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

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 desc...

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Hlavní autoři:	Lindenstrauss, Joram, Preiss, David, Tišer, Jaroslav
Médium:	E-kniha Kniha
Jazyk:	angličtina
Vydáno:	Princeton Princeton University Press 2012
Vydání:	1
Edice:	Annals of Mathematics Studies
Témata:	Approximation Auxiliary function Banach space Banach spaces Big O notation Borel measure Borel set Bounded operator Bounded set (topological vector space) Bump function Calculus of variations Chebyshev's inequality Compact space Complete metric space Continuous function Continuous function (set theory) Contradiction Convex function Convex hull Convex set Corollary Countable set Dense set Derivative Descriptive set theory Differentiable function Dimension Dimension (vector space) Dimensional analysis Directional derivative Divergence theorem Division by zero Estimation Frechet spaces Fubini's theorem Functional analysis Hilbert space Infimum and supremum Lebesgue measure Linear approximation Linear map Linear span Lipschitz continuity Lipschitz spaces MATHEMATICS MATHEMATICS / Applied MATHEMATICS / General MATHEMATICS / Mathematical Analysis Measurable function Measure (mathematics) Metric space Monotonic function Norm (mathematics) Null set Open set Parameter Perturbation function Porosity Porous set Projection (linear algebra) Rademacher's theorem Requirement Scientific notation Semi-continuity Separable space Sign (mathematics) Smoothness Sobolev space Special case Standard basis Subset Theorem Two-dimensional space Uniform continuity Unit sphere Unit vector Upper and lower bounds Variable (mathematics) Variational principle Perturbation function Open set Measurable function Subset Convex set Auxiliary function Bounded operator Rademacher's theorem Dense set Projection (linear algebra) Semi-continuity Chebyshev's inequality Hilbert space Divergence theorem Convex function Two-dimensional space Corollary Contradiction Monotonic function Approximation Estimation Continuous function Fubini's theorem Directional derivative Division by zero Requirement Dimension (vector space) Infimum and supremum Convex hull Big O notation Bump function Porous set Derivative Special case Unit sphere Complete metric space Sign (mathematics) Continuous function (set theory) Metric space Linear span Sobolev space Unit vector Null set Countable set Upper and lower bounds Lipschitz continuity Descriptive set theory Lebesgue measure Linear map Dimensional analysis Theorem Borel measure Differentiable function Bounded set (topological vector space) Compact space Standard basis Borel set Dimension Banach space Smoothness Variable (mathematics) Separable space Norm (mathematics) Linear approximation Variational principle Parameter Scientific notation Porosity Measure (mathematics) Uniform continuity
ISBN:	9780691153551, 0691153566, 0691153558, 9780691153568, 1400842697, 9781400842698
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Front Matter Table of Contents Chapter One: Introduction Chapter Two: Gâteaux differentiability of Lipschitz functions Chapter Three: Smoothness, convexity, porosity, and separable determination Chapter Four: ε-Fréchet differentiability Chapter Five: Γ-null and Γn-null sets Chapter Six: Fréchet differentiability except for Γ-null sets Chapter Seven: Variational principles Chapter Eight: Smoothness and asymptotic smoothness Chapter Nine: Preliminaries to main results Chapter Ten: Porosity, Γn- and Γ-null sets Chapter Eleven: Porosity and ε-Fréchet differentiability Chapter Twelve: Fréchet differentiability of real-valued functions Chapter Thirteen: Fréchet differentiability of vector-valued functions Chapter Fourteen: Unavoidable porous sets and nondifferentiable maps Chapter Fifteen: Asymptotic Fréchet differentiability Chapter Sixteen: Differentiability of Lipschitz maps on Hilbert spaces Bibliography Index Index of Notation
