Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Princeton
Princeton University Press
2012
|
| Schriftenreihe: | Annals of mathematics studies
no. 179 |
| Schlagworte: | |
| Online-Zugang: | FAW01 FAW02 Volltext |
| Beschreibung: | 14.7 Proof of Theorem 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 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 Includes bibliographical references and index |
| Beschreibung: | 1 Online-Ressource (436 pages) |
| ISBN: | 0691153558 0691153566 1400842697 9780691153551 9780691153568 9781400842698 |
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| 245 | 1 | 0 | |a Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces |
| 264 | 1 | |a Princeton |b Princeton University Press |c 2012 | |
| 300 | |a 1 Online-Ressource (436 pages) | ||
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| 490 | 0 | |a Annals of mathematics studies |v no. 179 | |
| 500 | |a 14.7 Proof of Theorem | ||
| 500 | |a 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 | ||
| 500 | |a 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 | ||
| 500 | |a 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 | ||
| 500 | |a 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 | ||
| 500 | |a 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 | ||
| 500 | |a 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 | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Mathematics | |
| 650 | 7 | |a MATHEMATICS / Calculus |2 bisacsh | |
| 650 | 7 | |a MATHEMATICS / Mathematical Analysis |2 bisacsh | |
| 650 | 7 | |a MATHEMATICS / Set Theory |2 bisacsh | |
| 650 | 7 | |a Banach spaces |2 fast | |
| 650 | 7 | |a Calculus of variations |2 fast | |
| 650 | 7 | |a Functional analysis |2 fast | |
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Banach spaces | |
| 650 | 4 | |a Calculus of variations | |
| 650 | 4 | |a Functional analysis | |
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| 700 | 1 | |a Preiss, David |e Sonstige |4 oth | |
| 700 | 1 | |a Tišer, Jaroslav |e Sonstige |4 oth | |
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Datensatz im Suchindex
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| author | Lindenstrauss, Joram |
| author_facet | Lindenstrauss, Joram |
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| author_sort | Lindenstrauss, Joram |
| author_variant | j l jl |
| building | Verbundindex |
| bvnumber | BV043070535 |
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| dewey-full | 515/.88 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.88 |
| dewey-search | 515/.88 |
| dewey-sort | 3515 288 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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| id | DE-604.BV043070535 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:16:32Z |
| institution | BVB |
| isbn | 0691153558 0691153566 1400842697 9780691153551 9780691153568 9781400842698 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-028494727 |
| oclc_num | 769343169 |
| open_access_boolean | |
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| owner_facet | DE-1046 DE-1047 |
| physical | 1 Online-Ressource (436 pages) |
| psigel | ZDB-4-EBA ZDB-4-EBA FAW_PDA_EBA |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Princeton University Press |
| record_format | marc |
| series2 | Annals of mathematics studies |
| spelling | Lindenstrauss, Joram Verfasser aut Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces Princeton Princeton University Press 2012 1 Online-Ressource (436 pages) txt rdacontent c rdamedia cr rdacarrier Annals of mathematics studies no. 179 14.7 Proof of Theorem 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 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 Includes bibliographical references and index Mathematics MATHEMATICS / Calculus bisacsh MATHEMATICS / Mathematical Analysis bisacsh MATHEMATICS / Set Theory bisacsh Banach spaces fast Calculus of variations fast Functional analysis fast Mathematik Banach spaces Calculus of variations Functional analysis Fréchet-Differenzierbarkeit (DE-588)4370829-8 gnd rswk-swf Lipschitz-Bedingung (DE-588)4167811-4 gnd rswk-swf Lipschitz-Bedingung (DE-588)4167811-4 s Fréchet-Differenzierbarkeit (DE-588)4370829-8 s 1\p DE-604 Preiss, David Sonstige oth Tišer, Jaroslav Sonstige oth http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=421487 Aggregator Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Lindenstrauss, Joram Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces Mathematics MATHEMATICS / Calculus bisacsh MATHEMATICS / Mathematical Analysis bisacsh MATHEMATICS / Set Theory bisacsh Banach spaces fast Calculus of variations fast Functional analysis fast Mathematik Banach spaces Calculus of variations Functional analysis Fréchet-Differenzierbarkeit (DE-588)4370829-8 gnd Lipschitz-Bedingung (DE-588)4167811-4 gnd |
| subject_GND | (DE-588)4370829-8 (DE-588)4167811-4 |
| title | Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces |
| title_auth | Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces |
| title_exact_search | Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces |
| title_full | Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces |
| title_fullStr | Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces |
| title_full_unstemmed | Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces |
| title_short | Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces |
| title_sort | frechet differentiability of lipschitz functions and porous sets in banach spaces |
| topic | Mathematics MATHEMATICS / Calculus bisacsh MATHEMATICS / Mathematical Analysis bisacsh MATHEMATICS / Set Theory bisacsh Banach spaces fast Calculus of variations fast Functional analysis fast Mathematik Banach spaces Calculus of variations Functional analysis Fréchet-Differenzierbarkeit (DE-588)4370829-8 gnd Lipschitz-Bedingung (DE-588)4167811-4 gnd |
| topic_facet | Mathematics MATHEMATICS / Calculus MATHEMATICS / Mathematical Analysis MATHEMATICS / Set Theory Banach spaces Calculus of variations Functional analysis Mathematik Fréchet-Differenzierbarkeit Lipschitz-Bedingung |
| url | http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=421487 |
| work_keys_str_mv | AT lindenstraussjoram frechetdifferentiabilityoflipschitzfunctionsandporoussetsinbanachspaces AT preissdavid frechetdifferentiabilityoflipschitzfunctionsandporoussetsinbanachspaces AT tiserjaroslav frechetdifferentiabilityoflipschitzfunctionsandporoussetsinbanachspaces |