a σ-slicely continuous map Ψ : (X, σ(X, F )) → (X, · ) such that Ψx − x < ε for all x ∈ X. In particular X admits an equivalent σ(X, F )-lower semicontinuous LUR norm. From the above theorem it follows as in [GTWZ83] that if X is a dual weakly compactly generated Banach space then X admits a dual LUR norm. Another application is the following Corollary 4.9. If ν is a probabilistic measure then L1 (ν) is LUR renormable. It is clear that Id : L2 (ν) → L1 (ν) satisﬁes the conditions of Theorem 4.8. Remark 4.10. M. Raja [Raj02] proved that if K is a σ(X, F )-compact subset of X, where F is a norming subspace of X ∗ and Id : (K, σ(X, F )) → (K, · ) is σ-slicely continuous then span K

·

is LUR renormable.

In order to prove Theorem 4.8 we need the following Lemma 4.11. Let D ⊂ A ⊂ X, Φ : A → Y a norm continuous map and F ⊂ X ∗ such that ΦD is σ(X, F ) σ-slicely continuous and for every x ∈ A and for every δ > 0, we have {λx : 1 ≤ λ ≤ 1 + δ} ∩ D = ∅ . Then Φ is σ(X, F ) σ-slicely continuous. Proof. Pick ε > 0. For every x ∈ A there exists δx > 0 such that for all y ∈ A with x − y < δx we have Φx − Φy < ε .

(4.19)

80

4 σ-Slicely Continuous Maps

There

exists λx ∈ [1, 1 + δx ) such that wx = λx x ∈ D. We can write D = m Dm in such a way that for every x ∈ A there exist fx ∈ F and θx = ±1 such that fx (wx ) > θx and Φwx − Φu < ε

(4.20)

whenever fx (u) > θx , u, wx ∈ Dm for some m ∈ N. We can ﬁnd jx ∈ N and qx ∈ Q ∩ [1, +∞) such that

and

fx (wx ) > θx + jx−1

(4.21)

qx < λx ≤ qx 1 + jx−1 .

(4.22)

Set Aj,m = {x ∈ A : jx = j, θx = θ, qx = q, wx ∈ Dm }. Evidently A = q,θ

j,m j,m,q Aq,θ . −1 −1 Pick x ∈ Aj,m λx . We get from (4.21) fx (x) = q,θ . Set µx = θ + jx −1 −1 −1 λx fx (wx ) > λx θ + jx = µx . Let y ∈ Aj,m q,θ and fx (y) > µx . Then fx (wy ) = λy fx (y) > λy µx . From (4.22) we deduce λy µx = λy λ−1 θ + jx−1 . x So fx (wy ) > θ. Since wy ∈ Dm from (4.20) we get Φwx − Φwy < ε. Since x − wx = λx − 1 < δx , y − wy < δy we deduce from (4.19) Φx − Φwx <

ε, Φx − Φwy < ε. Then Φx − Φy < 3ε. Proof of Theorem 4.8. Set K = {T y : y ∈ BY }. Clearly for every y ∈ Y T −1 T y ≤ T yK ≤ y ,

(4.23)

where · K is the Minkowski functional of K in T Y . From (4.23) we get {y ∈ BY : T yK = 1} ⊂ SY . This and (4.23) imply that · K is a LUR norm in T Y . So K satisﬁes the conditions of Proposition 4.5.

Set Dq = {u : dist(u, K) = q}, q ∈ Q+ , and D = q Dq . From Proposition 4.5 it follows that if Φ : X → K is a map such that x − Φx = dist(x, K) for all x ∈ X then ΦD is σ(X, F ) slicely continuous. According to Lemma 4.11 ΦX\K is σ(X, F ) σ-slicely continuous too. For q ∈ Q+ we deﬁne Φq : X → qK in such a way that x − Φq x = dist(x, qK) for all x ∈ X. Pick ε > 0. For every x ∈ X \ {0} we can ﬁnd q(x) ∈ Q+ such that 0 < dist(x, q(x)K) < ε. Set Ψ0 = 0 and Ψx = Φq(x) x for x = 0. Since ΦX\qK is σ(X, F ) σ-slicely continuous so is Ψ. From the choice of Φq and Ψ we get that for all x ∈ X

4.2 Joint σ-Slicely Continuity and Characterization of σ-Slicely

81

Ψx − x = Φq(x) x − x = dist(x, q(x)K) < ε. The existence of σ(X, F )-lower semicontinuous LUR norm follows from Corollary 3.10.

Deﬁnition 4.12. Let M be a compact space and L a compact subset of M . The symbol R : C(M ) → C(L) will denote the restriction map, i.e. Rf = f L , for f ∈ C(M ). We will say that the map Ξ : C(L) → C(M ) is an extension map when R ◦ Ξ = Id where Id is the identity map on C(L). if and only Corollary 4.13. If K ⊂ [0, 1]Γ then C(K) is LUR renormable if there exists an extension map Ξ : C(K) → C [0, 1]Γ which is σ-slicely continuous. Proof. To show that the condition is suﬃcient it is enough to prove that Ξ is co-σ-continuous. But this is trivial from the very deﬁnition of co-σ-continuous. To see that the condition is necessary let Ξ : C(K) → C [0, 1]Γ be the BartleGraves map [DGZ93, Lemma VII.3.2] associated to R, i.e. a continuous map such that R ◦ Ξ = Id where Id is the identity map on C(K). Obviously Ξ is a continuous extension map. Since C(K) is LUR-renormable, the identity map Id : C(K) → C(K) is σ-slicely continuous. Since Ξ is continuous and Ξ = Ξ ◦ Id to ﬁnish the proof it is enough to apply Corollary 2.42.

Remark 4.14. Corollary 4.13 does not hold if we replace σ-slicely continuous by bounded linear extension operator. Indeed if K is the Ciesielski-Pol compact then C(K) is LUR renormable, [Dev86], (see also [DGZ93, p. 305] and p. 101 here) and there is no bounded linear one-to-one map from C(K) into c0 (Γ) for any Γ, (see e.g. [DGZ93, p. 261]). Let I be a set such that K ⊂ [0, 1]I . Since there exists a set Γ and a bounded linear one-to[0, 1]I is a Valdivia compact one map Ξ : C [0, 1]I → c0 (Γ) [Val90]. Then if there would exist a bounded linear one-to-one map from Ψ : C(K) → C [0, 1]I then the composition Ξ ◦ Ψ : C (K) → c0 (Γ) would be a bounded linear one-to-one map which is a contradiction.

4.2 Joint σ-Slicely Continuity and Characterization of σ-Slicely Continuous Maps with Values in a Metric Space Let us recall that a subset A of a linear topological space X is said to be bounded if for every neighbourhood V of 0 in X there exists λ V > 0 such that A ⊂ λV V . We say that A is σ-bounded if we can write A = n∈N An in such a way that every An is bounded.

82

4 σ-Slicely Continuous Maps

Remark 4.15. If X is a normed space and F is a subspace of X ∗ then BX is σ(X, F )-bounded. In our applications X will be a normed space in which we consider a norm or a σ(X, F ) topology for some subspace F of X ∗ . So the requirement of the σ-boundedness of A is not restrictive. Theorem 4.16. Let A be a σ-bounded subset of a locally convex linear topological space X, let (Y, ) be a metric space and Φ : A → Y . The following are equivalent: i) Φ is σ-slicely continuous.

ii) for every ε > 0 we can write A = n An,ε in such a way that for all x ∈ An,ε and every Lipschitz 1 function g : ΦA → R we have ∂ε g ◦ Φ (x|An,ε ) = ∅ ∞

iii) there is a sequence {An }1 of subsets of A such that {An ∩ H : n ∈ N , H is an open-half space in X} is a function base of Φ (i.e. for every -open set V of Y and every x ∈ Φ−1 (V ) there is some n ∈ N and H open half space of X with x ∈ An ∩ H and Φ (An ∩ H) ⊂ V ). iv) If {Dγ : γ ∈ Γ} is a discrete family of subsets of (Y, ) then −1 Φ (Dγ ) : γ ∈ Γ is σ-slicely isolatedly decomposable in A. The next two well known assertions give us the tool for the proof of Theorem 4.16. Lemma 4.17. (see e.g. [Phe93, p. 48]) Let C a convex subset of a locally convex linear topological space X, and ψ a convex function on C. Let x ∈ C, ε > 0 and lim inf ψ (xα ) ≥ ψ(x) − ε α

whenever x, xα ∈ C, limα xα = x. Then ∂2ε ψ (x|C) = ∅. Proof. The proof is similar to that in [Phe93, p. 48] but we include it for the sake of completeness. Set Y = X × R and B = {(y, t) : y ∈ C, t ∈ R, ψ(y) ≤ t}. Since ψ is a convex function on the convex set C we have that B is a convex / B. Assume the contrary. set hence so is B. We show that (x, ψ(x) − 2ε) ∈ Then there exists a net (xα , tα ) ∈ B with limα xα = x, limα tα = ψ(x) − 2ε. Since ψ (xα ) ≤ tα we have ψ(x) − ε ≤ lim inf ψ (xα ) ≤ lim inf tα = ψ(x) − 2ε . α

A contradiction.

α

4.2 Joint σ-Slicely Continuity and Characterization of σ-Slicely

83

According to the Hahn-Banach theorem we can ﬁnd a continuous linear functional h on X and a ∈ R such that for all (y, t) ∈ B we have ' ( ' ( (h, a), (x, ψ(x) − 2ε) < (h, a), (y, t) , i.e. h(x) + a(ψ(x) − 2ε) < h(y) + at. Set y = x, t = ψ(x) we get a(ψ(x) − 2ε) < aψ(x). So a > 0. Set now f = −h/a, t = ψ(y) we get −f (x) + ψ(x) − 2ε < −f (y) + ψ(y) ,

which implies f ∈ ∂2ε ψ (x|C).

Deﬁnition 4.18. Given a real function ϕ deﬁned on a subset A of a linear space X, by ϕ˜ we denote the convex function deﬁned on conv(A) that minimizes ϕ on A, i.e. for x ∈ conv(A) λi xi , xi ∈ A , λi > 0 , λi = 1 . ϕ(x) ˜ = inf λi ϕ (xi ) : x =

Lemma 4.19. Let A be a subset of a locally convex linear topological space X, ε > 0, x ∈ A and ϕ : A → R a bounded below function such that lim inf ϕ˜ (xα ) ≥ ϕ(x) − ε α

(4.24)

whenever xα ∈ conv(A), limα xα = x. Then ∂3ε ϕ (x|A) = ∅. In particular if ϕ is a convex function which is lower semicontinuous at x we have ∂η ϕ(x|A) = ∅ for all η > 0. Proof. If we set in (4.24) xα = x we get ϕ˜ (x) ≥ ϕ(x) − ε. Having in mind that ϕ˜ is convex from Lemma 4.17 it follows that there exists a continuous linear functional f on X such that ϕ(y) ˜ ≥ ϕ(x) ˜ + f (y − x) − 2ε for all y ∈ conv(A). From the deﬁnition of ϕ˜ we have ϕ(y) ≥ ϕ(y) ˜ for all y ∈ A. Since ϕ(x) ˜ ≥ ϕ(x) − ε we get ϕ(y) ≥ ϕ(x) + f (y − x) − 3ε for all y ∈ A. So ∂3ε ϕ (x|A) = ∅.

Lemma 4.20. Let A be a subset of a linear topological space X, Φ : A → (Y, ), ε > 0, x ∈ A, h a continuous linear functional on X, with h(x) + ε > sup h , A

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4 σ-Slicely Continuous Maps

and (Φx, Φu) < ε whenever u ∈ A, h(x − u) < 1. Let g : ΦA → R be a Lipschitz 1 function and ϕ = g ◦ Φ. Then for all u ∈ conv(A) such that h(x − u) < ε (4.25) we have ϕ(u) ˜ > ϕ(x) − (1 + 2d)ε ,

(4.26)

where d =diam(ΦA). Proof. Pick & u ∈ conv(A) and satisfying (4.25). Let u = λi > 0, λi = 1. Set I = {i : h (x − ui ) < 1}. Since

&

λi ui , ui ∈ A,

h(x) ≥ sup h − ε ≥ h (ui ) − ε , A

from (4.25) we get ε > h(x − u) = λi h (x − ui ) + λi h (x − ui ) ≥ −ε λi + λi ≥ i∈I /

i∈I

≥ −ε +

i∈I

i∈I /

λi .

i∈I /

Hence

λi < 2ε .

(4.27)

i∈I /

Since g is Lipschitz 1 we have ϕ (ui ) = g (Φui ) > g (Φx) − (Φui , Φx) = ϕ(x) − (Φui , Φx) .

(4.28)

As (Φx, Φui ) ≤ d from (4.28) it follows that ϕ (ui ) ≥ ϕ (x) − d .

(4.29)

For i ∈ I we have h (x − ui ) < 1 hence (Φx, Φui ) < ε. From (4.28) it follows for i ∈ I (4.30) ϕ (ui ) ≥ ϕ(x) − ε . From (4.27), (4.29) and (4.30) we get λi ϕ (ui ) ≥ (ϕ(x) − ε) λi + (ϕ(x) − d) λi > ϕ(x) − (1 + 2d)ε . i∈I

So (4.26) is proved.

i∈I /

The next lemma shows that if Φ is σ-slicely continuous map on A then A can be split up into countable pieces An in such a way that for every x ∈ An the slice with small oscillation of Φ can be generated by a functional which almost attains at x its supremum on An .

4.2 Joint σ-Slicely Continuity and Characterization of σ-Slicely

85

Lemma 4.21. Let A be a σ-bounded subset of a linear topological space X, let Φ : A → (Y, ) be a σ-slicely continuous map and let ε > 0. Then for every η > 0 we can write An,η (4.31) A= n

in such a way that for every x ∈ An,η there exists a continuous linear functional hx on X such that hx (x) + η ≥ sup hx An,η

and (Φx, Φu) < ε whenever u ∈ An,η and hx (x − u) < 1 . In the case when X is a normed space, F is a norming subspace of X ∗ and Φ is σ(X, F ) σ-slicely continuous then the decomposition (4.31) can be obtained in such a way that hx ∈ F satisﬁes in addition hx (x) > 1 for all x ∈ An,η , x = 0 .

Proof. Without loss of generality we can assume that A is bounded. Then for every continuous linear functional f on X we have sup f < +∞ .

(4.32)

A

Since Φ is σ-slicely continuous we can write A = j Aj in such a way that for every j ∈ N we can ﬁnd maps ∆j from Aj into the space of all continuous linear functionals on X and maps θj : Aj → Q such that ∆j x(x) > θj (x) and

ε , (4.33) 2 whenever x, u ∈ Aj and ∆j x(u) > θj (x). According to (4.32) we can assume without loss of generality that (Φx, Φu) <

sup {∆j v(w) : j ∈ N , v, w ∈ Aj } < +∞ .

