Renormings in Banach Spaces A Toolbox 1st Edition by Antonio José Guirao, Vicente Montesinos, Václav Zizler ISBN 3031086554 9783031086540

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ISBN 10: 3031086554
ISBN 13: 9783031086540
Author: Antonio José Guirao, Vicente Montesinos, Václav Zizler

Renormings in Banach Spaces A Toolbox 1st Edition Table of contents:

Part I An Introductory Course in Renorming

Chapter 1 Norms, normed spaces, Banach spaces

1.1 Norms, normed and Banach spaces

1.2 Equivalent norms

1.2.1 Definition

1.3 Linear operators and linear functionals. Duality, weak topologies

1.4 A few basic examples

1.5 Finite-dimensional spaces

Chapter 2 Some basic definitions and tools

2.1 A short utility-grade approach to locally convex spaces

2.2 Extreme points

2.3 Some basic results in Banach space theory

Chapter 3 Equivalent norms

3.1 On the definition of equivalent norms

3.2 Finite-dimensionality and equivalent norms

3.3 Dual equivalent norms

3.3.1 Norms on the dual versus dual norms

3.4 Equivalent norms and norming subspaces

3.4.1 Norming and 1–norming subspaces

3.4.2 Separating subspaces

3.4.3 Characterizing norming subspaces

3.4.4 Getting norming subspaces

3.5 Some other results related to norming subspaces

3.6 The interplay between norms and their unit balls

3.6.1 Minkowski functionals

3.6.2 Support functions

3.6.3 Polarity

3.6.4 Fenchel conjugation

3.6.5 The infimal convolution of two convex functions

3.7 Norm and ball arithmetics; duality

3.7.1 Sum of two norms

3.7.2 Inverse summation of two norms

3.7.3 Supremum of two norms

3.7.4 The infimal convolution of two norms

3.7.5 A basic result

3.7.6 Dealing with seminorms

3.7.7 The predual norm of a p-sum of w*-lower semicontinuous seminorms

Chapter 4 Basic differentiability in Banach spaces

4.1 Gâteaux and Fréchet differentiability

4.1.1 The basic definitions

4.1.2 Lipschitz functions I

4.1.3 Convex functions

4.1.4 The case of the norm

4.1.5 The set of points of differentiability of general functions on a Banach space

4.2 Strengthening Gâteaux differentiability

4.3 Uniformities in differentiability

4.4 Norms defined by level sets of differentiable functions

Chapter 5 Basic rotundity

5.1 Strict convexity

5.2 Uniform convexity

5.2.1 Introduction

5.2.2 A consequence of the uniform convexity

5.2.3 A weak version of the uniform convexity

5.3 Local uniform convexity

5.3.1 Introduction

5.3.2 Some consequences of the local uniform convexity

5.3.3 A weak version of the local uniform rotundity property

5.4 Topological properties of the unit sphere

5.5 Variants of uniform convexity and local uniform convexity

5.5.1 2-rotundness and weakly 2-rotundness

5.5.2 Rotundness in directions

5.5.3 Near uniform convexity

5.5.4 Midpoint and weak midpoint local uniform convexity

5.6 Using the square of the norm

5.6.1 Some useful computations using ‖ · ‖2, and some tools

5.6.2 Describing rotundity

5.7 Extension of rotund norms

Chapter 6 Some structural properties of Banach spaces

6.1 Biorthogonal systems, Markushevich bases, Schauder bases, and projectional resolutions of the id

6.1.1 Biorthogonal systems

6.1.2 Markushevich bases

6.1.3 Schauder bases

6.1.4 Projectional resolutions of the identity I

6.1.5 Markushevich bases versus projectional resolutions of the identiy

6.1.6 Several classes of Banach spaces defined in terms of Markushevich bases or projectional resolu

6.1.7 Superreflexivity, finite representability

6.1.8 Generation and strong generation

6.2 Some classes of compact topological spaces

6.3 Extra definitions needed

Chapter 7 The use of Šmulyan’s tests

7.1 Some comments on the Šmulyan tests

7.2 An application of the Šmulyan lemma to w*-convergence

Chapter 8 Asplund averaging I

8.1 Asplund original approach

8.2 Applications

Chapter 9 Tools for renorming

9.1 Two simple facts

9.2 An elementary strictly convex renorming

9.2.1 A strictly convex renorming of the space

9.2.2 A strictly convex renorming of the dual space

9.2.3 A strictly convex and Gâteaux smooth renorming of the space

9.3 Transfer methods by using the square of the norm

9.3.1 Clarkson’s method

9.4 Some “ad-hoc” norms on classical spaces

9.4.1 Day’s norm I: Day’s norm on ℓ∞(Γ)

9.4.2 Some other particular norms on c0(Γ)

9.4.3 Phelps’ norm on ℓ1

9.5 Locally uniformly convex norms in separable spaces: Kadets renorming and homeomorphism theorems

9.5.1 Kadets renorming theorem

9.5.2 Some consequences of Kadets’ renorming theorem

