Abstract: There are two types of differentiability for Lipschitz functions between infinite dimensional Banach spaces. For Gateau differentiability of such functions there are satisfatory general existence theorems. However Gateaux derivatives are only weak linear approximations of a function. It is much more desirable to have points of Frechet differentiability. It turns out that it is very hard to prove existence of such points. One source for the difficulty is a lack of a proper notion of "almost everywhere".

In joint work with David Preiss we introduce a new notion of null sets which enables existence results to be proved for points of Frechet differentiability in some cases.