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Fréchet 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 Frechet 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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| Otros Autores: | , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Princeton :
Princeton University Press,
2012.
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| Colección: | Book collections on Project MUSE.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- 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 Frechet Smooth and Nonsmooth Renormings; 3.3 Frechet Differentiability of Convex Functions; 3.4 Porosity and Nondifferentiability.
- 3.5 Sets of Frechet Differentiability Points3.6 Separable Determination; Chapter 4. e-Frechet Differentiability; 4.1 e-Differentiability and Uniform Smoothness; 4.2 Asymptotic Uniform Smoothness; 4.3 e-Frechet 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. Frechet Differentiability Except for G-Null Sets; 6.1 Introduction.
- 6.2 Regular Points6.3 A Criterion of Frechet Differentiability; 6.4 Frechet 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-Frechet Differentiability; 11.1 Introduction; 11.2 Finite Dimensional Approximation; 11.3 Slices and e-Differentiability; Chapter 12. Frechet 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. Frechet Differentiability of Vector-Valued Functions; 13.1 Main Results; 13.2 Regularity Parameter; 13.3 Reduction to a Special Case; 13.4 Regular Frechet Differentiability; 13.5 Frechet 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.