On Frechet 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 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 IF-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 Ftechet 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-Nikodym property are Gateaux differentiable r-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 r-null. We show that this holds for C(K) with K countable compact, the Tsirelson space and for allsubspaces of c(0), but that it fails for Hilbert spaces.