(4.34)

Pick δk > 0, with δk → 0. For k, m ∈ N and q ∈ Q we set ak,m = δk (1 + (m − 1)η) , Aqj,k = {x ∈ Aj : θj (x) = q, ∆j x(x) > q + δk } ,

Aqj,k,m

=

x∈

Aqj,k

: q + ak,m < sup ∆j y(x) ≤ q + ak,m+1 y∈Aqj,k

.

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4 σ-Slicely Continuous Maps

Having in mind that for all x ∈ Aqj,k q + ak,1 = q + δk < ∆j x(x) ≤ sup ∆j y(x) y∈Aqj,k

and taking into account (4.34) we get q Aj,k,m . Aj = k,m,q

Pick x ∈ Aqj,k,m . There exists y ∈ Aqj,k such that ∆j y(x) > q + ak,m .

(4.35)

Since y ∈ Aqj,k for all z ∈ Aqj,k,m we have ∆j y(z) ≤ q + ak,m+1 . So from (4.35) we get ∆j y(x) + δk η > q + ak,m + δk η = q + ak,m+1 ≥

sup z∈Aqj,k,m

∆j y(z) .

(4.36)

On the other hand since ak,m ≥ δk > 0 from (4.35) we have ∆j y(x) > q = θj (y). Having in mind (4.33) we get (Φy, Φx) < Pick u ∈ Aqj,k,m with

ε . 2

∆j y(x − u) < δk .

(4.37)

Then again from (4.35) we get ∆j y(u) > q = θj (y). So (Φy, Φu) < ε/2. Therefore (Φx, Φu) < ε . (4.38) So the proof of the assertion when X is a linear topological space is complete. Now let A be a subset of a normed space X and let F be a norming subspace of X ∗ . Without loss of generality we can and do assume that F is 1-norming for X. Fix ε > 0. From the proof above and (4.37) it follows that we can ﬁnd ξj ∈ (0, 1) and write A= Aj j∈N

in such a way that for every x ∈ Aj there exists Ψj x ∈ SF such that (Φx, Φw) < ε/2 whenever u ∈ Aj and Ψj x(x − u) < 10ξj . Without loss of generality we can assume also that sup {v/w : v, w ∈ Aj } ≤ 1 + ξj j −1 ,

sup {v : v ∈ Aj } ≤ j . (4.39)

4.2 Joint σ-Slicely Continuity and Characterization of σ-Slicely

87

We will now prove that for every x ∈ Aj there exists Ξj x ∈ F such that Ξj x(x) > ξj and (Φx, Φu) <

(4.40) ε 2

(4.41)

whenever u ∈ Aj , Ξj x(x − u) < ξj . If Ψj x(x) > ξj we set Ξj x = Ψj x. Suppose on the contrary that Ψj x(x) ≤ ξj . Pick fx ∈ 1 + ξj j −1 BF such that fx (x) = x and set Ξj x = Ψj x +

(2ξj − Ψj x(x)) fx . x

Clearly Ξj x(x) = 2ξj > ξj . Pick u ∈ Aj with Ξj x(x − u) < ξj . From (4.39) we get 0 < 2ξj − Ψj x(x) ≤ 2 + j , fx (x − u) x − fx (u) ξj u −3ξj = >1− 1+ ≥ . x x j x j Hence (2ξj − Ψj x(x)) fx (x − u) −3 (2ξj − Ψj x(x)) ξj > > −9ξj . x j So Ψj x(x − u) = Ξj x(x − u) − (2ξj − Ψj x(x)) fx (x − u)x−1 < ξj + 9ξj = 10ξj . Therefore (Φx, Φu) < ε/2. Now the proof of (4.40) and (4.41) is complete. For x ∈ Aj pick θj (x) ∈ (Ξj x(x) − ξj , Ξj x(x)) ∩ Q+ . Clearly Ξj x(x) > θj (x). If u ∈ Aj and Ξj x(u) > θj (x) we have Ξj x(x − u) = Ξj x(x) − Ξj x(u) < Ξj x(x) − θj (x) < ξj . Hence (Φx, Φu) < ε/2. Now applying the initial proof with ∆j x = Ξj x we can write q Aj = Aj,k,m k,m,q

in such a way that for x ∈ Aqj,k,m we have q = θj (x) > 0 and we can ﬁnd y ∈ Aj such that

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4 σ-Slicely Continuous Maps

Ξj y(x) > q + δk (1 + (m − 1)η) ,

Ξj y(x) + δk η >

sup z∈Aqj,k,m

Ξj y(z)

(4.42)

and (Φx, Φu) < ε, whenever u ∈ Aqj,k,m and Ξj y(x − u) < δk . Set now hx = Ξj y/δk . From (4.42) we get hx (x) > q/δk + 1 + (m − 1)η > 1 ,

hx (x) + η > sup hx Aqj,k,m

and if hx (x − u) < 1 we have Ξj y(x − u) = δk hx (x − u) < δk . Hence (Φx, Φu) < ε.

Lemma 4.22 (Joint σ-Continuity). Let A be a σ-bounded subset of a linear topological space X, let Φ : A → (Y, ), Ψ : A → (Y, ) be σ-slicely continuous maps. Then for every ε > 0 we can write An,ε A= n

in such a way that for every x ∈ An,ε there exists a open half space H of X containing x with diam Φ (H ∩ An,ε ) < ε and diam Ψ (H ∩ An,ε ) < ε. In particular if Y is a normed space or a normed algebra then Φ + Ψ and ΦΨ are σ-slicely continuous.

Proof. Given ε > 0 according to the former lemma we can write A = n An in such a way that for every x ∈ An there exist continuous linear functionals gx and hx on X such that gx (x) +

1 ≥ sup gx , 2 An

hx (x) +

1 ≥ sup hx 2 An

(4.43)

and (Φx, Φy) < ε, (Ψx, Ψy) < ε whenever u, v ∈ An and gx (x − u) < 1, hx (x − v) < 1. Let z ∈ An and gx (x − z) + hx (x − z) <

1 . 2

(4.44)

Then from (4.43) we get gx (x−z) > −1/2, hx (x−z) > −1/2. Then from (4.44) we get gx (x − z) < 1, hx (x − z) < 1. Hence (Φx, Φz) < ε, (Ψx, Ψz) < ε. Assume that Y is a normed algebra and set Aj = {x ∈ A : Φx, Ψx ≤

j}. Given ε > 0 we can write Aj = n Aj,n in such a way that for every x ∈ Aj there exists a open half space H of X containing x with diam Φ (H ∩ Aj,n ) < ε/j, diam Ψ (H ∩ Aj,n ) < ε/j. Pick z ∈ Aj,n ∩ H we have ΦxΦz − ΨxΨz ≤ ΦxΨx − Ψz + ΨzΦx − Φz < 2ε .

4.2 Joint σ-Slicely Continuity and Characterization of σ-Slicely

89

Corollary 4.23. Let X be a normed space, let F be a norming subspace of X ∗ . Let Y be a normed space and let Φ : X → Y and let Φn : X → Y , n ∈ N a sequence of σ(X, F ) σ-slicely continuous maps. If Φx ∈ span {Φn x : n ∈ N} for every x ∈ X then Φ is σ-slicely continuous too.

·

In addition if Y is a normed algebra the former condition can be replaced ·

by Φx ∈ alg {Φn x : n ∈ N} for every x ∈ X where alg A is the subalgebra of Y generated by the subset A of Y . Proof. Fix x ∈ X. The countable set of Q-linear combinations of functions from {Φn x : n ∈ N} (respectively the Q-linear algebra generated by {Φn x : n ∈ N} when Y is a normed algebra) veriﬁes the condition of Proposition 2.23 since they are σ-slicely continuous according to the joint-σ-continuity lemma.

Proof of Theorem 4.16. The implication i)⇔iv) has been shown in Proposition 2.22. i)⇒ii). We may assume that diam(ΦA) < +∞ since we can decompose A into countable pieces if it is necessary. For simplicity

we suppose diam(ΦA) ≤ 1. According to Lemma 4.21 we can write A = An,ε in such a way that for every x ∈ An,ε there exists h∈F ,

h(x) + ε > sup h An,ε

such that (Φx, Φu) < ε, whenever u ∈ An,ε , h(x−u) < 1. Let g be a Lipschitz 1 function. Set ϕ(u) = g(Φu), u ∈ An,ε . We get from Lemma 4.20 lim inf ϕ˜ (xα ) ≥ ϕ(x) − 3ε α

whenever xα ∈ conv (An,ε ) and lim xα = x. From Lemma 4.19 we get that ∂9ε ϕ (x|An,ε ) = ∅. ii)⇒iii). The sequence An,p−1 : n, p = 1, 2, . . . provided by ii) gives us the family of subsets An,p−1 ∩ H : H is an open half space of X, n, p = 1, 2, . . . which is going to be a function base of Φ. Indeed, let V be an open subset of (Y, ) and let us consider the Lipschitz 1 function g : Y → R deﬁned by the formula g(y) = dist(y, Y \ V ) . Since Y \ V is closed we have V = g −1 ((0, +∞)). Let x ∈ A and Φx ∈ V . Find −1 p ∈ N such that g(Φx) > p−1 .According to ii) for ε = p we have n ∈ N with x ∈ An,p−1 and ∂p−1 g ◦ Φ x|An,p−1 = ∅. Pick f ∈ ∂p−1 g ◦ Φ x|An,p−1 and set H := z ∈ X : f (z) > f (x) + p−1 − g(Φx) . −1 Since g(Φx) > p we getx ∈ H ∩ An,p−1 . Having in mind that f ∈ ∂p−1 g ◦ Φ x|An,p−1 for every z ∈ H ∩ An,p−1 we have

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4 σ-Slicely Continuous Maps

g(Φz) ≥ g(Φx) + f (z − x) − p−1 > 0 . So Φz ∈ V . Hence Φ An,p−1 ∩ H ⊂ V . The condition iii) is fulﬁlled. iii)⇒i). Given ε > 0 we consider an open cover V of Y by sets of -diameter less than ε. If we deﬁne An,ε as the set of all x ∈ An for which there exists an open half space Hx of X and Vx ∈ V such that x ∈ Hx and Φ (An ∩ Hx ) ⊂ Vx then we have ∞ A= An,ε n=1

because V is a cover of Y and {An ∩ H : n ∈ N, H is an open halfspace of X} a function base of Φ. By its very deﬁnition it is clear that for x ∈ An,ε there is Hx , open half space of X, Vx ∈ V with x ∈ Hx and Φ (An ∩ Hx ) ⊂ Vx . Since An,ε ⊂ An we get oscHx ∩An,ε Φ ≤ diam Vx < ε .

Corollary 4.24. A normed space X is LUR renormable if and only if there ∞ is a sequence {Xn }1 of subsets of X such that for every x ∈ X and every ε > 0 there is n = n(x, ε) ∈ N such that x ∈ Xn and ∂ε g (x|Xn ) = ∅ for every Lipschitz 1 function g : X → R. Proof. If X is LUR renormable according to the characterization of LUR renormability (Theorem 1.1) we must have that the identity map Id : (X, σ(X, X ∗ )) → (X, · ) is σ-slicely continuous. Then from Theorem 4.16 it follows that for every ε > 0 we can write Xn,ε X= n∈N

in such a way that given n ∈ N for all x ∈ Xn,ε and every Lipschitz 1 function g : X → R we have ∂ε g (x|Xn,ε ) = ∅. Now the family Xn,p−1 : n, p ∈ N fulﬁlls the requirements of the statement. Conversely given ε > 0 let Xn be the sets of the statement. Write Xn,ε for the set of all x ∈ Xn such that ∂ε g (x|Xn,ε ) = ∅ whenever g : X → R is a Lipschitz 1 function. Evidently the sets Xn,ε satisfy the condition ii) from the former theorem for the identity map Id : (X, σ(X, X ∗ )) → (X, · ). Hence Id is σ-slicely continuous. Therefore from the characterization of LUR renormability it follows that X is LUR renormable.

4.2 Joint σ-Slicely Continuity and Characterization of σ-Slicely

91

Remark 4.25. Form the above corollary

it follows that if X is a LUR space then for every ε > 0 we can write X = n Xn,ε in such a way that for all x, z ∈ X, we have ∂ε ϕz (x|Xn,ε ) = ∅ where ϕz (u) = z + u, u ∈ X. The rest of the section is devoted to the proof of Theorem 1.1. The next lemma is a simple modiﬁcation of Lemma 4.20. Lemma 4.26. Let A a subset of the unit ball BX of a normed space X and let F be an 1-norming subspace of X ∗ . Assume that x ∈ A, h ∈ F , ε > 0 and c > 0 are such that h(x) + ε > sup h , A

c≤

h(x) − 1 , h

(4.45)

and x − u < ε whenever u ∈ A and h(x − u) < 1 . Let C = conv

σ(X,F )

(A ∪ cBX ). Then if u ∈ C and h(x − u) < ε

(4.46)

x − u ≤ 5ε .

(4.47)

we have

Proof. From (4.45) we get for all v ∈ A that h(x − v) > −ε .

(4.48)

h(x − v) ≥ h(x) − hv ≥ h(x) − hc ≥ 1 .

(4.49)

For v ∈ cBX we have

We ﬁrst prove that if u ∈ conv (A ∪ cBX ) and satisﬁes (4.46) then u must verify (4.47). Indeed let u= λi ui , & λi = 1. Set I = {i : ui ∈ A, h (x − ui ) < 1}. ui ∈ A ∪ cBX , λi > 0, According to (4.48) we have h (x − ui ) > −ε , i ∈ I .

(4.50)

h (x − ui ) ≥ 1 , i ∈ /I.

(4.51)

From (4.49) it follows Now from (4.50) and (4.51) we get

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4 σ-Slicely Continuous Maps

ε > h(x − u) =

λi h (x − ui ) +

λi +

λi ≥ −ε +

i∈I /

i∈I

Hence

λi h (x − ui ) ≥

i∈I /

i∈I

≥ −ε

λi .

i∈I /

λi < 2ε .

(4.52)

i∈I /

For i ∈ I we have h (x − ui ) < 1 and ui ∈ A so x − ui < ε. Then λi x − ui + λi (x + ui ) ≤ x − u ≤ i∈I /

i∈I

≤ε

λi + 2

i∈I

λi < 5ε .

i∈I /

Assume that u ∈ C satisﬁes (4.46) and x − u > 5ε. Then there exists a net {vα } ⊂ conv (A ∪ cBX ) converging to u in the σ(X, F )-topology. Since σ(X,F )

x − vα −→ x − u , h ∈ F and F is 1-norming we get lim inf x − vα ≥ x − u > 5ε ,

lim h (x − vα ) = h(x − u) < ε .

