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Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces

Auteur : Joram Lindenstrauss; David Preiss; Jaroslav Tišer
Éditeur : Princeton : Princeton University Press, 2012.
Collection : Annals of mathematics studies, no. 179.
Édition/format :   Livre électronique : Document : AnglaisVoir toutes les éditions et tous les formats
Base de données :WorldCat
Résumé :
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  Lire la suite...
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Genre/forme : Electronic books
Format – détails additionnels : Print version:
Lindenstrauss, Joram.
Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces.
Princeton : Princeton University Press, ©2012
Type d’ouvrage : Document, Ressource Internet
Format : Ressource Internet, Fichier informatique
Tous les auteurs / collaborateurs : Joram Lindenstrauss; David Preiss; Jaroslav Tišer
ISBN : 9781400842698 1400842697
Numéro OCLC : 769343169
Notes : 14.7 Proof of Theorem.
Description : 1 online resource (436 pages).
Contenu : 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 Points3.6 Separable Determination; Chapter 4. e-Fréchet Differentiability; 4.1 e-Differentiability and Uniform Smoothness; 4.2 Asymptotic Uniform Smoothness; 4.3 e-Fréchet Differentiability of Functions on Asymptotically Smooth Spaces; Chapter 5. G-Null and Gn-Null Sets; 5.1 Introduction; 5.2 G-Null Sets and Gâteaux Differentiability; 5.3 Spaces of Surfaces; 5.4 G- and Gn-Null Sets of low Borel Classes; 5.5 Equivalent Definitions of Gn-Null Sets; 5.6 Separable Determination; Chapter 6. Fréchet Differentiability Except for G-Null Sets; 6.1 Introduction. 6.2 Regular Points6.3 A Criterion of Fréchet Differentiability; 6.4 Fréchet Differentiability Except for G-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?n; 9.4 Divergence Theorem; 9.5 Some Integral Estimates. Chapter 10. Porosity, Gn- and G-Null Sets10.1 Porous and s-Porous Sets; 10.2 A Criterion of Gn-nullness of Porous Sets; 10.3 Directional Porosity and Gn-Nullness; 10.4 s-Porosity and Gn-Nullness; 10.5 G1-Nullness of Porous Sets and Asplundness; 10.6 Spaces in which s-Porous Sets are G-Null; Chapter 11. Porosity and e-Fréchet Differentiability; 11.1 Introduction; 11.2 Finite Dimensional Approximation; 11.3 Slices and e-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 Estimate12.4 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.
Titre de collection : Annals of mathematics studies, no. 179.

Résumé :

Focuses on the difficult question of existence of Frchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. This book provides a bridge between descriptive set theory and  Lire la suite...

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"The book is well written--as one would expect from its distinguished authors, including the late Joram Lindestrauss (1936-2012). It contains many fascinating and profound results. It no doubt will Lire la suite...

 
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