Renormings in Banach Spaces: A Toolbox

Contents
Preface
Introduction
Notation
Typography
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 identity
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 resolutions of the identity
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 uniformly rotund spaces
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