Fréchet differentiability of Lipschitz functions and porous sets in Banach spaces [[electronic resource] /] / Joram Lindenstrauss, David Preiss, Jaroslav Tiser
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| Author: |
Lindenstrauss Joram <1936->
|
| Title: |
Fréchet differentiability of Lipschitz functions and porous sets in Banach spaces [[electronic resource] /] / Joram Lindenstrauss, David Preiss, Jaroslav Tiser
|
| Publisher: | Princeton, : Princeton University Press, 2012 |
| Designation of edition : | Course Book |
| Physical description: | 1 online resource (436 p.) |
| Dewey: | 515/.88 |
| Topical subject: | Banach spaces |
| Calculus of variations | |
| Functional analysis | |
| Uncontrolled subject: | Asplund space |
| Banach space | |
| Borel sets | |
| Euclidean space | |
| Frechet differentiability | |
| Fréchet derivative | |
| Fréchet differentiability | |
| Fréchet smooth norm | |
| Gâteaux derivative | |
| Gâteaux differentiability | |
| Hilbert space | |
| Lipschitz function | |
| Lipschitz map | |
| Radon-Nikodým property | |
| asymptotic uniform smoothness | |
| asymptotically smooth norm | |
| asymptotically smooth space | |
| bump | |
| completeness | |
| cone-monotone function | |
| convex function | |
| deformation | |
| derivative | |
| descriptive set theory | |
| flat surface | |
| higher dimensional space | |
| infinite dimensional space | |
| irregular behavior | |
| irregularity point | |
| linear operators | |
| low Borel classes | |
| lower semicontinuity | |
| mean value estimate | |
| modulus | |
| multidimensional mean value | |
| nonlinear functional analysis | |
| nonseparable space | |
| null sets | |
| perturbation function | |
| perturbation game | |
| perturbation | |
| porosity | |
| porous sets | |
| regular behavior | |
| regular differentiability | |
| regularity parameter | |
| renorming | |
| separable determination | |
| separable dual | |
| separable space | |
| slice | |
| smooth bump | |
| subspace | |
| tensor products | |
| three-dimensional space | |
| two-dimensional space | |
| two-player game | |
| variational principle | |
| variational principles | |
| Γ-null sets | |
| ε-Fréchet derivative | |
| ε-Fréchet differentiability | |
| σ-porous sets | |
| Classification: | SI 830 |
| Other authors: |
PreissDavid
TišerJaroslav <1957->
|
| General notes: | Description based upon print version of record. |
| Bibliography note: | Includes bibliographical references and indexes. |
| Formatted content note: | Frontmatter -- Contents -- Chapter One: Introduction -- Chapter Two: Gâteaux differentiability of Lipschitz functions -- Chapter Three: Smoothness, convexity, porosity, and separable determination -- Chapter Four: ε-Fréchet differentiability -- Chapter Five: Γ-null and Γn-null sets -- Chapter Six: Férchet differentiability except for Γ-null sets -- Chapter Seven: Variational principles -- Chapter Eight: Smoothness and asymptotic smoothness -- Chapter Nine: Preliminaries to main results -- Chapter Ten: Porosity, Γn- and Γ-null sets -- Chapter Eleven: Porosity and ε-Fréchet differentiability -- Chapter Twelve: Fréchet differentiability of real-valued functions -- Chapter Thirteen: Fréchet differentiability of vector-valued functions -- Chapter Fourteen: Unavoidable porous sets and nondifferentiable maps -- Chapter Fifteen: Asymptotic Fréchet differentiability -- Chapter Sixteen: Differentiability of Lipschitz maps on Hilbert spaces -- Bibliography -- Index -- Index of Notation |
| Summary, etc: | 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. |
| Preferred title for the work: | Fréchet differentiability of Lipschitz functions and porous sets in Banach spaces |
| ISBN: | 1-283-37995-3 |
| 9786613379955 | |
| 1-4008-4269-7 | |
| Format: | Language material  |
| Bibliographic level | Monograph |
| Language: | English |
| Record Nr.: | 9910789737103321 |
| You will find it: | Univ. Federico II |
| Opac: | Check copies here |
Series:
Annals of mathematics studies ; ; no. 179.