Frechet 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 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 desc...

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Bibliographic Details
Main Authors:	Lindenstrauss, Joram, Preiss, David, Tišer, Jaroslav
Format:	eBook Book
Language:	English
Published:	Princeton Princeton University Press 2012
Edition:	1
Series:	Annals of Mathematics Studies
Subjects:	Approximation Auxiliary function Banach space Banach spaces Big O notation Borel measure Borel set Bounded operator Bounded set (topological vector space) Bump function Calculus of variations Chebyshev's inequality Compact space Complete metric space Continuous function Continuous function (set theory) Contradiction Convex function Convex hull Convex set Corollary Countable set Dense set Derivative Descriptive set theory Differentiable function Dimension Dimension (vector space) Dimensional analysis Directional derivative Divergence theorem Division by zero Estimation Frechet spaces Fubini's theorem Functional analysis Hilbert space Infimum and supremum Lebesgue measure Linear approximation Linear map Linear span Lipschitz continuity Lipschitz spaces MATHEMATICS MATHEMATICS / Applied MATHEMATICS / General MATHEMATICS / Mathematical Analysis Measurable function Measure (mathematics) Metric space Monotonic function Norm (mathematics) Null set Open set Parameter Perturbation function Porosity Porous set Projection (linear algebra) Rademacher's theorem Requirement Scientific notation Semi-continuity Separable space Sign (mathematics) Smoothness Sobolev space Special case Standard basis Subset Theorem Two-dimensional space Uniform continuity Unit sphere Unit vector Upper and lower bounds Variable (mathematics) Variational principle Perturbation function Open set Measurable function Subset Convex set Auxiliary function Bounded operator Rademacher's theorem Dense set Projection (linear algebra) Semi-continuity Chebyshev's inequality Hilbert space Divergence theorem Convex function Two-dimensional space Corollary Contradiction Monotonic function Approximation Estimation Continuous function Fubini's theorem Directional derivative Division by zero Requirement Dimension (vector space) Infimum and supremum Convex hull Big O notation Bump function Porous set Derivative Special case Unit sphere Complete metric space Sign (mathematics) Continuous function (set theory) Metric space Linear span Sobolev space Unit vector Null set Countable set Upper and lower bounds Lipschitz continuity Descriptive set theory Lebesgue measure Linear map Dimensional analysis Theorem Borel measure Differentiable function Bounded set (topological vector space) Compact space Standard basis Borel set Dimension Banach space Smoothness Variable (mathematics) Separable space Norm (mathematics) Linear approximation Variational principle Parameter Scientific notation Porosity Measure (mathematics) Uniform continuity
ISBN:	9780691153551, 0691153566, 0691153558, 9780691153568, 1400842697, 9781400842698
Online Access:	Get full text
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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 bibliography (p. [415]-418) and indexes
ISBN:	9780691153551 0691153566 0691153558 9780691153568 1400842697 9781400842698
DOI:	10.1515/9781400842698