α

α

Therefore there exists vα ∈ conv (A ∪ cBX ) for which we have h (x − vα ) < ε

and x − vα > 5ε, a contradiction. The ﬁrst geometrical proof of the next assertion goes back to M. Raja [Raja99]. We are able to pressure on the unit sphere even though its points are not assumed to be denting points. The optimization lemmas in the section are the main tool for the new proof. The convexiﬁcation process concentrates on the origin together with sets in the unit sphere instead of the iteration of Bourgain-Namioka superlemma. Proposition 4.27. Let X be a normed space and F a norming subspace of X ∗ . Let the identity map Id : (SX , σ(X, F )) → (SX , · ) be σ-slicely continuous. Then X has an equivalent σ(X, F )-lower semicontinuous LUR norm. Proof. By H we denote the family of all σ(X, F )-open half spaces H of X. Fix ε > 0. According to Lemma 4.26 and Lemma 4.21 we can ﬁnd numbers cn,ε ∈ (0, 1] and a decomposition SX = An,ε n

4.2 Joint σ-Slicely Continuity and Characterization of σ-Slicely

93

satisfying that for every x ∈ An,ε there exists H ∈ H such that diam (H ∩ Cn,ε ) < ε where Cn,ε = convσ(X,F ) (An,ε ∪ cn,ε BX ) . Set Dn,ε = Cn,ε \ {x ∈ Cn,ε : ∃H ∈ H , x ∈ H , diam (H ∩ Cn,ε ) < ε, H ∩ Cn,ε ∩ 2−1 BX = ∅ . Evidently Dn,ε is a σ(X, F )-closed convex subset of Cn,ε and for ε < 1/2 An,ε ∩ Dn,ε = ∅ .

(4.53)

Let · n,p and | · |n,p be the Minkowski functionals of Cn,p−1 and Dn,p−1 respectively, n, p ∈ N, p > 1. We can ﬁnd an,p > 0 such that an,p | · |n,p ≤ 2−n−p · . Set |·|=

1/2 . an,p · 2n,p + | · |2n,p

k

Let |xk |, |(xk + x) /2| −→ |x|, and x = 1. We will show that lim sup xk − x ≤ 2p−1

(4.54)

k

for all p > 1. Indeed ﬁx p > 1. Since x ∈ SX there exists n ∈ N such that x ∈ An,p−1 . From (4.53) we get xn,p = 1 < |x|n,p .

(4.55)

By the convex arguments (Lemma 3.36) we have lim (xk + x) /2n,p = lim xk n,p = xn,p , k

k

xk + x lim k 2

(4.56)

= |x|n,p . n,p

Set yk = xn,p xk /xk n,p . From (4.56) we get lim xk − yk = 0 , k

and according to (4.55) for k suﬃciently big we have

yk + x

≤ 1 < yk + x .

2 2 n,p n,p

(4.57)

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4 σ-Slicely Continuous Maps

Hence

yk + x yk + x ∈ Cn,p−1 , ∈ / Dn,p−1 . 2 2 From the deﬁnition of Dn,p−1 it follows that there exists H ∈ H with (yk + x) /2 ∈ H and diam H ∩ Cn,p−1 < p−1 . Then either x ∈ H or yk ∈ H. Since yk n,p = xn,p = 1 we get that either x ∈ H ∩ Cn,p−1 or yk ∈ H ∩ Cn,p−1 . So either x − (yk + x) /2 < p−1 or yk − (yk + x) /2 < p−1 , this together with (4.57) imply (4.54). Let us deﬁne the symmetric convex and σ(X, F )-lower semicontinuous function

2

2

F (x)2 = |x| + |−x|

for every x ∈ X. The Minkowski functional of the set B := {x ∈ X : F (x) ≤ 1} is the norm we are looking for. The proof of Theorem 1.1 is a straightforward consequence of the proposition above, Lemma 4.7 and the next assertion. Lemma 4.28. Let Y be a LUR normed space. Then for every ε > 0 there exists a decomposition Y = Yn,ε n∈N

and δn > 0 such that x − y < ε whenever x, y ∈ Yn,ε and x − (x + y)/2 < δn . In particular the identity map Id : (Y, σ(Y, G)) → (Y, · ) is σ-slicely continuous for any 1-norming subspace G of Y ∗ . Proof. Fix ε > 0. Since Y is LUR for every u ∈ Y there exists θu > 0 such that u − v < ε/2 whenever u = v and u − (u + v)/2 < θu . For m ∈ N and q ∈ Q+ we set Ym,q = u ∈ Y : q ≤ u ≤ q + m−1 , θu > m−1 . Evidently 2 + Y \ {0} = Ym,q : m ∈ N , m > , q ∈ Q . ε Set δm = (2m)−1 . Pick x, y ∈ Ym,q , m > 2/ε, q > 0 and x − (x + y)/2 < δm . Set w = xy/y. Then w − y = x − y ≤ m−1 . Since x + w ≥ x + y − w − y ≥ x + y − m−1 , we get

x + w

≤ x − x + y + w − y < m−1 < θx . x −

2 2

2

4.3 Characterization of σ-Slicely Continuous Maps

95

Then x − w < ε/2 and x − y ≤ x − w + w − y < ε/2 + m−1 ≤ ε. Now let G be a 1-norming subspace of Y ∗ . Pick again x ∈ Ym,q with m > 2/ε, q > 0. We can ﬁnd g ∈ G, g = 1, g(x) > x − δm /2. Let y ∈ Ym,q with g(x − y) < δm . We have

x + y

< g(x) + δm − g(x + y) = g(x − y) + δm < δm . x − 2 2 2 2 2 Hence x − y < ε.

4.3 Characterization of σ-Slicely Continuous Maps with Values in a LUR Normed Space Now taking advantage of the fact that the image space of a σ-slicely continuous map is a LUR space we simplify the characterization which makes it easier to check. Theorem 4.29. Let A be a σ-bounded subset of a locally convex linear topological space X, let Y be a LUR space, let G be a subspace of Y ∗ which is 1-norming for Y and let Φ : A → Y . The following are equivalent: i) Φ is σ-slicely continuous.

ii) for every ε > 0 we can write A = n An,ε in such a way that for all x ∈ An,ε and every Lipschitz 1 function g : ΦA → R we have ∂ε g ◦ Φ (x|An,ε ) = ∅

iii) for every ε > 0 we can write A = n An,ε in such a way that for all x ∈ An,ε and g ∈ BG ∂ε g ◦ Φ (x|An,ε ) = ∅ ∞

iv) there is a sequence {An }1 of subsets of A such that for every x ∈ A and every σ(Y, G)-open half space L of Y with Φx ∈ L there exist n ∈ N and an open half space H of X such that x ∈ H ∩ An ⊂ Φ−1 (L). v) If {Dγ : γ ∈ Γ} is a discrete family of subsets of Y then Φ−1 (Dγ ) : γ ∈ Γ is σ-slicely isolatedly decomposable in A. Proof. From Theorem 4.16 we have that i)⇔ii)⇔v). Evidently ii)⇒iii). In order to complete the proof of the theorem it is enough to prove iii)⇒iv)⇒i). ∞ iii)⇒iv). Let {An,ε }n=1 verifying iii). We show that An,p−1 : n, p ∈ N veriﬁes iv). Indeed let L = {z ∈ Y : g(z) > a}, g ∈ SG , a ∈ R and Φx ∈ L. −1 There exists p ∈ N such that g(Φx) > a+ p . We can ﬁnd n ∈ N such that x ∈ An,p−1 . Pick f ∈ ∂p−1 g ◦ Φ x|An,p−1 and set H = z ∈ X : f (z) > f (x) + a + p−1 − g(Φx) .

96

4 σ-Slicely Continuous Maps

Since g(Φx) > a + p−1 we get x ∈ H. Moreover for any z ∈ H ∩ An,p−1 we have g(Φz) ≥ g(Φx) + f (z − x) − p−1 > a . iv)⇒i). Since Y is LUR and G is 1-norming from Lemma 4.28 it follows

that given ε > 0 we can write Y = n Yn,ε in such a way that for every y ∈ Yn,ε there is an open half-space L of (Y, σ(Y, G)) with y ∈ L ∩ Yn,ε and diam (L ∩ Yn,ε ) < ε. If x ∈ Φ−1 (Yn,ε ) and L is an open half space of (Y, σ(Y, G)) such that Φx ∈ L ∩ Yn,ε and diam (L ∩ Yn,ε ) < ε the condition iv) gives us an integer m and an open half space H in X with x ∈ H ∩ Am ⊂ Φ−1 (L). Evidently we have that x ∈ H ∩ Am ∩ Φ−1 (Yn,ε ) and oscH∩Am ∩Φ−1 (Yn,ε ) Φ < ε . −1 (Yn,ε ) for Consequently if Am n,ε is the set made up by all x ∈ Am ∩ Φ which there exists an open half space H, with x ∈ H, and

we have A =

oscH∩Am ∩Φ−1 (Yn,ε ) Φ < ε ,

m,n

Am n,ε . So Φ is σ-slicely continuous.

Remark 4.30. Every bounded linear operator T : X → Y satisﬁes iii) or iv) above, thus T is always σ-slicely continuous if the range space Y is LUR renormable. We use that Y is LUR only for the proof of the implication iv)⇒i). It is essential since for a normed space which is not LUR renormable the identity map satisﬁes iii) and iv) but it does not satisfy i). We have also Theorem 4.31. Let A be a σ-bounded subset of a locally convex linear topological space X and let Y be a LUR space. The map Φ : A →

Y is σ-slicely continuous if, and only if, for every ε > 0 we can write A = n An,ε in such a way that for all x ∈ An,ε ∂ε ϕx (x|An,ε ) = ∅ ,

(4.58)

where ϕx (u) = Φu + Φx, u ∈ A. Proof. Let Φ be a σ-slicely continuous map. According to Theorem 4.29 given ∞ ε > 0 we can write A = n=1 An,ε in such a way that for all x ∈ An,ε and every Lipschitz 1 function g : ΦA → Y we have ∂ε g ◦ Φ (x|An,ε ) = ∅ .

(4.59)

Taking into account that g(v) = v + Φx is a 1-Lipschitz function deﬁned in A and ϕx = g ◦ Φ we get (4.58) from (4.59). Assume now that Φ satisﬁes (4.58). Pick ε > 0. According to Lemma 4.28 there exists a decomposition

4.3 Characterization of σ-Slicely Continuous Maps

A=

97

Anε

n

and δn > 0 such that Φx − Φy < ε whenever x, y ∈ Anε and Φx − (Φx + Φy)/2 < δn .

m Fix n ∈ N. We can write A = m Am n in such a way that for every x ∈ An there exists a continuous linear functional f on X such that for all y ∈ Am n Φx + Φy ≥ 2Φx + f (y − x) − δn . Let now y ∈ Anε ∩ Am n and f (y) > f (x) − δn . Then we have Φx + Φy > 2Φx − 2δn . Hence Φx − Φy < ε. So oscS Φ ≤ 2ε where S =

{y ∈ Anε ∩ Am n : f (x − y) < δn }. When A is a norm open convex subset we can apply Theorem 4.31 whenever ϕx (u) = Φu + Φx is locally bounded and convex at x since in this case ∂ϕx (x|A) = ∅ (see e.g. [Phe93, p. 6]) and so we arrive to the following Corollary 4.32. Let A be a norm open convex subset of a normed space X and Φ a locally bounded map from A into a LUR normed space Y such that (4.60) 2Φx ≤ λi Φx + Φxi whenever x, xi ∈ A, λi > 0,

&

λi = 1, x =

&

λi xi , in particular if

Φ(λx + µy) ≤ λΦx + µΦy ,

(4.61)

whenever x, y ∈ A, λ, µ ≥ 0, λ + µ = 1. Then Φ is σ-slicely continuous. In the case when Y is a LUR lattice (4.61) is fulﬁlled if Φ is a lattice positive convex map, i.e. 0 ≤ Φ(λx + µy) ≤ λΦx + µΦy

(4.62)

for all x, y ∈ A, λ, µ ≥ 0, λ + µ = 1. Proof. First we show that (4.61) implies (4.60). Indeed set ψ(x) = Φx. From (4.61) we get that ψ is convex and

Φx + Φy

, x, y ∈ A .

(4.63) ψ ((x + y)/2) ≤

2 & & Let x = λi xi , λi > 0, λi = 1, xi ∈ A. Taking into account that ψ is convex from (4.63) we get x + xi x + xi Φx + Φxi λi Φx = ψ . ≤ λi ψ ≤ λi 2 2 2 So it is proved that (4.61) implies (4.60).

98

4 σ-Slicely Continuous Maps

Now we show that Φ is σ-slicely continuous provided Φ satisﬁes (4.60). Pick x ∈ A and set ϕ(u) = Φx + Φu, u ∈ A. Let ϕ˜ be from Deﬁnition 4.18. Since for all u ∈ A, 0 ≤ ϕ(u) ˜ ≤ ϕ(u) and Φ is locally bounded we get that ϕ˜ is locally bounded on A. In order to apply Lemma 4.19 we will show that ϕ˜ is weakly lower semicontinuous at x. Since ϕ˜ is convex it is enough to prove that ϕ˜ is norm lower semicontinuous at x. In turn for this it suﬃces to show that there exist δ > 0 and L > 0 such that ϕ(x) ˜ − ϕ(y) ˜ ≤ Lx − y

(4.64)

for all y ∈ A with x − y < δ. Indeed since ϕ˜ is locally bounded and A is norm open there exist δ > 0 and M > 0 such that the closed ball B(x, δ) centered in x with radius δ is included in A and ϕ(y) ˜ ≤ M for all y ∈ B(x, δ). Pick y ∈ B(x, δ) and set z = x + δ(x − y)/d where d = x − y. Clearly z ∈ B(x, δ). Since d δ y+ z x= d+δ d+δ is a convex combination of y and z we have ϕ(x) ˜ ≤ Hence ϕ(x) ˜ − ϕ(y) ˜ ≤

d δ ϕ(y) ˜ + ϕ(z) ˜ . d+δ d+δ

d M (ϕ(z) ˜ − ϕ(y)) ˜ ≤ x − y . d+δ δ

So (4.64) is proved. Now from Lemma 4.19 we get ∂ε ϕ(x|A) ˜ = ∅. From (4.60) we get that ϕ(x) ˜ = ϕ(x). This and the fact that ϕ(u) ≥ ϕ(u), ˜ for all u ∈ A imply ∂ε ϕ(x|A) ⊃ ∂ε ϕ(x|A). ˜ Hence ∂ε ϕ(x|A) = ∅. From Theorem 4.31 we get that Φ is σ-slicely continuous.

