Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (eBook)

Joram Lindenstrauss, David Preiss, Jaroslav Tišer (Autoren)

eBook Download: EPUB

2012
Princeton University Press (Verlag)
978-1-4008-4269-8 (ISBN)

Infos

This book makes a significant inroad into the unexpectedly difficult question of existence of Frechet 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 Frechet 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 Frechet 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.

Autor

Joram Lindenstrauss is professor emeritus of mathematics at the Hebrew University of Jerusalem. David Preiss is professor of mathematics at the University of Warwick. Jaroslav Tier is associate professor of mathematics at Czech Technical University in Prague.

Produkt-Details

• Erscheint lt. Verlag: 26.2.2012
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Finanz- / Wirtschaftsmathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
• Themenwelt: Technik
• Schlagworte: Addition • Approximation • Asplund space • Asymptote • asymptotically smooth norm • asymptotically smooth space • asymptotic uniform smoothness • Auxiliary function • Banach space • Basis (linear algebra) • Big O notation • Borel measure • Borel set • Borel sets • bounded operator • Bounded set (topological vector space) • Bump • Bump function • Calculation • Chebyshev's inequality • Compact space • complete metric space • Completeness • cone-monotone function • conjecture • continuous function • Continuous function (set theory) • Contradiction • convex function • convex hull • convex set • corollary • Countable set • counterexample • Deformation • Dense set • Derivative • descriptive set theory • Differentiable function • Dimension • Dimensional Analysis • Dimension (vector space) • Directional derivative • direct sum • Divergence Theorem • Division by zero • dot product • dual space • Equation • estimation • Euclidean space • flat surface • _-Frchet derivative • Frchet derivative • _-Frchet differentiability • Frchet differentiability • Frchet smooth norm • Fréchet derivative • Frechet differentiability • Fréchet differentiability • Fréchet smooth norm • Fubini's Theorem • Gâteaux derivative • Gâteaux differentiability • Gteaux derivative • Gteaux differentiability • Hahn–Banach Theorem • higher dimensional space • hilbert space • Infimum and supremum • infinite dimensional space • irregular behavior • irregularity point • Lebesgue measure • Linear Approximation • Linear Function • Linear map • Linear Operators • Linear span • Linear subspace • Lipschitz continuity • Lipschitz function • Lipschitz map • low Borel classes • lower semicontinuity • Maxima and minima • mean value estimate • mean value theorem • Measurable function • Measure (mathematics) • Metric Space • Modulus • Monotonic Function • multidimensional mean value • Natural number • nonlinear functional analysis • nonseparable space • Norm (mathematics) • Null set • _-null sets • null sets • Open set • Operator norm • Parameter • Perturbation • perturbation function • perturbation game • Pointwise • Porosity • Porous set • _-porous sets • porous sets • Product topology • Projection (linear algebra) • Pseudometric • Quantity • Rademacher's theorem • Radon-Nikod_m property • Radon-Nikodým property • Real variable • regular behavior • regular differentiability • regularity parameter • Renorming • Requirement • scientific notation • Semi-continuity • separable determination • separable dual • separable space • Sign (mathematics) • slice • smooth bump • Smoothness • Sobolev Space • Special case • Standard basis • Subderivative • subsequence • Subset • Subspace • Tangent vector • Tensor Products • Theorem • three-dimensional space • Three-dimensional space (mathematics) • Topology • Transfinite induction • two-dimensional space • two-player game • uniform continuity • Unit sphere • Unit vector • Upper and lower bounds • Variable (mathematics) • Variational Principle • Variational Principles • Weak Derivative • Γ-null sets • ε-Fréchet derivative • ε-Fréchet differentiability • σ-porous sets
• ISBN-10: 1-4008-4269-7 / 1400842697
• ISBN-13: 978-1-4008-4269-8 / 9781400842698