9.5.3 Kadets homeomorphism theorem for separable spaces

9.5.4 Some statements concerning homeomorphic normed spaces

Part II An Intermediate Course in Renorming

Chapter 10 Locally uniformly convex renorming of nonseparable spaces

10.1 Weakly compactly generated spaces I: Some introductory results

10.2 Lindenstrauss–Troyanski tools for renorming, and developments

10.2.1 A basic Lindenstrauss lemma

10.2.2 An application to reflexive spaces

10.2.3 A glimpse of the weakly compactly generated case and Hahn–Banach extension operators

10.3 Projectional resolutions of the identity II: A tool

10.4 Troyanski’s renorming theorem, and extensions

10.4.1 Troyanski’s original proof

10.4.2 Moltó–Orihuela–Troyanski transfer method

10.4.3 Godefroy–Fabian transfer results

10.5 Two more renormings for weakly compactly generated spaces

10.6 Weakly compactly generated spaces II

10.7 Day’s norm II: Its locally uniformly convex behaviour on c0(Γ)

10.8 Asplund averaging II

Chapter 11 Norm-attaining operators, variational principles, and Asplund spaces

11.1 Lindenstrauss’ extremal structure of ℓ1

11.2 Lindenstrauss’ results on norm-attaining operators

11.3 Norm-attaining operators and strongly exposed points

11.4 Farthest points

11.5 Properties α and β.

11.6 Asplund spaces I

11.6.1 Introduction

11.6.2 Some equivalences

11.6.3 Some tools and proofs on rough norms

11.7 The use of variational principles

11.7.1 Ekeland’s and smooth variational principles

11.7.2 Other geometric statements equivalent to the Ekeland variational principle

11.7.3 The compact variational principle and related results

Chapter 12 Projectional resolutions of the identity III

12.1 Introduction

12.2 Four useful properties of projectional resolutions of the identity

12.3 Examples of spaces having projectional resolutions of the identity

12.4 Examples of spaces without projectional resolutions of the identity

12.5 Results on projectional resolutions of the identity in the 1970s and 1980s

Chapter 13 Smooth approximation of norms by norms

13.1 Smooth approximations in separable spaces

13.2 Smooth approximation in nonseparable spaces

Chapter 14 Smooth partitions of unity in nonseparable spaces

Chapter 15 Smooth norms in dense subspaces

Chapter 16 Miscellaneous applications

16.1 Difference of two convex continuous functions

16.2 Some topological issues

16.3 An application of locally uniformly convex renorming to lower semicontinuous functions

Chapter 17 Bumps depending locally on finitely many coordinates

Chapter 18 Summary on renorming for uniformly rotund in every direction, strictly convex, and weakly

Chapter 19 Examples on Examples on C1-smoothness

19.1 Uniformly Gâteaux differentiable norms and related results

19.2 Uniformly Kadets–Klee smooth norms

Chapter 20 Examples on Rotundity

20.1 Normal structure

20.2 M.A. Smith’s renormings of ℓ2

20.3 M.A. Smith’s renormings of c0 and ℓ1

20.4 A chart

Part III Advances and Developments in Renorming, and Applications

Chapter 21 Nonlinear transfer techniques

21.1 Deville’s master lemma and applications

21.2 Nonlinear transfer

Chapter 22 Lipschitz functions II

22.1 The Rademacher theorem

22.2 Preiss’ differentiation of Lipschitz functions

Chapter 23 Spaces isomorphic to Hilbert spaces

Chapter 24 Superreflexive spaces

Chapter 25 The Kingdom of Tsirelson’s space

Chapter 26 The L∞ spaces

Chapter 27 Higher-order smoothness

27.1 Higher-order Gâteaux smooth norms

27.2 Higher-order smoothness

27.3 Smooth norms in C(K) spaces

27.4 Survey on higher-order smooth norms on C(K) spaces

Chapter 28 James boundaries

28.1 Introduction

28.2 The boundary problem and the strong boundary problem

28.3 A glimpse of some techniques for the boundary problem and James’ theorem

Chapter 29 The Radon–Nikodým property

Chapter 30 Strongly subdifferentiable norms

Chapter 31 The Banach–Saks property

Chapter 32 Transitive norms

Chapter 33 Norms with the Mazur intersection property

Chapter 34 Nicely smooth Banach spaces

Chapter 35 Weak Hadamard differentiability

35.1 Introduction

35.2 Sequential convergence in X*, boundedness, and differentiability

Chapter 36 Lipschitz Asplund spaces

Chapter 37 Lipschitz-free spaces

37.1 Introduction

37.2 Lipschitz-free spaces

37.3 The extremal structure of BF(M)

37.4 Lipschitz-free spaces on Banach spaces

Chapter 38 Polyhedral spaces

38.1 Polyhedral spaces

38.2 Tilings

Chapter 39 Smooth functions on c0(Γ)

