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Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

Book Description

This book makes a significant inroad into the unexpectedly difficult question of existence of Fréchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. Because the question turns out to be closely related to porous sets in Banach spaces, it provides a bridge between descriptive set theory and the classical topic of existence of derivatives of vector-valued Lipschitz functions. The topic is relevant to classical analysis and descriptive set theory on Banach spaces. The book opens several new research directions in this area of geometric nonlinear functional analysis.

The new methods developed here include a game approach to perturbational variational principles that is of independent interest. Detailed explanation of the underlying ideas and motivation behind the proofs of the new results on Fréchet differentiability of vector-valued functions should make these arguments accessible to a wider audience. The most important special case of the differentiability results, that Lipschitz mappings from a Hilbert space into the plane have points of Fréchet differentiability, is given its own chapter with a proof that is independent of much of the work done to prove more general results. The book raises several open questions concerning its two main topics.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Dedication
  5. Contents
  6. 1 Introduction
    1. 1.1 Key notions and notation
  7. 2 Gâteaux differentiability of Lipschitz functions
    1. 2.1 Radon-Nikodým property
    2. 2.2 Haar and Aronszajn-Gauss null sets
    3. 2.3 Existence results for Gâteaux derivatives
    4. 2.4 Mean value estimates
  8. 3 Smoothness, convexity, porosity, and separable determination
    1. 3.1 A criterion of differentiability of convex functions
    2. 3.2 Fréchet smooth and nonsmooth renormings
    3. 3.3 Fréchet differentiability of convex functions
    4. 3.4 Porosity and nondifferentiability
    5. 3.5 Sets of Fréchet differentiability points
    6. 3.6 Separable determination
  9. 4 ε-Fréchet differentiability
    1. 4.1 ε-differentiability and uniform smoothness
    2. 4.2 Asymptotic uniform smoothness
    3. 4.3 ε-Fréchet differentiability of functions on asymptotically smooth spaces
  10. 5 Γ-null and Γn-null sets
    1. 5.1 Introduction
    2. 5.2 Γ-null sets and Gâteaux differentiability
    3. 5.3 Spaces of surfaces, and Γ- and Γn-null sets
    4. 5.4 Γ- and Γn-null sets of low Borel classes
    5. 5.5 Equivalent definitions of Γn-null sets
    6. 5.6 Separable determination
  11. 6 Fréchet differentiability except for Γ-null sets
    1. 6.1 Introduction
    2. 6.2 Regular points
    3. 6.3 A criterion of Fréchet differentiability
    4. 6.4 Fréchet differentiability except for Γ-null sets
  12. 7 Variational principles
    1. 7.1 Introduction
    2. 7.2 Variational principles via games
    3. 7.3 Bimetric variational principles
  13. 8 Smoothness and asymptotic smoothness
    1. 8.1 Modulus of smoothness
    2. 8.2 Smooth bumps with controlled modulus
  14. 9 Preliminaries to main results
    1. 9.1 Notation, linear operators, tensor products
    2. 9.2 Derivatives and regularity
    3. 9.3 Deformation of surfaces controlled by ωn
    4. 9.4 Divergence theorem
    5. 9.5 Some integral estimates
  15. 10 Porosity, Γn- and Γ-null sets
    1. 10.1 Porous and σ-porous sets
    2. 10.2 A criterion of Γn-nullness of porous sets
    3. 10.3 Directional porosity and Γn-nullness
    4. 10.4 σ-porosity and Γn-nullness
    5. 10.5 Γ1-nullness of porous sets and Asplundness
    6. 10.6 Spaces in which σ-porous sets are Γ-null
  16. 11 Porosity and ε-Fréchet differentiability
    1. 11.1 Introduction
    2. 11.2 Finite dimensional approximation
    3. 11.3 Slices and ε-differentiability
  17. 12 Fréchet differentiability of real-valued functions
    1. 12.1 Introduction and main results
    2. 12.2 An illustrative special case
    3. 12.3 A mean value estimate
    4. 12.4 Proof of Theorems 12.1.1 and 12.1.3
    5. 12.5 Generalizations and extensions
  18. 13 Fréchet differentiability of vector-valued functions
    1. 13.1 Main results
    2. 13.2 Regularity parameter
    3. 13.3 Reduction to a special case
    4. 13.4 Regular Fréchet differentiability
    5. 13.5 Fréchet differentiability
    6. 13.6 Simpler special cases
  19. 14 Unavoidable porous sets and nondifferentiable maps
    1. 14.1 Introduction and main results
    2. 14.2 An unavoidable porous set in ℓ1
    3. 14.3 Preliminaries to proofs of main results
    4. 14.4 The main construction, Part I
    5. 14.5 The main construction, Part II
    6. 14.6 Proof of Theorem 14.1.3
    7. 14.7 Proof of Theorem 14.1.1
  20. 15 Asymptotic Fréchet differentiability
    1. 15.1 Introduction
    2. 15.2 Auxiliary and finite dimensional lemmas
    3. 15.3 The algorithm
    4. 15.4 Regularity of f at x∞
    5. 15.5 Linear approximation of f at x∞
    6. 15.6 Proof of Theorem 15.1.3
  21. 16 Differentiability of Lipschitz maps on Hilbert spaces
    1. 16.1 Introduction
    2. 16.2 Preliminaries
    3. 16.3 The algorithm
    4. 16.4 Proof of Theorem 16.1.1
    5. 16.5 Proof of Lemma 16.2.1
  22. Bibliography
  23. Index
  24. Index of Notation