TAILIEUCHUNG - Đề tài "On Fr´echet differentiability of Lipschitz maps between Banach spaces "

A well-known open question is whether every countable collection of Lipschitz functions on a Banach space X with separable dual has a common point of Fr´chet differentiability. We show that the answer is positive for e some infinite-dimensional X. Previously, even for collections consisting of two functions this has been known for finite-dimensional X only (although for one function the answer is known to be affirmative in full generality). Our aims are achieved by introducing a new class of null sets in Banach spaces (called Γ-null sets), whose definition involves both the notions of category and measure, and showing. | Annals of Mathematics On Fr echet differentiability of Lipschitz maps between Banach spaces By Joram Lindenstrauss and David Preiss Annals of Mathematics 157 2003 257-288 On Frechet differentiability of Lipschitz maps between Banach spaces By Joram Lindenstrauss and David Preiss Abstract A well-known open question is whether every countable collection of Lipschitz functions on a Banach space X with separable dual has a common point of Frechet differentiability. We show that the answer is positive for some infinite-dimensional X. Previously even for collections consisting of two functions this has been known for finite-dimensional X only although for one function the answer is known to be affirmative in full generality . Our aims are achieved by introducing a new class of null sets in Banach spaces called r-null sets whose definition involves both the notions of category and measure and showing that the required differentiability holds almost everywhere with respect to it. We even obtain existence of Frechet derivatives of Lipschitz functions between certain infinite-dimensional Banach spaces no such results have been known previously. Our main result states that a Lipschitz map between separable Banach spaces is Frechet differentiable r-almost everywhere provided that it is regularly Gateaux differentiable r-almost everywhere and the Gateaux derivatives stay within a norm separable space of operators. It is easy to see that Lipschitz maps of X to spaces with the Radon-Nikodým property are Gateaux differentiable r-almost everywhere. Moreover Gateaux differentiability implies regular Gateaux differentiability with exception of another kind of negligible sets so-called ơ-porous sets. The answer to the question is therefore positive in every space in which every ơ-porous set is T-null. We show that this holds for C K with K countable compact the Tsirelson space and for all subspaces of c0 but that it fails for Hilbert spaces. 1. Introduction One of the main aims of .

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