Chapter 40 Kottman-type results on separated sets

40.1 Whitley’s results

40.2 Separated and symmetrically separated sets

40.3 Equilateral sets in infinite-dimensional spaces

Chapter 41 Three-space properties

Chapter 42 Polynomials on Banach spaces

Chapter 43 Szlenk derivation and applications

Chapter 44 Further miscellaneous applications

44.1 Fabian–Preiss intermediate differentiation of Lipschitz functions

44.2 Bates’ results, onto mappings, ranges of derivatives

Chapter 45 Miscellaneous topics

45.1 Phelps’ property U

45.2 Pointwise uniformly rotund spaces

45.3 Spaces with w*-sequentially compact dual balls

45.4 Injections I

45.5 Vašák spaces II

45.6 Uniformly Gâteaux differentiable norms II

45.7 Strongly uniformly Gâteaux differentiable norms

45.8 “Pathologies” in weakly compactly generated spaces

45.9 Unconditional bases

45.10 Fundamental biorthogonal systems

Chapter 46 Weakly compactly generated spaces and their relatives III

46.1 Asplund spaces II

46.2 Scattered compacta and Asplund spaces II

46.3 σ-Fréchet smooth norms

46.4 The Daugavet property

Chapter 47 Valdivia compacta

Chapter 48 Checking renormability in some classical spaces

48.1 Spaces of bounded functions

48.1.1 Strict convexity

48.1.2 Smoothness

48.1.3 Spaces of bounded functions with countable support

48.2 “Haydon’s forest”

48.3 Injections II

48.4 Johnson–Lindenstrauss spaces

48.5 Markushevich bases II

48.6 Weakly Lindelöf determined spaces I

48.7 Double arrow space

48.8 Space of continuous functions on ordinals

48.9 Weakly Lindelöf determined spaces II

48.10 Kunen compact space

48.11 Strongly weakly compactly generated spaces

48.12 Effros–Borel structure

Chapter 49 Symmetric norms

Chapter 50 Strictly convex renorming

50.1 Introduction

50.2 Topological sufficient conditions for strictly convex renorming and a characterization

Chapter 51 A concise list of coordinates for some relationships

51.1 Differentiability

51.1.1 Gâteaux differentiability (G)

51.1.2 Uniform Gâteaux differentiability (UG)

51.1.3 Very smoothness (VS)

51.1.4 Strong uniform Gâteaux differentiability (SUG)

51.1.5 Nice smoothness (NS)

51.1.6 Fréchet differentiability (F)

51.1.7 C2-smoothness

51.1.8 C∞-smoothness

51.2 Rotundity

51.2.1 Strict convexity (R)

51.2.2 Local uniform convexity (LUR)

51.2.3 2-rotundness (2R)

51.2.4 Weak 2-rotundness (W2R)

51.2.5 Average local uniform convexity (ALUR)

51.2.6 Midpoint local uniform convexity (MLUR)

51.2.7 Near uniform convexity (NUC)

51.2.8 Uniformly rotund in every direction (URED)

51.2.9 Weak uniform convexity (WUR)

51.3 Some extra properties

51.3.1 Kadets–Klee and sequential Kadets–Klee (KK) (SKK)

51.3.2 Uniform Kadets–Klee (UKK)

51.3.3 Normal structure

51.3.4 Asplundness

51.3.5 Mazur intersection property (MIP)

51.4 Weak compact generation (WCG) and relatives

51.4.1 Weak Lindelöf determinacy (WLD)

51.5 Injections

51.6 Markushevich bases

51.7 Miscellanea

Chapter 52 Open questions spread along the book and some additional ones

52.1 Differentiability

52.1.1 Gâteaux differentiability

52.1.2 Fréchet differentiability

52.1.3 Higher-order smoothness

52.1.4 Space c0

52.1.5 Space ℓ1

52.1.6 Space

52.1.7 C(K) and related spaces

52.2 Asplund and weak Asplund spaces

52.3 Rotundity

52.4 Weakly compactly generated spaces and relatives

52.5 Auerbach and Markushevich bases

52.6 Miscellanea

52.7 Distortion, hereditarily indecomposable spaces

52.8 Mazur intersection property

52.9 Lipschitz functions and Lipschitz Asplund spaces

52.9.1 Lipschitz functions

52.9.2 Lipschitz-free spaces

52.9.3 Lipschitz Asplund spaces

Bibliography

List of Figures

How to use the indices (and some notations)

General Index

Index of Symbols

Index of Authors

Renormings

Impossible Renormings

General Index

Index of Symbols

Index of Authors

Renormings

Impossible Renormings

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