06944nam 2201525 a 450 991082339820332120200520144314.01-283-37995-397866133799551-4008-4269-710.1515/9781400842698(CKB)2670000000133884(EBL)827806(OCoLC)769343169(SSID)ssj0000575876(PQKBManifestationID)11396459(PQKBTitleCode)TC0000575876(PQKBWorkID)10553953(PQKB)11008932(StDuBDS)EDZ0001756336(DE-B1597)447361(OCoLC)979582934(DE-B1597)9781400842698(Au-PeEL)EBL827806(CaPaEBR)ebr10521870(CaONFJC)MIL337995(PPN)199244979(PPN)187959625(FR-PaCSA)88838074(MiAaPQ)EBC827806(EXLCZ)99267000000013388420111017d2012 uy 0engur|n|---|||||txtccrFréchet differentiability of Lipschitz functions and porous sets in Banach spaces /Joram Lindenstrauss, David Preiss, Jaroslav TiserCourse BookPrinceton Princeton University Press20121 online resource (436 p.)Annals of mathematics studies ;no. 179Description based upon print version of record.0-691-15355-8 0-691-15356-6 Includes bibliographical references and indexes. 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 NotationThis 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.Annals of mathematics studies ;no. 179.Banach spacesCalculus of variationsFunctional analysisAsplund 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.Banach spaces.Calculus of variations.Functional analysis.515/.88SI 830rvkLindenstrauss Joram1936-41187Preiss David515729Tišer Jaroslav1957-515783MiAaPQMiAaPQMiAaPQBOOK9910823398203321Fréchet differentiability of Lipschitz functions and porous sets in Banach spaces854568UNINA