| Description |
1 online resource (436 pages) |
| Series |
Annals of mathematics studies ; no. 179 |
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Annals of mathematics studies ; no. 179.
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| Contents |
880-01 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 |
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880-01/(S Machine generated contents note: 1. Introduction -- 1.1. Key notions and notation -- 2. Gateaux differentiability of Lipschitz functions -- 2.1. Radon-Nikodym property -- 2.2. Haar and Aronszajn-Gauss null sets -- 2.3. Existence results for Gateaux derivatives -- 2.4. Mean value estimates -- 3. Smoothness, convexity, porosity, and separable determination -- 3.1. 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 points -- 3.6. Separable determination -- 4. ε-Frechet differentiability -- 4.1. ε-differentiability and uniform smoothness -- 4.2. Asymptotic uniform smoothness -- 4.3. ε-Frechet differentiability of functions on asymptotically smooth spaces -- 5. Γ-null and Γn-null sets -- 5.1. Introduction -- 5.2. Γ-null sets and Gateaux differentiability -- 5.3. Spaces of surfaces, and Γ- and Γn-null sets -- 5.4. Γ- and Γn-null sets of low Borel classes -- 5.5. Equivalent definitions of Γn-null sets -- 5.6. Separable determination -- 6. Frechet differentiability except for Γ-null sets -- 6.1. Introduction -- 6.2. Regular points -- 6.3. criterion of Frechet differentiability -- 6.4. Frechet differentiability except for Γ-null sets -- 7. Variational principles -- 7.1. Introduction -- 7.2. Variational principles via games -- 7.3. Bimetric variational principles -- 8. Smoothness and asymptotic smoothness -- 8.1. Modulus of smoothness -- 8.2. Smooth bumps with controlled modulus -- 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 -- 10. Porosity, Γn-and Γ-null sets -- 10.1. Porous and σ-porous sets -- 10.2. 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 -- 11. Porosity and ε-Frechet differentiability -- 11.1. Introduction -- 11.2. Finite dimensional approximation -- 11.3. Slices and ε-differentiability -- 12. Frechet differentiability of real-valued functions -- 12.1. Introduction and main results -- 12.2. illustrative special case -- 12.3. mean value estimate -- 12.4. Proof of Theorems 12.1.1 and 12.1.3 -- 12.5. Generalizations and extensions -- 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 -- 14. Unavoidable porous sets and nondifferentiable maps -- 14.1. Introduction and main results -- 14.2. unavoidable porous set in l1 -- 14.3. Preliminaries to proofs of main results -- 14.4. main construction, Part I -- 14.5. main construction, Part II -- 14.6. Proof of Theorem 14.1.3 -- 14.7. Proof of Theorem 14.1.1 -- 15. Asymptotic Frechet differentiability -- 15.1. Introduction -- 15.2. Auxiliary and finite dimensional lemmas -- 15.3. algorithm -- 15.4. Regularity of f at x[∞] -- 15.5. Linear approximation of f at x[∞] -- 15.6. Proof of Theorem 15.1.3 -- 16. Differentiability of Lipschitz maps on Hilbert spaces -- 16.1. Introduction -- 16.2. Preliminaries -- 16.3. algorithm -- 16.4. Proof of Theorem 16.1.1 -- 16.5. Proof of Lemma 16.2.1 |
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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 |
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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 |
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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 |
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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 |
| 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 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 |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Print version record |
| Subject |
Banach spaces.
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Calculus of variations.
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Functional analysis.
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MATHEMATICS -- Calculus.
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MATHEMATICS -- Mathematical Analysis.
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MATHEMATICS -- Set Theory.
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Banach spaces
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Calculus of variations
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Functional analysis
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| Form |
Electronic book
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| Author |
Preiss, David.
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Tišer, Jaroslav, 1957-
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| ISBN |
9781400842698 |
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1400842697 |
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1283379953 |
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9781283379953 |
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