Remark 4.33. Actually from the conditions of the above corollary it follows that ∂ϕ(x|A) = ∅. In [DGL81] it is shown that a Banach lattice Y admits an equivalent LUR norm if and only if Y is order continuous. Any order continuous Banach lattice can be presented as a direct sum of Banach spaces of measurable functions with respect to some probabilistic measure (see e.g. [LT77, Chap. I]). So in this case the positive cone of Y has a natural description. We mention that in general the condition (4.62) is stronger than (4.61). Taking into account that Day’s norm in c0 (Γ) is a lattice LUR norm (see Proposition 3.38) from Corollary 4.32 and Theorem 4.29 it follows the proof of Corollary 1.21 in the introduction:

4.3 Characterization of σ-Slicely Continuous Maps

99

Corollary 4.34. i) Let Φ be a locally bounded map from a normed space X into c0 (Γ) for some Γ such that for every γ ∈ Γ the real function δγ ◦ Φ on X is non-negative and convex, where δγ is the Dirac measure on Γ at γ. Then Φ is σ-slicely continuous. ii) Let Φ be a map from a normed space X into c0 (Γ) for some Γ such that for every γ ∈ Γ, δγ ◦ Φ is non-negative, convex and σ(X, F ) lower semicontinuous for some subspace F of X ∗ which separates the points of X then Φ is σ-slicely continuous in (X, σ(X, F )). Proof. i) It follows directly from Corollary 4.32. ii) Let · be a lattice LUR norm on c0 (Γ). Assume that for all γ ∈ Γ, δγ ◦ Φ is non negative, convex and σ(X, F ) lower semicontinuous. Given x ∈ A ⊂ X we show that ϕ(u) = Φu + Φx is σ(X, F ) lower semicontinuous on A provided ΦA is bounded. Pick u, uα ∈ A with σ(X, F ) − lim uα = u. α

Set y = Φu + Φx , yα = Φuα + Φx. ∗

Then for every h ∈ (c0 (Γ), · ) , h ≥ 0, we have lim inf h (yα ) ≥ h(y) . α

Pick

(4.65)

∗

f ∈ (c0 (Γ), · ) , f = 1 , f (y) = y . Set h = f ∨ 0. Since · is a lattice norm and y ≥ 0 we have h ≤ f ≤ 1 , h(y) ≥ f (y) . Then from(4.65) it follows lim inf ϕ (uα ) ≥ lim inf h (yα ) ≥ h(y) ≥ f (y) = ϕ(u) . α

α

So ϕ is σ(X, F ) lower semicontinuous. Since δγ ◦ Φ is non-negative and convex we get that Φ satisﬁes (4.62). Following the proof of Corollary 4.32 we get that Φ satisﬁes (4.60) . Hence ϕ is convex at x. From Lemma 4.19 we get ∂ε ϕ(x|A) = 0 for all ε > 0. Now from Theorem 4.31 it follows that Φ is σslicely continuous in (X, σ(X, F )).

5 Some Applications

5.1 Three Space Property for LUR Renorming Let us remember that given two Banach spaces X, Y and a bounded linear operator T from X onto Y there exists a continuous (nonlinear in general) mapping B of Y into X such that T By = y for every y ∈ Y [DGZ93, p. 299]. The operator B is called the Bartle-Graves selector of T −1 . We start with the following Theorem 5.1. Let Z be a subspace of a Banach space X and W = X/Z. Assume that Z and W admit LUR norms which are lower semicontinuous with respect to the topologies σ(Z, F ) and σ(W, G) where F and G are subspaces of X ∗ and W ∗ respectively. Let Q be the canonical quotient map from X onto W and B be the Bartle-Graves norm-to-norm continuous selector of the multivalued map Q−1 . Then the map Φ given by the formula Φx = (x − BQx, Qx)

(5.1)

is a norm-to-norm homeomorphism from X into Z × W which is σ(X, E) σ-slicely continuous where E = F + Q∗ G. In particular X admits a σ(X, E) lower semicontinuous LUR norm. The applications of the above theorem are based on the following simple and well known fact. Lemma 5.2. Let T be a bounded linear operator from a Banach space X onto a Banach space Y . Let Q be the canonical quotient map from X onto X/Ker T . Then there exists an isomorphism U : X/Ker T → Y such that U Qx = T x for all x ∈ X. K. Ciesielski and R. Pol gave an example of a compact space K with K = ∅ for which there is no bounded linear one-to-one map from C(K) into c0 (Γ) for any Γ, (see e.g. [DGZ93, p. 261]). However C(K) is LUR renormable, A. Molt´ o et al., A Nonlinear Transfer Technique for Renorming. Lecture Notes in Mathematics 1951, c Springer-Verlag Berlin Heidelberg 2009

101

102

5 Some Applications

[Dev86], (see also [DGZ93, p. 305]). This example showed that there exist LUR renormings that cannot be obtained from a “linear” transfer from some c0 (Γ). Nevertheless it turns out that for this space there exists an “enough good map” that is “almost linear”. The formula (5.1) gives an easy way to construct a norm-to-norm homeomorphism Φ from C(K) into c0 (K) which is σ-slicely continuous. Indeed taking into account that K = ∅ and setting in the above theorem X = C(K), Z = c0 (K \ K ) we get X/Z is isometric to C(K ) which is isomorphic to c0 (K ). So Z × (X/Z) is isomorphic to c0 (K). From Theorem 5.1 we get a new proof for the three space problem for LUR renorming [GTWZ85] (see also [DGZ93, p. 299], [JM97]). A particular case of the three space problem for LUR renorming was done in [Alex83]. In [JM97] it was proved that if X/Z is LUR renormable and Z has a Kadec norm then X has a Kadec norm too. In [Alex89] it was shown that if X/Z is LUR renormable and Z is MLUR renormable then X is MLUR renormable. In [Rib87] it was proved that if X/Z and Z are σ-fragmentable then X is σ-fragmentable too. In all these results the formula (5.1)) is somehow used. In order to prove Theorem 5.1 we need the following assertion which is a corollary of Theorem 4.29. Lemma 5.3. Let X be a normed space and let E be a subspace of X ∗ which separates points of X. Let Y1 be a subspace of a Banach space Y such that Y1 admits a σ (Y1 , G)-lower semicontinuous LUR norm for some subspace G of Y ∗ . Let Ψ : X → Y be a map which is σ(X, E) σ-slicely continuous and let T : X → Y be a bounded linear operator such that Φ = Ψ + T maps X into Y1 . Then Φ is σ(X, E) σ-slicely continuous provided T ∗ G ⊂ E. In particular every bounded linear operator U : X → Y1 with U ∗ G ⊂ E is σ(X, G) σ-slicely continuous. Proof. We shall consider in this proof X as a linear topological space endowed with the topology

σ(X, E). According to Theorem 4.29 for every ε > 0 we can write X = n Xn,ε in such a way that for all x ∈ Xn,ε and g ∈ BY ∗ we have ∂ε g ◦ Ψ (x|Xn,ε ) = ∅. Let us mention that for the proof of the implication i)⇒iii) in Theorem 4.29 we do not need that Y is LUR. Since T is a bounded linear operator we have for x ∈ D ⊂ X and ε > 0 that ∂ε g ◦ Φ (x|D) = ∂ε g ◦ Ψ (x|D) + T ∗ g.

(5.2)

Taking into account that T ∗ G ⊂ E and that E is linear we get from (5.2) that for all g ∈ BG∗ ∂ε g ◦ Φ (x|Xn,ε ) = ∅. (5.3) Having in mind Lemma 4.7 without loss of generality we may assume that the norm in Y1 is LUR and G is 1-norming. Then we get from Theorem 4.29 that Φ is σ-slicely continuous.

5.2 The Weakly Countably Determined Case

103

Proof of the Theorem 5.1 For all w ∈ W we have QBw = w. Hence for all x ∈ X we get QBQx = Qx. (5.4) Set Y = X × W, Y1 = Z × W, Y2 = {(0, w) ∈ Y : w ∈ W }, G2 = {(0, g) ∈ Y ∗ : g ∈ G} . We deﬁne T : X → Y, U : X → Y2 , Ξ : Y2 → Y, Ψ : X → Y, Φ : X → Y1 by the formulae T x = (x, Qx), U x = (0, Qx) , Ξ (0, w) = (−Bw, 0) , Ψ = ΞU, Φ = T + Ψ. We have Φx = T x + ΞU x = (x − BQx, Qx). Since Q(x − BQx) = 0 we get x − BQx ∈ Z. So Φx ∈ Y1 . Since Y2 admits a σ (Y2 , G2 ) lower semicontinuous LUR norm and U is a bounded linear operator with U ∗ G2 ⊂ E we get from Lemma 5.3 that U is σ(X, E) σ-slicely continuous. Since Ξ is norm continuous from Proposition 1.14 and Corollary 2.42 we obtain that Ψ is σ(X, E) σ-slicely continuous. Since Y1 admits a σ (Y1 , F × G) lower semicontinuous LUR norm from Lemma 5.3 we get that Φ is σ(X, E) σ-slicely continuous. Evidently Φ is norm-to-norm continuous. We show that Φ−1 is normto-norm continuous too. We introduce in Y the norm | · | by the formula | · | = · + | · |, where · , | · | are the norms in X and W respectively. Fix x ∈ X and pick ε > 0. Since B is continuous we can ﬁnd δ > 0 such that BQx − Bw < ε whenever w ∈ W and |Qx − w| < δ. Let |Φx − Φy| = (x − BQx) − (y − BQy) + |Qx − Qy| < min(ε, δ). Then |Qx − Qy| < δ so BQx − BQy < ε . Hence x − y ≤ (x − BQx) + (y − BQy) + BQx − BQy < 2ε.

5.2 The Weakly Countably Determined Case In section 2.6 we studied the automatic co-σ-continuity of every linear continuous one-to-one map deﬁned on a weakly countably determined Banach space. More can be said and the inverse map is going to be σ-slicely continuous because of the following Theorem 5.4. Let X be a weakly countably determined Banach space and weak∗ = X ∗ . Then the let F ⊂ X ∗ a separating points subspace of X ∗ , i.e. F identity map Id : (X, σ(X, F )) → (X, · ) is σ-slicely continuous.

104

5 Some Applications

Proof. We shall use the fact that X is LUR renormable to reduce the proof to show that there exists a sequence {An : n ∈ N} of subsets of X such that for every open half space H of X and x ∈ H there exists a σ(X, F )-open half space L and an integer p with x ∈ L ∩ Ap ⊂ H (Theorem 4.29.(iv)). Let {Kn : n ∈ N} be a sequence of weak∗ compact subsets of X ∗∗ which countably determines X in X ∗∗ (see deﬁnitions in Sect. 2.6). Given H := {u ∈ X : g(u) > µ},

f ∈ X∗

and x ∈ X with g(x) > µ + δ, δ > 0, we select σ ∈ Σ ⊂ NN so that x ∈ ∞ weak∗ = X the Mackey-Arens Theorem (see e.g. i=1 Kσ(i) ⊂ X. Since F [HHZ96, p. 164]) ensure us that for some g ∈ F we will have ∞ Kσ(i) < δ/2. (5.5) sup |f (y) − g(y)| : y ∈ i=1

We claim that there exists an integer p such that δ x ∈ Kσ(1) ∩ Kσ(2) ∩ . . . ∩ Kσ(p) ∩ u ∈ X : g(u) > µ + ⊂ H. 2 Indeed, if it is not the case we could select a sequence xn ∈ Kσ(1) ∩ Kσ(2) ∩ . . . ∩ Kσ(n) with g (xn ) > µ + δ/2 and f (xn ) ≤ µ, n ∈ N. Since {xn : n ∈ N} should have ∞ a cluster point y for the weak topology with y ∈ i=1 Kσ(i) we arrive to a contradiction since g(y) ≥ µ + δ/2 and f (y) ≤ µ, so |g(y) − f (y)| ≥ δ/2 but ∞

y ∈ i=1 Kσ(i) and (5.5) must be true. Corollary 5.5. If X is a weakly countably determined Banach space and T : X → Y is a one-to-one linear continuous map with values in the normed space Y then T −1 : T X → X is σ-slicely continuous. Corollary 5.6. (Fabian [Fab91], [DGZ93, Theorem 2.3, p. 290]) If a dual Banach space X ∗ is WCD then X ∗ admits an equivalent dual LUR norm. Proof. The identity map from (X ∗ , weak∗ ) into (X ∗ , · ) is σ-slicely continuous and X is a norming subspace of X ∗∗ so X ∗ admits an equivalent

weak∗ -lower semicontinuous LUR norm.

5.3 Product of σ-Slicely Continuous Maps It is well known that Y = c0 (Γ) Yγ is LUR renormable provided {Yγ }γ∈Γ is a family of LUR renormable normed spaces. The next assertion is a generalization of this result.

5.3 Product of σ-Slicely Continuous Maps

105

Theorem 5.7. Let A be a σ-bounded subset of a linear topological space X, let {Yγ · γ }γ∈Γ be a family of normed spaces and Y = c0 (Γ) Yγ . Let Ψ : A → c0 (Γ), Ψγ : A → Yγ , γ ∈ Γ, ∈ ∞ (Γ) for every x ∈ A. be σ-slicely continuous maps such that Ψγ xγ γ∈Γ

Then the map Φ : A → Y deﬁned by the formula Φx = (Ψx(γ)Ψγ x)γ∈Γ is σ-slicely continuous too. If X is a normed space, Yγ ⊂ X for γ ∈ Γ and for all x ∈ A we have ·

x ∈ span {Ψγ x : γ ∈ supp Ψx}

then Φ is co-σ-continuous.

Proof. For y = {yγ }γ∈Γ ∈ Y we set |y|∞ = supγ yγ γ . Without loss of generality we assume that set, A is a bounded set Γ is a well ordered and for every x ∈ A sup Ψγ xγ : γ ∈ Γ ≤ 1. For a ﬁxed γ we have that ψγ (x) = Ψx(γ) is a real function deﬁned on A and therefore ψγ is σ-slicely continuous. Then the map Φγ : A → Yγ deﬁned by the formula Φγ x = ψγ (x)Ψγ x = Ψx(γ)Ψγ x is σ-slicely continuous as a product of two σ-slicely continuous maps. (See Lemma 4.22.) Fix ε > 0. For x ∈ A we set L(|Ψx|) = {γ ∈ Γ : |Ψx(γ)| ≥ ε}. Since Φγ is σ-slicely continuous, from Lemma 4.21 it follows that for ∈ N we can write ∞

A=

Am,γ

m=1

in such a way that, for every x ∈ Am,γ there exists a linear continuous

functional hx, ,m,γ on X such that hx, ,m,γ (x) +

1 ≥ sup hx, ,m,γ Am,γ

(5.6)

and Φγ x − Φγ uγ < ε whenever u ∈ Am,γ and hx, ,m,γ (x − u) < 1.

For x ∈ A we deﬁne = (x) ∈ N, p = p(x) ∈ N, r = r(x) ∈ N, mi = mi (x) ∈ N, γi = γi (x) ∈ Γ, i = 1, 2, . . . , in such a way that

sup {|Ψx(γ)| : γ ∈ / L (|Ψx|)} < ε − 1/p, #L (|Ψx|) = , L (|Ψx|) = {γi }1 , mi ,γi γ1 < γ2 < . . . < γ , x ∈ A

and

(5.7) sup hx, ,mi ,γi ≤ r A i=1

For , p, r ∈ N, m =

(mi )1

∈ N we set

Am

,p,r = {x ∈ A : = (x), p = p(x), r = r(x), mi = mi (x), i = 1, 2, . . . , } . Evidently A = {x ∈ A : L (|Ψx|) = ∅} ∪

Am

,p,r : , p, r ∈ N, m ∈ N

.

