The Resource Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces
Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces
 Summary
 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 vectorvalued 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 vectorvalued 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
 Language
 eng
 Extent
 1 online resource (436 pages).
 Contents

 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 RadonNikodým Property; 2.2 Haar and AronszajnGauss 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 Points3.6 Separable Determination; Chapter 4. eFréchet Differentiability; 4.1 eDifferentiability and Uniform Smoothness; 4.2 Asymptotic Uniform Smoothness; 4.3 eFréchet Differentiability of Functions on Asymptotically Smooth Spaces; Chapter 5. GNull and GnNull Sets; 5.1 Introduction; 5.2 GNull Sets and Gâteaux Differentiability; 5.3 Spaces of Surfaces; 5.4 G and GnNull Sets of low Borel Classes; 5.5 Equivalent Definitions of GnNull Sets; 5.6 Separable Determination; Chapter 6. Fréchet Differentiability Except for GNull Sets; 6.1 Introduction
 6.2 Regular Points6.3 A Criterion of Fréchet Differentiability; 6.4 Fréchet Differentiability Except for GNull 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?n; 9.4 Divergence Theorem; 9.5 Some Integral Estimates
 Chapter 10. Porosity, Gn and GNull Sets10.1 Porous and sPorous Sets; 10.2 A Criterion of Gnnullness of Porous Sets; 10.3 Directional Porosity and GnNullness; 10.4 sPorosity and GnNullness; 10.5 G1Nullness of Porous Sets and Asplundness; 10.6 Spaces in which sPorous Sets are GNull; Chapter 11. Porosity and eFréchet Differentiability; 11.1 Introduction; 11.2 Finite Dimensional Approximation; 11.3 Slices and eDifferentiability; Chapter 12. Fréchet Differentiability of RealValued Functions; 12.1 Introduction and Main Results; 12.2 An Illustrative Special Case
 12.3 A Mean Value Estimate12.4 Proof of Theorems; 12.5 Generalizations and Extensions; Chapter 13. Fréchet Differentiability of VectorValued 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
 Isbn
 9781400842698
 Label
 Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces
 Title
 Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces
 Language
 eng
 Summary
 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 vectorvalued 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 vectorvalued 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
 Cataloging source
 EBLCP
 Index
 index present
 Literary form
 non fiction
 Nature of contents

 dictionaries
 bibliography
 Series statement
 Annals of mathematics studies
 Series volume
 no. 179
 Label
 Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces
 Bibliography note
 Includes bibliographical references and index
 Carrier category
 online resource
 Carrier category code
 cr
 Carrier MARC source
 rdacarrier
 Content category
 text
 Content type code
 txt
 Content type MARC source
 rdacontent
 Contents

 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 RadonNikodým Property; 2.2 Haar and AronszajnGauss 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 Points3.6 Separable Determination; Chapter 4. eFréchet Differentiability; 4.1 eDifferentiability and Uniform Smoothness; 4.2 Asymptotic Uniform Smoothness; 4.3 eFréchet Differentiability of Functions on Asymptotically Smooth Spaces; Chapter 5. GNull and GnNull Sets; 5.1 Introduction; 5.2 GNull Sets and Gâteaux Differentiability; 5.3 Spaces of Surfaces; 5.4 G and GnNull Sets of low Borel Classes; 5.5 Equivalent Definitions of GnNull Sets; 5.6 Separable Determination; Chapter 6. Fréchet Differentiability Except for GNull Sets; 6.1 Introduction
 6.2 Regular Points6.3 A Criterion of Fréchet Differentiability; 6.4 Fréchet Differentiability Except for GNull 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?n; 9.4 Divergence Theorem; 9.5 Some Integral Estimates
 Chapter 10. Porosity, Gn and GNull Sets10.1 Porous and sPorous Sets; 10.2 A Criterion of Gnnullness of Porous Sets; 10.3 Directional Porosity and GnNullness; 10.4 sPorosity and GnNullness; 10.5 G1Nullness of Porous Sets and Asplundness; 10.6 Spaces in which sPorous Sets are GNull; Chapter 11. Porosity and eFréchet Differentiability; 11.1 Introduction; 11.2 Finite Dimensional Approximation; 11.3 Slices and eDifferentiability; Chapter 12. Fréchet Differentiability of RealValued Functions; 12.1 Introduction and Main Results; 12.2 An Illustrative Special Case
 12.3 A Mean Value Estimate12.4 Proof of Theorems; 12.5 Generalizations and Extensions; Chapter 13. Fréchet Differentiability of VectorValued 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
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 Extent
 1 online resource (436 pages).
 Form of item
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 Isbn
 9781400842698
 Media category
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 Media MARC source
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 Media type code
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 Specific material designation
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