On Fréchet differentiability of Lipschitz maps between Banach spaces

Details On Fréchet differentiability of Lipschitz maps between Banach spaces On Fréchet differentiability of Lipschitz maps between Banach spaces

2004-02-10

arxiv.org/abs/math/0402160v1

Ann. of Math. (2), Vol. 157 (2003), no. 1, 257--288

Mathematics

Functional Analysis

32 pages, published version

Scientific paper

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 $\Gamma$-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 Fr\'echet 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 Fr\'echet differentiable $\Gamma$-almost everywhere provided that it is regularly Gateaux differentiable $\Gamma$-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-Nikodym property are Gateaux differentiable $\Gamma$-almost everywhere. Moreover, Gateaux differentiability implies regular Gateaux differentiability with exception of another kind of negligible sets, so-called $\sigma$-porous sets. The answer to the question is therefore positive in every space in which every $\sigma$-porous set is $\Gamma$-null. We show that this holds for $C(K)$ with $K$ countable compact, the Tsirelson space and for all subspaces of $c_0$, but that it fails for Hilbert spaces.

Affiliated with

Also associated with

No associations

Rating

On Fréchet differentiability of Lipschitz maps between Banach spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On Fréchet differentiability of Lipschitz maps between Banach spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On Fréchet differentiability of Lipschitz maps between Banach spaces will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-168072