106

5 Some Applications

Fix , p,r ∈ N, m ∈ N . Since Ψ is σ-slicely continuous from Lemma 4.21 it follows that we can write ∞

Am

,p,r =

Ak,m

,p,r

k=1

in such a way that for every x ∈ Ak,m

,p,r there exists a linear continuous functional gx on X such that gx (x) +

1 ≥ sup gx 8r Ak,m

(5.8)

,p,r

and Ψx − Ψu∞ < 1/p whenever u ∈ Ak,m

,p,r and gx (x − u) < 1. Fix now x ∈ Ak,m

,p,r . Then = (x), p = p(x), r = r(x), mi = mi (x), γi = γi (x),

i = 1, 2, . . . , , γ1 < γ2 < . . . < γ and L (|Ψx|) = {γi }1 . Set hx =

hx, ,mi ,γi ,

fx = gx +

i=1

1 hx . 3r

Pick u ∈ Ak,m

,p,r with fx (x − u) <

1 . 8r

From (5.6) we get 1 hx, ,mi ,γi (x − u) > − , This implies hx, ,mi ,γi (x − u) −

i = 1, 2, . . . , .

−1 < hx (x − u).

From (5.8) we get hx, ,mi ,γi (x − u) < hx (x − u) + < 3r

−1 −1 = 3r (fx (x − u) − gx (x − u)) +

1 1 + 8r 8r

+

−1 < 1.

Hence Φγi x − Φγi uγi < ε.

(5.9)

For γ ∈ / L (|Ψx|) = L (|Ψu|) we have Φγ xγ = |Ψx(γ)| Ψγ xγ < ε,

Φγ uγ < ε .

This and (5.9) imply |Φx − Φγ u|∞ < 2ε. So Φ is σ-slicely continuous.

5.3 Product of σ-Slicely Continuous Maps

107

Now let X be a normed space let Yγ ⊂ X for γ ∈ Γ and ·

x ∈ span {Ψγ x : γ ∈ supp Ψx}

. ·

We will show that Φ is co-σ-continuous. Set Zx = span {Ψγ x : γ ∈ supp Ψx} . Evidently Zx is separable, and supp Φx ⊂ supp Ψx. Since Ψγ x = 0 whenever ·

γ ∈ supp Ψx\supp Φx we get Zx = span {Ψγ x :

γ ∈ supp Φx} . Let xn , x ∈ A and |Φxn − Φx|∞ → 0. Then supp Φx ⊂ n supp Φxn . Hence x ∈

· Zx ⊂ span n Zxn . From Theorem 1.7 we get that Φ is co-σ-continuous.

Corollary 5.8. Let {Yγ , · γ }γ∈Γ be a family of normed spaces and Y = c0 (Γ) Yγ . Assume that for every γ ∈ Γ the norm · γ is σ (Yγ , Fγ ) lower semicontinuous LUR where Fγ is a subspace ofYγ∗ . Then Y admits a σ(Y, F ) lower semicontinuous LUR norm where F = 1 (Γ) Fγ . ∈ c0 (Γ), Pγ y = yγ , Proof. For y = {yγ }γ∈Γ ∈ Y we set Ψy = yγ γ γ∈Γ

Ψγ y = ψγ (y)Pγ y where

ψγ (y) =

−1

yγ

if yγ = 0 ;

0

if yγ = 0.

From Corollary 4.34 we get that the map Ψ : Y → c0 (Γ) is σ(Y, F ) σ-slicely continuous. Now we shall show that Ψγ : Y → Yγ is σ(Y, F ) σ-slicely continuous too for every γ ∈ Γ. Indeed taking into account that ψγ is a real valued function we get that it is σ(Y, F ) σ-slicely continuous. Since ·γ is a σ (Yγ , Fγ ) lower semicontinuous LUR norm, from Corollary 2.42 we get that Pγ : Y → Yγ is σ(Y, F ) σ-slicely continuous. Hence Ψγ = ψγ Pγ is σ(Y, F ) σ-slicely continuous as a product of two σ(Y, F ) σ-slicely continuous maps (see Lemma 4.22). Since for the identity map IdY we have IdY y = {Ψy(γ)Ψγ y}γ∈Γ from the previous proposition we get that IdY is σ(Y, F ) σ-slicely continuous. Hence Y admits a σ(Y, F ) lower semicontinuous LUR norm.

The following corollary is a nonlinear version of the main tool for LUR renorming C0 (Υ) where Υ is a tree in which this renorming is possible [Hay99, Proposition 4.2.]. Corollary 5.9. Let L be a locally compact space and let {Uγ }γ∈Γ be a family of clopen subsets of L such that, for each γ, there is an equivalent LUR norm · γ on C0 (Uγ ). Let Ψ : C0 (L) → c0 (Γ) be a σ-slicely continuous map such that for every x ∈ C0 (L) supp x ⊂ {Uγ : γ ∈ supp Ψx} then there is an equivalent LUR norm on C0 (L).

108

5 Some Applications

Proof. Following Theorem 5.7 let Ψγ : C0 (L) → C0 (Uγ ) deﬁned by Ψγ x = continuous. x Uγ for γ ∈ Γ. Since C0 (Uγ ) is LUR renormable Ψγ is σ-slicely

∈ ∞ (Γ) for Without loss of generality we may assume that x Uγ γ γ∈Γ

every x ∈ C0 (L). We have that Φ deﬁned as in Theorem 5.7 is σ-slicely continuous. Now we prove that Φ is co-σ-continuous. For every x ∈ C0 (L) and γ ∈ Γ we can ﬁnd compact and open sets Ux,m,γ ⊂ Uγ , m ∈ N, such that limm x1lUγ \Ux,m,γ ∞ = 0 Let Zx be the algebra generated by x, 1lUx,m,γ , m ∈ N and γ ∈ supp Ψx. Clearly Zx is separable, x ∈ Zx , and x ∈ span

{Zxn : n ∈ N}

·∞

whenever limn |Φxn − Φx| = 0. According to Theorem 1.7 Φ is co-σcontinuous.

Another consequence of Theorem 5.7 is a result of Zizler [Ziz84] which in turn has more applications, see for instance [JNR95]. Corollary 5.10. Let X be a normed space and let Tγ : X → X,γ ∈ Γ, a family bounded linear operators with the following properties: i) For every x ∈ X we have (Tγ x)γ∈Γ ∈ c0 (Γ); ii) for every x ∈ X we have x ∈ span· {Tγ x : γ ∈ Γ}; iii) For every γ ∈ Γ the normed space Tγ X admits an equivalent LUR norm. Then X admits an equivalent LUR norm. Proof. Since for every γ ∈ Γ the function ψγ (x) = Ψx(γ) = Tγ x is non-negative and convex, from Corollary 1.21.i it follows that Ψ is a σ-slicely continuous map. Applying now Theorem 5.7 with Ψλ := Tλ , we get that X is LUR renormable because the map Φ is going to be σ-slicely and co-σ continuous with a LUR renormable range space.

A well known criterion for the invertibility of a linear continuous map T : a Banach space is that Id − T < 1 because the series &∞ X → X on n (Id − T ) does converge in norm of operators to T −1 (von Neumann n=0 criterion, see [FHHMPZ01, p. 210]). In that case T is an isomorphism and it will be σ-slicely continuous if, and only if, X admits an equivalent LUR norm. We shall now consider the question of σ-slicely continuity of the identity map when we have a “close” σ-slicely continuous map Φ : X → X in the sense that (Id − Φ)n goes to zero when n goes to ∞, so we will have a nonlinear version of the von Neumann criterion for our σ-slicely continuous maps. Actually the following result goes back implicitly in the proof of LUR renormability of Banach spaces C(K) where K is a σ-discrete compact, i.e. C(K)∗ has an equivalent dual LUR norm [Raj02], by R. Haydon [Hay].

5.3 Product of σ-Slicely Continuous Maps

109

Theorem 5.11. Let X be a normed space and let F ⊂ X ∗ a norming subspace for it. Let Φ : (X, σ(X, F )) → X a σ-slicely continuous map such that 0 ∈ {(Id − Φ)n x : n ∈ N}

weak

, for all x ∈ X.

Then the identity map from (X, σ(X, F )) into X is σ-slicely continuous and consequently X admits an equivalent σ(X, F )-lower semicontinuous LUR norm. Proof. Let us consider S(X) := {Ψ : X → X : Ψ is σ-slicely continuous from (X, σ(X, F )) to (X, · )} that we know it is non-void since Φ ∈ S(X). Given Ψ ∈ S(X) we know that Ψ ∈ S(X) and it veriﬁes (i)⇒(ii)⇒(iii) of Theorem 4.29 since for these implications the LUR condition is not used at all. Since the identity map on X clearly veriﬁes (iii) we have that (Id − Ψ) veriﬁes (iii) of Theorem 4.29 too. For the implication (iii)⇒(iv) again the LUR condition is not necessary at all and we already have that (Id − Ψ) veriﬁes (iv) of Theorem 4.29 with σ(X, F )-open half spaces in both sides. If we apply now Corollary 2.42 we see that Φ ◦ (Id − Ψ) is σ-slicely continuous too and we can deﬁne a map S : S(X) → S(X) by S(Ψ) := Ψ + Φ(Id − Ψ) because the sum of σ-slicely continuous maps is σ-slicely continuous too by Lemma 4.22. We can now iterate the map S beginning with Φ and observe that if we call the iterates Φ0 := Φ, Φ1 := S(Φ), Φ2 := S(S(Φ)) = S (Φ1 ) = Φ1 + Φ(Id − Φ1 ), . . . , Φn+1 := S n+1 (Φ) = S (Φn ) = Φn + Φ (Id − Φn ) , . . . then we have (Id − Φn ) = (Id − Φ)

n+1

, n ∈ N.

Indeed for n = 0 it is clear, and if we assume (Id − Φn ) = (Id − Φ)n+1 then we have Id − Φn+1 = Id − S (Φn ) = Id − (Φn + Φ ◦ (Id − Φn )) = = Id − Φn − Φ ◦ (Id − Φn ) = (Id − Φ) ◦ (Id − Φn ) = = (Id − Φ) ◦ (Id − Φ)

n+1

= (Id − Φ)

n+2

.

110

5 Some Applications

Therefore our hypothesis tells us 0 ∈ {(Id − Φn ) x : n ∈ N}

weak

, for all x ∈ X,

and consequently x ∈ span {Φn x : n ∈ N}

·

, for all x ∈ X.

Since Φn ∈ S(X), n ∈ N we have Id ∈ S(X) by Corollary 4.23 and the proof is over.

Remark 5.12. The same proof of Theorem 5.11 shows that its statement holds if we replace Φ by a sequence Φm , i.e. Let X be a normed space and let F ⊂ X ∗ a norming subspace for it. Let Φm : (X, σ(X, F )) → X, m ∈ N, σ-slicely continuous maps such that n

0 ∈ {(Id − Φm ) x : m, n ∈ N}

weak

, for all x ∈ X.

Then the identity map from (X, σ(X, F )) into X is σ-slicely continuous and consequently X admits an equivalent σ(X, F )-lower semicontinuous LUR norm. As a consequence we can deduce the following Theorem 5.13. (Haydon-Rogers [HR90]) Let K be a compact space such that K (ω1 ) = ∅ then C(K) admits an equivalent pointwise-lower semicontinuous LUR norm. Proof. We know that there exists β < ω1 such that K (β) is ﬁnite. Moreover for any γ < β we have that the points of K (γ) \ K (γ+1) are isolated in K (γ) therefore for each t ∈ K (γ) \ K (γ+1) we can choose a clopen set Vt such that Vt ∩ K (γ) \ K (γ+1) = {t}. Fix an ε > 0. We are going to deﬁne a (nonlinear) map Φ = Φε , Φ : C(K) → C(K). For this take x ∈ C(K) and let us consider Λ(x, ε) := γ < β : x K (γ+1) ⊂ (−ε, ε) and x K (γ) ⊂ (−ε, ε) . If Λ(x, ε) = ∅ we have x(K) ⊂ (−ε, ε) and we set Φx = 0. Suppose now that x (K) ⊂ (−ε, ε) then Λ(x, ε) = ∅ and it makes sense to consider α = α(x, ε) := min Λ(x, ε). Any point t of accumulation of the set t ∈ K (α) : x(t) ∈ / (−ε, ε) must be contained in K (α+1) and must satisfy x(t) ∈ / (−ε, ε). Then by the

5.4 Totally Ordered Compacta

111

choice of α we have that the accumulation of the set Lε (|x|) ∩ K (α) = / (−ε, ε) must be empty so the set is ﬁnite. t ∈ K (α) : x(t) ∈ Set Φx :=

x(t)1lVt .

t∈Lε (|x|)∩K (α)

Since Φx is zero in K (α) \ Lε (|x|) we have that either Lε (|x − Φx|) = ∅ or α((Id − Φ)x, ε) < α(x, ε). Then the sequence α((Id − Φ)n x, ε) is decreasing so it must be eventually constant, i.e. there exists m ∈ N such that ((Id − Φ)n x)(K) ⊂ (−ε, ε). The same proof as in Proposition 4.3, p. 75, gives that the map Φ is σ-slicely continuous . Now to ﬁnish the proof it is enough to apply Remark 5.12

to the sequence Φm = Φ1/m . Remark 5.14. Recently R. Haydon [Hay] has proved the following impressive result: X is LUR renormable whenever X ∗ has an equivalent dual LUR norm. (5.10) He ﬁrst showed that C(K) has such a norm when K is σ-discrete. Finally R. Haydon proved that for a compact space (K, T ) with a lower semicontinuous metric on it such that the identity map from (K, T ) → (K, ) is σ-continuous we have that C(K) admits an equivalent Tp -lower semicontinuous LUR norm (which implies (5.10)).

5.4 Totally Ordered Compacta Let L and M be totally ordered spaces. In this section by L×M we denote their lexicographic product, i.e. if (x1 , x2 ), (y1 , y2 ) ∈ L × M we say that (x1 , x2 ) < (y1 , y2 ) if x1 < y1 or x1 = y1 and x2 < y2 . More generally, let {Lγ }0≤γ<µ be a ) transﬁnite sequence of totally ordered spaces by 0≤γ<µ Lγ we denote their ) lexicographical product i.e. if x = {xγ }0≤γ<µ , y = {yγ }0≤γ<µ ∈ 0≤γ<µ Lγ we say that x < y if there exists an ordinal γ0 such that xγ = yγ for γ < γ0 and xγ0 < yγ0 . By Lµ we denote the lexicographical product of µ copies of L. In a totally ordered space L we consider the order topology, i.e. the open intervals (u, v) containing x form a base of neighbourhoods of x. It is straight) forward to check that the lexicographical product 0≤γ<µ Kγ is compact provided {Kγ }0≤γ<µ is a transﬁnite sequence of totally ordered spaces which are compact in their order topology. Theorem 5.15. [HJNR00] Let µ be a countable ordinal and let {Kγ }0≤γ<µ be a transﬁnite sequence of compact totally ordered spaces such that C (Kγ ) admits a (pointwise lower semicontinuous) LUR norm for every γ ∈ [0, µ). ) admits a (pointwise lower semicontinuous) LUR Then C 0≤γ<µ Kγ norm too.

