eBook

Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

Tišer, Jaroslav, Lindenstrauss, Joram, Preiss, David

2012

Overview

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.

• Publisher: Princeton University Press
• Subject: 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
• ISBN: 9780691153551, 0691153566, 0691153558, 9780691153568, 1400842697, 9781400842698