Renormings in Banach Spaces: A Toolbox (Monografie Matematyczne, 75)

This monograph presents an up-to-date panorama of the different techniques and results in the large field of renorming in Banach spaces and its applications. The reader will find a self-contained exposition of the basics on convexity and differentiability, the classical results in building equivalent norms with useful properties, and the evolution of the subject from its origin to the present days. Emphasis is done on the main ideas and their connections. The book covers several goals. First, a substantial part of it can be used as a text for graduate and other advanced courses in the geometry of Banach spaces, presenting results together with proofs, remarks and developments in a structured form. Second, a large collection of recent contributions shows the actual landscape of the field, helping the reader to access the vast existing literature, with hints of proofs and relationships among the different subtopics. Third, it can be used as a reference thanks to comprehensive lists and detailed indices that may lead to expected or unexpected information. Both specialists and newcomers to the field will find this book appealing, since its content is presented in such a way that ready-to-use results may be accessed without going into the details. This flexible approach, from the in-depth reading of a proof to the search for a useful result, together with the fact that recent results are collected here for the first time in book form, extends throughout the book. Open problems and discussions are included, encouraging the advancement of this active area of research. 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