112

5 Some Applications

Corollary 5.16. [HJNR00] For any countable ordinal µ the space C ([0, 1]µ ) admits a pointwise lower semicontinuous LUR norm. In [Alex88] and [JNR95] independently the above corollary is proved in the case when µ is ﬁnite. In [HJNR00] is shown that it is necessary to require that µ is countable even for the existence of a LUR norm in C ([0, 1]µ ). First we consider the case of the lexicographical product of two totally ordered compacta. Proposition 5.17. [HJNR00] Let K and L be compact totally ordered spaces such that C(K) and C(L) admit (pointwise lower semicontinuous) LUR norms. Then C(K × L) admits a (pointwise lower semicontinuous) LUR norm too. Proof. We apply Theorem 5.1 and Lemma 5.2. Denote by 0L and 1L the least and the biggest elements of L respectively. Let C0 (L) be the subspace of C(L) consisting of all y ∈ C(L) such that y (0L ) = 0. Set * Y = C0 (L)k c0 (K)

i.e. Y is the c0 (K) sum of K identical copies of C0 (L). For x ∈ C(K × L), k ∈ K and ∈ L we set xk () = x(k, ) − x (k, 0L ) . Evidently xk ∈ C0 (L). We show that {xk }k∈K ∈ Y . Assume the contrary. ∞ Then there exists ε > 0 and a strictly monotone sequence {kn }1 in K which converges in the order topology to some s ∈ K such that xkn ∞ > ε. So ∞ there exists a sequence {n }1 of elements of L with xkn (n ) > ε. It is easy ∞ ∞ to see that the sequences {(kn , n )}1 and {(kn , 0L )}1 have the same limit in the order topology of the lexicographical product (s, t) which is (s, 0L ) if it is strictly increasing and (s, 1L ) if it is strictly decreasing. Then lim xkn (n ) = x (s, t) − x (s, t) = 0 n

we get a contradiction. Let us deﬁne the linear bounded operator T : C(K × L) → Y by the formula T x = {xk }k∈K . Claim. The operator T is onto. Proof. Pick y = {yk } ∈ Y with supk yk ∞ ≤ 1. For n ∈ N set Kn = k ∈ K : 2−n < yk ∞ ≤ 21−n . Fix n ∈ N. We construct xn ∈ C(K × L) such that xn ∞ ≤ 21−n , xn (k, )−xn (k, 0L ) = yk () for k ∈ Kn , ∈ L and xn (k, )−xn (k, 0L ) = 0 for k ∈ K \ Kn and ∈ L.

5.4 Totally Ordered Compacta

113

i

Indeed since Kn is ﬁnite we can write Kn = {ki,n }0n in such a way that k0,n < k1,n < . . . < kin ,n . For simplicity we suppose that k0,n = min K and kin ,n = max K. According to Urysohn’s Lemma for each i, 1 ≤ i ≤ in , there exists a continuous function zi,n on [ki−1,n , ki,n ] such that zi,n (ki−1,n

) = yki−1 (1L ), zi,n (ki,n ) = yki (0L ) = 0 and zi,n ∞ ≤ max yki−1 ∞ , yki ∞ ≤ 21−n . Now we deﬁne ⎧ ⎨ zi,n (k) if (k, ) ∈ [(ki−1,n , 1L ) , (ki,n , 0L )] , 0 ≤ i ≤ in ; xn (k, ) = ⎩ yki () if (k, ) = (ki , ) , 0 ≤ i ≤ in . It is easy to see that xn ∈ C(K × L). Setting x =

&∞ n=1

xn we get T x = y.

Set X = C(K × L), Z = Ker T . Evidently Z is isomorphic to C(K). From Lemma 5.2 it follows that there exists an isomorphism U : X/Z → Y such that U Qx = T x (5.11) for all x ∈ X, where Q is the canonical quotient map from X onto X/Z. For k ∈ K and ∈ L we deﬁne a linear functional gk, on Y by the formula gk, (y) = yk () where y = {yk }k∈K . Denote by G the subspace of Y ∗ generated by gk, , k ∈ K, ∈ L. Assume now that C(K) and C(L) admit pointwise lower semicontinuous LUR norms. According to Corollary 5.8 we get that Y admits a σ(Y, G)-lower semicontinuous LUR norm. Now taking into account (5.11) from Theorem 5.1 we get that X = C(K × L) admits a pointwise lower semicontinuous LUR norm.

Proof of Theorem 5.15 Assume that for each γ < µ the space ) C (Kγ ) admits a pointwise lower semicontinuous LUR norm. Set Lβ = 0≤γ<β Kγ for β ≤ µ. By transﬁnite induction we show that C (Lβ ) admits a pointwise lower semicontinuous LUR norm for every β ≤ µ. Fix β ≤ µ and assume that C (Lα ) admits pointwise lower semicontinuous LUR norm for every α < β. If β is not a limit ordinal we get from the former proposition that C (Lβ ) admits a pointwise lower semicontinuous LUR norm. Assume now that β is a limit ordinal. Set 0γ = min Kγ , 1γ = max Kγ , 0 = {0γ }γ<β , 1 = {1γ }γ<β and tα = {tα,γ }γ<β where 1γ if γ ≤ α; tα,γ = for α, γ < β. 0γ if γ > α.

114

5 Some Applications

We consider 0, 1, tα as elements of Lβ . We deﬁne the map Φα : C (Lβ ) → C (Lβ ) by the formula x(t) if t ≤ tα ; Φα x(t) = x (tα ) if t > tα . Set Cα = Φα C (Lβ ). Evidently Cα is a subspace of C (Lβ ) which is isometric to C [0, tα ]. Since [0, tα ] is homeomorphic to Lα we get that Cα is isometric to C (Lα ). So Cα has a pointwise lower semicontinuous LUR norm. Since Φα Cα = IdCα we get that Φα is σ-slicely continuous in (C (Lβ ) , pointwise). We have for all x ∈ C (Lβ ) Φα x − x∞ ≤ osc x [tα ,1] . Taking into account that limα→β tα = 1 in the order topology of Lβ we that for all x ∈ C (Lβ ) (5.12) lim Φα x − x∞ = 0. α→β

Since {Φα }α<β is a countable family of σ-slicely continuous maps in C (Lβ ) endowed with the pointwise topology, we get from Corollary 4.23 that C (Lβ ) admits a pointwise lower semicontinuous LUR norm.

Following [HJNR00] we study the LUR renormability of C(L), L totally ordered compacta also in a diﬀerent way. Deﬁnition 5.18. [HJNR00] Let L be a totally ordered space taken with its order topology. A family I of intervals of L is called a dyadic atomization of L if, for some ordinal µ, I has the form I = {I(γ) : 0 ≤ γ < µ}, where the families I(γ), 0 ≤ γ < µ, satisfy the conditions. a) I(0) = {L}. b) For each I in I, I is a closed interval [a, b] of c) If 0 ≤ γ < µ and [a, b] ∈ I(γ) with (a, b) c ∈ (a, b), I(γ + 1) contains the two intervals no other interval that meets (a, b). If 0 ≤ γ (a, b) = ∅ then [a, b] ∈ / I(γ + 1). d) If τ is a limit ordinal and

L with a < b. = ∅, then for some c with [a, c] and [c, b], but contains < µ and [a, b] ∈ I(γ) with

I(θ) ∈ I(θ) for 0 ≤ θ < τ, with I(0) ⊃ I(1) ⊃ . . . ⊃ I(θ) ⊃ . . . , then J(τ ) =

0 ≤ θ < τ,

{I(θ) : 0 ≤ θ < τ }

is either a single point, or τ < µ and J(τ ) belongs to I(τ ).

5.4 Totally Ordered Compacta

115

In any totally ordered space that is compact in its order topology there exists a dyadic atomization [HJNR00, Theorem 2.2.]. Deﬁnition 5.19. [HJNR00] A family I (i0 , i1 , . . . , in−1 ) , ij = 0 or 1, for 0 ≤ j < n, n ≥ 0, of closed non-empty intervals of a totally ordered space L, will be called a dyadic interval system in L, if I (i0 , i1 , . . . , in−1 , 0) and I (i0 , i1 , . . . , in−1 , 1) are disjoint subintervals of I (i0 , i1 , . . . , in−1 ) for n ≥ 0. Deﬁnition 5.20. [HJNR00] A function is called a decreasing interval function on the totally ordered space L, if is a real-valued function deﬁned on all the closed intervals I of L and (I) ≤ (J) when I ⊃ J. Theorem 5.21. [HJNR00] Let L be a totally ordered space that is compact in its order topology. If there is a bounded decreasing interval function which is constant on no dyadic interval system on L then there is an equivalent LUR norm on C(L). Proof. Let I = ∪ {I(γ) : 0 ≤ γ < µ} be a dyadic atomization of L. Let an interval function Without loss of generality we can and do assume that takes its values in the real interval [0, 1], (I) = 1 when I is a two-point interval and (I) ≤ 1/2 otherwise [HJNR00]. Let ([HJNR00]) σ(J) := (J) − sup {(I) : I ∈ I, I ⊃ J, I = J} , for J ∈ I and σ(L) = 0 ; J := {I ∈ I : σ(I) = 0} and Γ := J ∪ {∅}. Then let us consider the map T : C(L) → ∞ (Γ) deﬁned by T x(∅) = x(0) ; T x(J) = σ(J)(x(b) − x(a)), for J = [a, b] ∈ J .

(5.13)

In [HJNR00] it is shown that T C(L) ⊂ c0 (Γ). For x ∈ C(L) and δ > 0 let J (x, δ) the (ﬁnite) family of all the intervals J ∈ J such that T x(J)∞ > δ. Let Dx,δ be the set of all end-points of the intervals in J (x, δ). Fix now x ∈ C(L). In a previous version of [HJNR00] it is proved that For all ε > 0, there exists δ := δ(x, ε) > 0 such that

(5.14)

|x(a) − x(b)| < ε, whenever a, b ∈ L, (a, b) ∩ Dx,δ = ∅, and [a, b] ∈ / J (x, δ).

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5 Some Applications

Now set n(x,ε)

Dx,ε ∪ {0, 1} = {ai (x, ε)}i=0

where ai−1 (x, ε) < ai (x, ε), 1 ≤ i ≤ n(x, δ),

then |x(a) − x(b)| < ε ,

(5.15) for any a, b ∈ [ai−1 (x, ε), ai (x, ε)) , 1 ≤ i ≤ n(x, ε).

Set Ai (x, ε) = [ai−1 (x, ε), ai (x, ε)) for 1 ≤ i ≤ n(x, ε) and An+1 (x, ε) = an(x,ε) (x, δ) . Let Zx,ε := 1lAi (x,ε) : 1 ≤ i ≤ n + 1 . Let us rewrite (5.14) in terms of Zx,ε : m

Let {bj }j=0 , bj−1 < bj , 1 ≤ j ≤ m, such that {ai (x, ε) : 0 ≤ i ≤ n} ⊂ {bj : 0 ≤ j ≤ m} . Then from (5.14) we get dist x, span 1l{bn } , 1l[b0 ,b1 ) , 1l[b1 ,b2 ) , . . . , 1l[bn−1 ,bn ) < ε. (5.16)

Set Zx := p>0 Zx,1/p . Now we are going to deduce from Proposition 2.33 that ∞ T is co-σ-continuous. Let us consider a sequence {xn }n=1 , of elements of C(L), and x ∈ C(L) such that T xn − T x → 0. Given ε > 0 let δ = δ(x, ε) be from (5.14) and p ∈ N such that 1/p < δ. According to the choice of T in (5.13) there exists m ∈ N such that Dx,δ ⊂ Dx,1/p ⊂ Dxm ,1/p . Then from (5.16) it ·∞ follows that distance x, Zxn ,1/p < 1/k therefore x ∈ span {Zxn : n ∈ N} .

6 Some Open Problems

We have extensively considered here the use of Stone’s theorem on the paracompactness of metric spaces in order to build up new techniques to construct an equivalent locally uniformly rotund norm on a given normed space X. The discreetness of the basis for the metric topologies gives us the necessary rigidity condition that appears in all the known cases of existence of such a renorming property [Hay99, MOTV06]. Our approximation process is based on co-σ-continuous maps using that they have separable ﬁbers, see Sect. 2.2. We present now some problems that remain open in this area. Some of them are classical and have been asked by diﬀerent authors in conferences, papers and books. Others have been presented in schools, workshops, conferences and recent papers on the matter and up to our knowledge they remain open. The rest appear here for the ﬁrst time. We apologize for any fault assigning authorship to a given question. Rather than to formulate precise evaluation for the ﬁrst time the problems were proposed, our aim is to provide good questions for young mathematicians entering in the ﬁeld, we think they deserve all our attention to complete the state of the art in renorming theory.