Cover -- Title Page -- Copyright Page -- Table of Contents -- Chapter 1. Introduction -- 1.1 Key notions and notation -- Chapter 2. Gâteaux Dfferentiability of Lipschitz Functions -- 2.1 Radon-Nikodým Property -- 2.2 Haar and Aronszajn-Gauss Null Sets -- 2.3 Existence Results for Gâteaux Derivatives -- 2.4 Mean Value Estimates -- Chapter 3. Smoothness, Convexity, Porosity, and Separable Determination -- 3.1 A criterion of Differentiability of Convex Functions -- 3.2 Fréchet Smooth and Nonsmooth Renormings -- 3.3 Fréchet Differentiability of Convex Functions -- 3.4 Porosity and Nondifferentiability -- 3.5 Sets of Fréchet Differentiability Points -- 3.6 Separable Determination -- Chapter 4. ε-Fréchet Differentiability -- 4.1 ε-Differentiability and Uniform Smoothness -- 4.2 Asymptotic Uniform Smoothness -- 4.3 ε-Fréchet Differentiability of Functions on Asymptotically Smooth Spaces -- Chapter 5. Γ-Null and Γn-Null Sets -- 5.1 Introduction -- 5.2 Γ-Null Sets and Gâteaux Differentiability -- 5.3 Spaces of Surfaces -- 5.4 Γ- and Γn-Null Sets of low Borel Classes -- 5.5 Equivalent Definitions of Γn-Null Sets -- 5.6 Separable Determination -- Chapter 6. Fréchet Differentiability Except for Γ-Null Sets -- 6.1 Introduction -- 6.2 Regular Points -- 6.3 A Criterion of Fréchet Differentiability -- 6.4 Fréchet Differentiability Except for Γ-Null Sets -- Chapter 7. Variational Principles -- 7.1 Introduction -- 7.2 Variational Principles via Games -- 7.3 Bimetric Variational Principles -- Chapter 8. Smoothness and Asymptotic Smoothness -- 8.1 Modulus of Smoothness -- 8.2 Smooth Bumps with Controlled Modulus -- Chapter 9. Preliminaries to Main Results -- 9.1 Notation, Linear Operators, Tensor Products -- 9.2 Derivatives and Regularity -- 9.3 Deformation of Surfaces Controlled by ωn -- 9.4 Divergence Theorem -- 9.5 Some Integral Estimates
Chapter 10. Porosity, Γn- and Γ-Null Sets -- 10.1 Porous and σ-Porous Sets -- 10.2 A Criterion of Γn-nullness of Porous Sets -- 10.3 Directional Porosity and Γn-Nullness -- 10.4 σ-Porosity and Γn-Nullness -- 10.5 Γ1-Nullness of Porous Sets and Asplundness -- 10.6 Spaces in which σ-Porous Sets are Γ-Null -- Chapter 11. Porosity and ε-Fréchet Differentiability -- 11.1 Introduction -- 11.2 Finite Dimensional Approximation -- 11.3 Slices and ε-Differentiability -- Chapter 12. Fréchet Differentiability of Real-Valued Functions -- 12.1 Introduction and Main Results -- 12.2 An Illustrative Special Case -- 12.3 A Mean Value Estimate -- 12.4 Proof of Theorems -- 12.5 Generalizations and Extensions -- Chapter 13. Fréchet Differentiability of Vector-Valued Functions -- 13.1 Main Results -- 13.2 Regularity Parameter -- 13.3 Reduction to a Special Case -- 13.4 Regular Fréchet Differentiability -- 13.5 Fréchet Differentiability -- 13.6 Simpler Special Cases -- Chapter 14. Unavoidable Porous Sets and Nondifferentiable Maps -- 14.1 Introduction and Main Results -- 14.2 An Unavoidable Porous Set in ℓ1 -- 14.3 Preliminaries to Proofs of Main Results -- 14.4 The Main Construction -- 14.5 The Main Construction -- 14.6 Proof of Theorem -- 14.7 Proof of Theorem -- Chapter 15. Asymptotic Fréchet Differentiability -- 15.1 Introduction -- 15.2 Auxiliary and Finite Dimensional Lemmas -- 15.3 The Algorithm -- 15.4 Regularity of f at x∞ -- 15.5 Linear Approximation of f at x∞ -- 15.6 Proof of Theorem -- Chapter 16. Differentiability of Lipschitz Maps on Hilbert Spaces -- 16.1 Introduction -- 16.2 Preliminaries -- 16.3 The Algorithm -- 16.4 Proof of Theorem -- 16.5 Proof of Lemma -- Bibliography -- Index -- Index of Notation
Index of Notation
Chapter One: Introduction
Chapter Fifteen: Asymptotic Fréchet differentiability
Chapter Six: Férchet differentiability except for Γ-null sets
Chapter Two: Gâteaux differentiability of Lipschitz functions
Index
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Chapter Four: ε-Fréchet differentiability
Chapter Five: Γ-null and Γn-null sets
Chapter Seven: Variational principles
Chapter Eight: Smoothness and asymptotic smoothness
Chapter Nine: Preliminaries to main results
Contents
Chapter Thirteen: Fréchet differentiability of vector-valued functions
Frontmatter --
Chapter Ten: Porosity, Γn- and Γ-null sets
Chapter Sixteen: Differentiability of Lipschitz maps on Hilbert spaces
Chapter Fourteen: Unavoidable porous sets and nondifferentiable maps
Chapter Three: Smoothness, convexity, porosity, and separable determination
Chapter Twelve: Fréchet differentiability of real-valued functions
Bibliography
Chapter Eleven: Porosity and ε-Fréchet differentiability