6.1 More on the Nonlinear Transfer Technique for Renorming After the results presented in this monograph it comes up that the study of strictly convex, Kadec or midpoint locally uniformly rotund norms, [Hay99, MOTV01] will be of great interest in the near future. Let us remind the reader here that a normed space X (or the norm on it) is said to be midpoint locally uniformly rotund (MLUR) if the following assertion holds lim uk − vk = 0 k

whenever lim x − (uk + vk ) /2 = 0 , uk , vk ≤ x = 1 . k

A. Molt´ o et al., A Nonlinear Transfer Technique for Renorming. Lecture Notes in Mathematics 1951, c Springer-Verlag Berlin Heidelberg 2009

117

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6 Some Open Problems

Namely, we propose the following: Question 6.1. The study of nonlinear maps Φ : X −→ Y transferring a strictly convex (respectively a Kadec, or a MLUR) norm from a strictly convex normed space Y (respectively Kadec Y , or MLUR Y ) into X. For the case of strict convexity we refer to the recent paper [MOTZ07] where a linear topological characterization of the property of strictly convex renorming for Banach spaces is presented. For dual norms in spaces C(K)∗ , where K is a scattered compact space, and in particular for the compactiﬁcation of trees, a recent result has been obtained by R. Smith [Smi06, Smit07]. The corresponding problem for Kadec renormings is completely open. The main reason is that there is no example of a Banach with a σ-isolated network for the weak topology but without an equivalent Kadec norm, see Sect. 3.2. Moreover, in every descriptive normed space (X, · ) for every ε > 0 there exists a positively homogeneous function G : X −→ R such that · ≤ G(·) ≤ (1 + ε) · and the weak and norm topologies do coincide on the set {x ∈ X : G(x) = 1} , [Raj99]. There are still some open questions on the existence of additional properties of the function G or on the possibility to develop a convexiﬁcation process like in Sect. 4.2 to get an equivalent norm, in particular: Question 6.2. Let X be a Banach space and F a norming subspace of X ∗ such that the identity map Id : (X, σ(X, F )) −→ (X, · ) is σ-continuous. Which additional property of Raja‘s function G above can be deduced in this case? For instance, is it possible to convexify the construction to obtain that G veriﬁes the triangle inequality: G(x + y) ≤ G(x) + G(y) for all x, y ∈ X giving in this way an equivalent σ(X, F )-Kadec norm? A positive answer is known when X is a dual space and F is a predual, in that case we have a dual locally uniformly rotund renorming [Raj02]. It is known for instance that G can be constructed to be norm continuous too, [Raja03]. Moreover we know that it is possible to obtain such a G satisfying that G(x + y) ≤ k(G(x) + G(y)) for all x, y ∈ X with a constant k ≥ 1, but it seems that reducing the value of k down to 1 is far away from our methods. Let X be a Banach space X with a Kadec norm, let us recall that if X has a rotund equivalent norm, or the Krein-Milman property then X has an equivalent LUR norm too [MOTV00], so it is natural to propose:

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Question 6.3. Let X be a Banach space and F a norming subspace of X ∗ such that the identity map Id : (X, σ(X, F )) −→ (X, · ) is σ-continuous. Let us assume that X admits a strictly convex renorming, does X admit a σ(X, F )-lower semicontinuous LUR norm? What happens if in addition we suppose that X has the Krein-Milman property? For a weakly uniformly rotund Banach space the unit sphere is metrizable as we have seen in [MOTV99], so the next one is a particular case of Question 6.3. Question 6.4. Let X be a Banach space with a strictly convex norm and a metrizable unit sphere for the weak topology. Does X admits an equivalent LUR norm? For MLUR renormings the papers [Hay99] and [MOTV01] provide characterizations which could be of some use to obtain conditions which allow to ‘transfer’ this class of norms. The ﬁrst cases have been just studied in [LPT07]. It is interesting to answer the following: Question 6.5. Let X be a Banach space such that every point in the unit sphere of X is an extreme point of the bidual unit ball BX ∗∗ ; i.e. what is called a weakly MLUR norm. Doest it follow that X admits an equivalent MLUR norm? Let us recall that for a descriptive normed space the family of weak Borel sets coincides with the norm Borel sets, [Han01]. Nevertheless it is unknown whether the reverse is true assuming some axiom or axioms independent of the usual ones of ZFC set theory, so following L. Oncina [Onc00], we may ask: Question 6.6. Let X be a Banach space such that every norm open set is a countable union of sets which are diﬀerences of closed sets for the weak topology. Doest it follow that the identity map Id : (X, σ(X, X ∗ )) −→ (X, · ) is σ-continuous ? Since every σ-continuous map is the pointwise limit of a sequence of maps which are locally constant up to a countable partition [MOTV06], a positive answer to the former question can be seen as a non separable inﬁnite dimensional version of the Lebesgue Hausdorﬀ theorem on the coincidence of Baire and Borel ﬁrst level maps. Based on a sophisticated construction of S. Todorcevic [Tod06], R. Pol has kindly informed us that it is consistent with the axioms of ZFC set theory the existence of a compact scattered space K, such that in the function space C(K) each norm open set is an Fσ -set with respect to the weak topology but the identity map Id : (C(K), σ(C(K), C(K)∗ )) −→ (C(K), · ∞ ) is not σ-continuous [MP07]. Descriptive Banach spaces are Souslin sets made up with σ(X ∗∗ , X ∗ ) open or closed subsets of the bidual. These spaces are

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ˇ called weakly Cech analytic and are exactly those that can be represented with a Souslin scheme of Borel subsets in their σ(X ∗∗ , X ∗ ) biduals. The ˇ main result in [JNR93] asserts that every weakly Cech analytic Banach space is σ-fragmentable. The reverse implications are open question considered in [JNR92], see Sect. 3.2, and we do remind here the next problem: Question 6.7. Is there any gap between the classes of descriptive Banach spaces and that of σ-fragmented Banach spaces? After the seminal paper of R. Hansell [Han01] we know that the covering property of hereditarily weakly θ-reﬁnability for the weak topology of the Banach space is a necessary and suﬃcient condition for the coincidence of both classes. In fact all known examples of Banach spaces which are not weakly θ-reﬁnable are not σ-fragmentable, [DJP97, DJP]. For the spaces C0 (Υ ) with Υ a tree, R. Haydon has proved that there is no gap between σ-fragmentability and the pointwise Kadec renormability property of the space, [Hay99]. We may consider a particular case of the former question as follows. ˇ Question 6.8. Let X be a weakly Cech analytic Banach space where every norm open set is a countable union of sets which are diﬀerences of closed sets for the weak topology. Doest it follow that the identity map Id : (X, σ(X, X ∗ )) −→ (X, · ) is σ-continuous ? In the particular case of a Banach space X with the Radon-Nikodym property it is still an open whether such X has an equivalent rotund norm. Therefore, according to our results in [MOTV00], the existence of a LUR and a Kadec equivalent norm are indeed equivalent. Summarizing: Question 6.9. If the Banach space X has the Radon-Nikodym property, i.e. every bounded closed convex subset of X has slices of arbitrarily small diameter, does X have an equivalent Kadec norm? Does it have an equivalent rotund norm? Let us remark here that a result of A. Plichko and D. Yost [PY01] shows that the Radon-Nikodym property doest not imply the separable complementation property. Thus to solve the former question, it is not possible to use any projectional resolution of the identity like in [FG88] for dual spaces with the Radon-Nikodym property. In that direction it is interesting to consider the following: Question 6.10. Let X be a Banach space without copies of 1 (N) such that the dual space X ∗ is not separable. Does X ∗ admit an equivalent locally uniformly rotund norm? Let us stress the fact that for X separable the question is widely open too. Indeed, all concrete examples has been already checked.

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121

The question above is a particular case of the renorming problems in dual spaces by non dual norms. Indeed any dual space of the form C(K)∗ for an arbitrary compact space K has an equivalent LUR norm since it is isomorphic to an L1 space, moreover this renorming is a dual norm if, and only if, K is σ-discrete. Any dual space X ∗ with the Radon-Nikodym property has an equivalent LUR norm which can be a dual norm too if, and only if, the identity map Id : (X ∗ , σ(X ∗ , X)) −→ (X ∗ , · ) is σ-continuous. So we may ask: Question 6.11. Describe the general class of dual Banach spaces X ∗ admitting an equivalent locally uniformly rotund norm but not necessarily σ(X ∗ , X)lower semicontinuous. Since we are considering linear topological characterizations of renorming properties it is quite natural to study the invariance of those properties by uniform, Lipschitz or weak homeomorphisms. A weak homeomorphism is of the ﬁrst Baire class for the norm in both directions, see Sect. 2.1, and therefore preserves descriptive and σ-fragmented normed spaces. We do not know if a uniform homeomorphism, or even a Lipschitz homeomorﬁsm Φ between the Banach spaces X and Y preserves the property of LUR renormability. Question 6.12. Let X be a LUR Banach space (respectively with a Kadec, strictly convex or MLUR norm) and Φ : X −→ Y a Lipschitz homeomorphism, or a uniform homeomorphism, or even a homeomorphism for the weak topologies onto the Banach space Y . Is Y LUR (respectively Kadec, strictly convex or MLUR) renormable? We have seen here that for a Banach space X with a Fr´echet diﬀerentiable norm, which has a Gˆ ateaux diﬀerentiable dual norm, the duality map ∂ provides a σ-slicely continuous and co-σ-continuous map between the unit spheres of X and the dual space X ∗ , and therefore X admits an equivalent LUR norm, see Proposition 4.4. It seems possible to relax the requirement on the dual norm, perhaps some σ-continuous maps from X ∗ into X could be obtained so that their compositions with the duality map provide enough σ-slicely continuous maps from X into X to approximate the identity map, if so we could deduce the LUR renormability of the space X. Then we propose to study the following. Question 6.13. Let X be a Banach space with a Fr´echet diﬀerentiable norm. Is it possible to construct a sequence of σ-continuous maps for the norm topologies Φn : X ∗ −→ X such that the sequence {Φn ◦ ∂ : n ∈ N} will provide a way to approximate the identity map on the unit sphere of X? A starting point to look for the sequence (Φn ◦ ∂) could be modiﬁcations of the Toru´ nczyk homeomorphism between the dual X ∗ and X for every Asplund space X since the density characters of X and X ∗ coincide. If it is so, the Banach space must be LUR renormable, deducing a positive answer to the following old question.

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Question 6.14. Let X be a Banach space with a Fr´echet diﬀerentiable norm. Does X admit an equivalent LUR norm? R. Haydon has recently showed that if the dual space X ∗ has a dual LUR norm then X admits an equivalent LUR norm, showing that these spaces have C 1 -partitions of unity [Hay]. The former problem has been raised by R. Haydon in [Hay99], it appears in the book by R. Deville, G. Godefroy and V. Zizler, [DGZ93, Problem VII.1, Chap. VIII, Theorems 3.2, 3.12] as well as in Zizler’s renorming paper [Ziz03]. In the other direction, taking into account the results of Sect. 4.2, we can ask: Question 6.15. Let X be a LUR Banach space and g a real Lipschitz function on X. Does there exist a norm dense subset D of X such that D = Dn and ∂g (x|Dn ) = ∅ for x ∈ Dn , n ∈ N? Let us mention that in the LUR renormable space 1 (Γ), #Γ > ℵ0 , the norm is nowhere Gˆ ateaux diﬀerentiable. So, for a Lipschitz function on a LUR Banach space it is natural to look only for nonempty subdiﬀerentials. For diﬀerentiability properties of Lipschitz function see [Pre90] and [LP00] where it is proved that in Asplund spaces every locally Lipschitz function is Fr´echet diﬀerentiable on a dense subset of its domain. We mention that in [JLPS02] the ε–Fr´echet diﬀerentiability of Lipschitz maps is studied. In relation with our studies on pointwise LUR norms contained in Sect. 3.3, we may consider an inﬁnite set Γ, the space l∞ (Γ) and look for properties that guarantee the pointwise LUR renormability of a given subspace X ⊂ l∞ (Γ). If Γ is a countably determined topological space and X is made up by continuous functions on Γ, a result of Mercourakis asserts that this is indeed the case, [Mer87], see Proposition 3.42. Another situation in which this happens has been proved by M. Raja [Raj03] considering dual spaces of the form C(K)∗ . Question 6.16. We propose the study of nonlinear transfer techniques for pointwise LUR norms on subspaces of l∞ (Γ). For such study it is necessary to ﬁnd good linear topological characterizations of the property to have a pointwise LUR renorming on a subspace X ⊂ l∞ (Γ). The Mercourakis space [Mer87], as well as the DashiellLindenstrauss classes [DL73], are important examples for this question. We already know that the Dashiell-Lindenstrauss spaces are pointwise LUR subspaces of l∞ ([0, 1]), [Gui06]. We also know that if X is a normed space with a norming subspace F in the dual space X ∗ , then X has an equivalent σ(X, F ) LUR norm on X if, and only if, the topology σ(X, F ) has a σ-isolated network. Our proof of the fact that a weakly LUR norm on the Banach space X implies the LUR renormability of X opens the gate to ask some questions like: Question 6.17. Let X be a Banach space in which we can associate to each x ∈ X a separable subspace Zx ⊂ X such that given sequences (xn ), (yn ) and x in X we have · x ∈ span{Zyn : n ∈ N}

6.2 Renormings of C(K) Spaces

123

whenever the two following conditions hold 2 2 2 lim 2 xn + 2 x − xn + x = 0 ; n

2 2 2 lim 2 xn + 2 yn − xn + yn = 0 . n

Does X admit an equivalent LUR norm? Let us mention here Arhangel’ski˘ı’s theorem asserting that a semimetrizable space (X, s) is metrizable whenever the conditions lim s(xn , x) = 0 and lim s(xn , yn ) = 0 n

n

imply that lim s(yn , x) = 0 , n

[Gru84]. We conjecture that something similar happens with the symmetric 2 2 2 s(x, y) := 2 x + 2 y − x + y in the former problem, if so a metric could be constructed and our transfer technique could be applied.

6.2 Renormings of C(K) Spaces The general question goes back to J. Lindenstrauss [Lin72, problem 11] who asked for internal characterizations, i.e. only in terms of the topological properties of the compact space K, of the existence of an equivalent rotund norm on the Banach space C(K). Of course other renormings, in which we are interested are related with this and we can formulate the following fully open question. Question 6.18. Is there any topological property P of the compact space K such that C(K) is LUR renormable (respectively Kadec, strictly convex or MLUR renormable) if, and only if, the compact space K has P? If K is scattered, is there any property for the family of clopen sets of K which characterizes the former renorming properties? Let us remark here that a result obtained independently by R. Haydon and by I. Namioka and R. Pol states, for a scattered compact space K, that the identity map Id : Cp (K) −→ (C(K), · ∞ ) is σ-continuous if, and only if, the family of clopen sets A of the compact space K can be written as A = {An : n ∈ N} and every An is isolated in the pointwise topology of {0, 1}K , where we identify a clopen set with its characteristic function. Thus this fact and the second part of Question 6.18 above leads us to present the following.

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Question 6.19. Let K be a scattered compact space such that its family of clopen sets is σ-isolated in the pointwise topology. Is there any equivalent Kadec norm on the Banach space C(K)? According to Haydon’s dyadic tree Υ in [Hay99], it follows that the family of clopen sets can be σ-slicely isolated in Cp (Υ ) but the Banach space C0 (Υ ) fails to have an equivalent LUR norm. On the other hand, the following problem proposed by R. Haydon [Hay99] seems to be the key point for the relation with the σ-fragmentability. Question 6.20. Let K be a compact space and let A be a subset of C(K) which is scattered for the pointwise topology. Is A necessarily a countable union of sets which are discrete for that topology? Is the answer diﬀerent if we assume K to be scattered? A compact space K is σ-discrete if, and only if, it is scattered with a σisolated network, [Raj02]. Compact spaces with a σ-isolated network are studied and called descriptive compact spaces in [Han01, OR04]. For a descriptive scattered compact space K the dual C(K)∗ admits an equivalent dual LUR norm, [Raj02], so according to a recent result of R. Haydon [Hay] C(K) admits an equivalent pointwise lower semicontinuous and LUR norm. These considerations lead us to present: Question 6.21. Let K be a descriptive compact space. Is the Banach space C(K) σ-fragmentable? Does C(K) admit an equivalent pointwise Kadec or pointwise lower semicontinuous LUR equivalent norm? Does C(K) admit an equivalent rotund norm? The Helly space is an example of a class of compacta where there are still some interesting open questions. We say that a compact space K is a Rosenthal compact if there exists a Polish space Γ and a homeomorphism from K onto a subset of the space B1 (Γ) of Baire–1 functions on Γ, equipped with the pointwise topology. As Todorcevic has recently observed, there is a scattered compactiﬁcation H of a tree which is a Rosenthal compact, such that C(H) has no LUR renorming, [Tod06]. Since that example H is non-separable and, as a previous work of Todorcevic [Tod99] has shown, it is only from separable Rosenthal compacta that we should expect really good behaviour, following [HMO07] we ask: Question 6.22. If K is a separable Rosenthal compact then doest it follow that C(K) admits an equivalent LUR renorming? A positive answer is conjectured in [HMO07], if so this gives as an immediate corollary that X ∗ is LUR renormable whenever X is a separable Banach space with no subspace isomorphic to 1 . Indeed, in this case, we may take Γ to be the unit ball of the dual space X ∗ , which is compact and metrizable (so certainly Polish) under the weak* topology σ(X ∗ , X) and K to be the unit ball of X ∗∗ under the weak* topology σ(X ∗∗ , X ∗ ). By some results of [OR75],

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125

the elements of K are then of the ﬁrst Baire class when we look at them as functions on Γ. Moreover, K is separable, since the unit ball of X (which is assumed to be separable) is dense in K by Goldstine’s theorem. Finally, of course, X ∗ embeds as a closed subspace of C(K). Thus a positive answer to the former question implies a positive answer to Question 6.10 too. The main theorem in the recent paper [HMO07] proves the LUR renormability of C(K) only for a subclass of separable Rosenthal compacta K, namely those representable as spaces of functions with only countably many discontinuities in the Polish space Γ. When K is not assumed to be separable but each element of K has only countably many discontinuities the space Cp (K) is σ-fragmented by the norm metric, [Kor00, HMO07]. Nevertheless we do not know the answer to the following: Question 6.23. Let K ⊂ RΓ be a compact subset of Baire one functions on a Polish space Γ such that every function of K has at most countably many discontinuities. Do the Borel sets for the pointwise and the norm topologies coincide in C(K)? Is the identity map Id : Cp (K) −→ (C(K), · ∞ ) σ-continuous? Does the Banach space C(K) admit an equivalent LUR norm? A negative answer to the second question above implies an answer to question 6.7 above too. We can now formulate a possible approach for Question 6.10 in the following way: Question 6.24. Let X be a separable Banach space without copies of 1 (N). Is there a representation of the bidual unit ball BX ∗∗ as a weak ∗ compact of Baire one functions on a Polish space Γ such that every element of BX ∗∗ has only countably many discontinuities? A related question is: Question 6.25. Let K be a compact space such that C(K) admits an equivalent LUR (respectively Kadec, or strictly convex, or MLUR) norm. Let T be the compact space of Radon measures BC(K)∗ with the weak ∗ topology. Is there an equivalent LUR (respectively Kadec, or strictly convex, or MLUR) norm on the space C(T ) To solve Question 6.18 for an arbitrary compact space K, it could be of some help the study of the possible reduction arguments linked to the so called Ditor space associated to K. Let us remind that for a metric compact space K there is continuous onto map Φ : ∆ −→ K and a regular averaging operator T : C(∆) −→ C(K), where ∆ denotes the Cantor set, i.e. T (f ◦ Φ) = f for every f ∈ C(K), T (Id ∆ ) = Id K and T is a positive linear operator, [Pel68]. That construction can be extended to an arbitrary compact space K looking for a totally disconnected compact space D, with the same topological weight that K, instead of the Cantor set ∆, together with a continuous onto map Φ : D −→ K which admits a regular averaging operator T : C(D) −→ C(K). Such a totally disconnected space D is called a Ditor space for K. The following problem has been proposed to us by R. Haydon.

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Question 6.26. Is there a Ditor space D associated to a compact space K, with C(K) LUR renormable (respectively Kadec, or MLUR), such that the Banach space C(D) is LUR (respectively Kadec, or MLUR) renormable too? Is there a Ditor space D of a compact space K with the identity map Id : Cp (K) −→ (C(K), · ∞ ) being σ-continuous (respectively σ-fragmented) such that the same happens for C(D)? Let us remark here that we do not know if a Ditor space of a descriptive compact space can be constructed to be descriptive too. Recent research on the Ditor space of particular classes of compact spaces has been done in [AA06, AA02]. On the other hand, there are some open problems related with linear extension operators that are relevant to our topic, see Sect. 4.1. Indeed, since the cube [0, 1]Γ is a compact space for which C([0, 1]Γ ) admits a LUR renorming (Corollary 2.68), it is quite natural to reformulate Question 6.18 as follows: Question 6.27. Find an internal characterization of compact spaces admitting a σ-slicely continuous (respectively σ-continuous or σ-fragmented) extension map Ξ : Cp (K) −→ C [0, 1]Γ , where K is a subset of the cube [0, 1]Γ . An internal characterization spaces admitting a linear exten of compact sion operator T : C(K) −→ C [0, 1]Γ was obtained by R. Haydon, [Tod97]. If the compact space K has a good structure of metrizable closed subsets the former question leads to the following. Question 6.28. Let K be a compact space such that there is an upper semicontinuous onto multivalued map ζ : P −→ 2K from a separable metrizable space M to the set of a compact metric subsets

of K, i.e. ζ(p) is a compact metric subset of K for every p ∈ M and K = {ζ(p) : p ∈ M }. Without loss of generality we can assume that K embeds in the cube [0, 1]Γ . Which conditions should be imposed to obtain a σ-slicely continuous extension map (respectively σ-continuous or σ-fragmented) Ξ : Cp (K) −→ C [0, 1]Γ , gluing all the linear extension operators we have from the metrizable subsets ζ(p)? A.V. Arhangel’ski˘ı studied those compact spaces K which can be covered by a countable family of closed subsets F so that all sections, i.e. the sets deﬁned by K(x, F) := {F : F ∈ F, x ∈ F } for every x ∈ K, have some ﬁxed property P, which are called P-approximable. Since then this class of compact spaces has been considered by diﬀerent authors, [Tkace91]. When P is metrizability we get the same compact spaces as in the former question, [Tkach94] and [KOS06]. In the last paper is observed that every compact set of functions with countably many points of discontinuity on a Polish space is metrizable-approximable, which shows the relation between Question 6.23 and this one above. It must be said that to obtain the σ-slicely continuous extension map, satisfying the requirements of the question above, some conditions should be

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imposed. In fact, W. Kubi´s has communicated to us that there exist compact spaces K for which C(K) admits no LUR equivalent norm despite there exists a map ζ as above, [Kub07]. The study of the following problems has been proposed by R. Haydon in [Hay99] after his seminal paper on trees in renorming theory. Question 6.29. Let K be a scattered compact space: Is there any logical connection between LUR renormability of C(K) and Fr´echet-smooth renormability of that space? If C(K) admits a strictly convex renorming, does it necessarily admit a MLUR renorming? If C(K) admits an strictly convex renorming, does it admit a Gˆ ateaux-smooth renorming? The last questions have found positive answers for trees in [Smi07]. As R. Haydon says in [Hay99], in the area of non-separable renorming theory the most important open problems are those concerning bump functions and partitions of unity. Question 6.30. Does every Asplund space admit a C 1 -bump function? In the special case of spaces C(K), for K scattered, of course, we may ask whether there always exist a C ∞ -bump and C ∞ -partition of unity. P. H´ ajek and R. Haydon have recently proved that a space C(K) with a C 1 -bump function necessarily admits C 1 -partitions of unity [HH06]. They also show that for a σ-discrete compact space K the Banach space C(K) admits an equivalent C ∞ -norm and a C ∞ -partition of unity. The construction is based on that of the so called Talagrand operator (possibly nonlinear). A continuous linear map T : C(K) −→ c0 (K × Γ) is a Talagrand operator if for each non zero x ∈ C(K) there exist t ∈ K, u ∈ Γ such that |f (t)| = f ∞ and (T x)(t, u) = 0. In general a Talagrand operator Φ : C(K) −→ c0 (K × Γ) is a (possibly nonlinear) map where each coordinate function x → (Φx)(t, u) is assumed to be of class C m on the set where it is not zero to have a Talagrand operator of class C m . Question 6.31. If K is a scattered compact with a Talagand operator Φ of class C 1 , when Φ can be forced to be σ-slicely continuous from Cp (K) into c0 (K × Γ) i.e. is it possible to obtain another Talagrand operator Φ from Cp (K) into c0 (K × Γ) such that Φ is σ-slicely continuous? When Φ can be forced to be co-σ-continuous too? When both things can be done we have a pointwise lower semicontinuous LUR renorming on C(K), see Theorems 2.32 and 4.29. In the case of trees the existence of a linear Talagrand operator is equivalent to the LUR renormability [Hay99]. In general, if we have a linear Talagrand operator in C(K) then K is a scattered compact and the Banach space C(K) does admit and equivalent pointwise lower semicontinuous LUR norm, [Hay07].

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6.3 Measures of Non Compactness For a bounded set B in a metric space X, the Kuratowski index of noncompactness of B is deﬁned by α(B) := inf{ε > 0 : B ⊆

m

Ai , diam(Ai ) ≤ ε and m ∈ N}

i=1

The main results in the recent work [GOOT04] provide extensions of Theorem 1.1 when the Kuratowski index of non-compactness is used to measure slices instead of the diameter. Indeed, the following result is proved there: Theorem 6.32. Let X be a normed space and let F be a norming subspace of its dual. Then X admits an equivalent σ(X, F )-lower semi-continuous LUR

∞ norm if, and only if, for every ε > 0 we can write X = n=1 Xn,ε in such a way that for every integer n and x ∈ Xn,ε , there exists a σ(X, F )-open half space H containing x with α (H ∩ Xn,ε ) < ε. From the topological point of view the normed space X admits an equivalent σ(X, F )-lower semi-continuous LUR norm if, and only if, the norm

∞ topology has a network N = n=1 Nn , such that for every n ∈ N and for every x ∈ Nn , there is a σ(X, F )-open half space H, with x ∈ H, such that H meets only a ﬁnite number of elements from Nn , therefore turning the isolated condition appearing in Chap. 3 into a locally ﬁnite one. These results open the door to the study of the situation for diﬀerent measures of non compactness. Indeed, the index α measures the distance of a set B to a norm compact subset of the normed space X. We can introduce the index β, studied by S. Troyanski in [Tro94], to measure the distance to a uniform Ebelein compacta, i.e. we set β(B) := inf{ε > 0 : B ⊆ K + εBX where K is a uniform Eberlein compact}, or the distance to a weakly compact subset of the Banach space, χ(B) := inf{ε > 0 : B ⊆ K + εBX where K is a weakly compact subset}, or the separability index measuring the distance to separable subsets, ∞ Ai : diam(Ai ) ≤ ε . λ(B) := inf ε > 0 : B ⊆ i=1

Question 6.33. Given a normed space X that veriﬁes the conditions of Theorem 6.32 with the index λ (respectively β, χ) instead of α. Is it true that the Borel sets for the norm and the σ(X, X ∗ ) topology coincide? Is the identity map Id : (X(σ(X, X ∗ )) −→ (X, · ) σ-continuous? Does X admit an equivalent LUR norm?

6.3 Measures of Non Compactness

129

In case of positive answers to any of the above questions what can be said about the network characterizations or the σ(X, F ) lower semicontinuity of the equivalent norm? The main result in [Tro94] asserts that for a Banach space X with a unit ball BX such that for every point x ∈ SX , and every ε > 0, there is an open half space H, with x ∈ H, and β(H ∩ BX ) < ε, it follows that X admits an equivalent LUR norm. The proof involves martingale calculus, nevertheless M. Raja [Raj07] has just provided a geometrical proof of that result, answering the former question for the β index in full generality. Indeed, M. Raja applies the methods of [GOOT04] to deal with general measures of non compactness and for the β measure provides the LUR renorming. Thus, it seems to be the right time to study the above question with the index χ at least. With the index λ the question seems more diﬃcult, nevertheless we know examples of compact spaces K such that the space C(K) can be decomposed as in Theorem 6.32, with the index λ instead of the index α, and nothing is known about the descriptiveness of the space C(K). For instance, that happens for the possibly non separable Rosenthal compact of Question 6.23 that has been studied in [HMO07]. We certainly know that the space X[σ(X, X ∗ )] is σ-fragmented by its norm, [JNR93], when it satisﬁes the conditions of Theorem 6.32 with the index λ instead of α. Thus a negative answer to that question would shed light on Question 6.7 too.

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Index

σ-fragmentable normed space, 18 MLUR, 117 approximable P-approximable, 126 metrizable-approximable, 126 Baire map, 4 barely continuous map, 17 Bartle-Graves map, 81, 101 basis for a family, 28 Bing-Nagata-Smirnov metrization theorem, 49

descriptive compact space, 124 discrete family σ-discrete family, 13 discrete family, 9 discretely decomposable family σ-discretely decomposable family, 13 σ-slicely discretely decomposable family, 22 Ditor space, 125 duality mapping, 76 dyadic atomization, 114 dyadic interval system, 115 extension map, 81

Cech-analytic ˇ Cech-analytic space, 54 ˇ weakly Cech-analytic space, 120 Ciesielski-Pol compact, 81, 101 class P, 6 σ-continuous map, 13 co-σ-continuous map, 4, 23 controls set that controls a function, 41 convex arguments, 67, 68 countable cover by sets of small local diameter, 54 countably determined K-countably determined, 39 Day’s map, 73 Day’s norm, 67 decreasing interval function, 115 descriptive descriptive Banach space, 55

ﬁrst Borel class map, 17 fragmentable σ-fragmentable normed space, 18 fragmented σ-fragmented by a metric, 53 σ-fragmented map, 18 fragmented by the metric d, 53 fragmenting metric, 56, 57 function base, 16 good point, 35 Helly space, 44 hereditarily weakly θ-reﬁnable, 17 isolated σ-isolated family, 13 σ-isolatedly decomposable family, 13 σ-slicely isolated family, 22 141

142

Index

isolated (continued ) σ-slicely isolatedly decomposable family, 9, 22 isolated family, 9 slicely isolated family, 22

point of continuity, 17 PRI, 32 projectional resolution of the identity, 32 property P (V, W), 28

James boundary, 65 joint σ-continuity lemma, 88

radial set, 30 Radon-Nikodym property, 120 restriction map, 81 Rosenthal compact, 124 rotund norm, 1

Kadec norm, 54 Kuratowski index of non-compactness, 128 lattice LUR norm, 70 lexicographic product, 111 locally uniformly rotund, 1 LUR T LUR normed space, 5 σ(X, F ) LUR norm, 5 LUR, 1 lattice LUR norm, 70 weak LUR norm, 5 weak∗ LUR norm, 5 M-basis, 33 Markushevich basis, 33 MLUR Midpoint locally uniformly rotund norm, 117 Midpoint locally uniformly rotund space, 117 Namioka-Phelps compact, 57 network, 10 norming 1-norming, 77 M-basis, 34 order topology, 111 oscillation map, 42

slicely continuous ε-σ-slicely continuous, 52 σ-slicely continuous map, 7 slicely continuous map, 7 slicely discrete family, 9, 22 σ-slicely discrete family, 22 slicely isolated family, 9 strictly convex norm, 1 subdiﬀerential, 9 ε-subdiﬀerential, 9 symmetric, 58, 60 symmetrizable topological space, 58 Talagrand operator, 127 theorem of Mibu, 41 three space property for LUR renorming, 101 transfer theorem, 7 tree, 35 ever branching, 35 Hausdorﬀ, 35 von Neumann criterion, 108 weakly LUR norm, 51 weakly MLUR, 51

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