C(K) spaces that could be Gateaux Differentiability Spaces and which are not weak Asplund
by
Dean J. Ives
University College London
Coauthors: David Preiss (University College London)

We obtain partial results that could eventually lead to showing that there are Banach spaces that are GDS spaces but not weak Asplund spaces. Let K be a first countable Hausdorff compact space. A topology \tau is defined on C(K) that is finer than the norm topology but with which C(K) is still a Baire space.

It is shown that for a continuous convex function f on C(K) such that every subdifferential, \partialf(j), of f contains a measure of finite support, f is Gâteaux differentiable on a \tau residual set. An example of such a function is the distance from a finite dimensional subspace of C(K). A non-trivial example of such a space is the double arrow space D of functions on [0, 1] that are right continuous at every 0 <= x < 1, left continuous at x=1, and have left limits at at every 0 < x < 1, equipped with the supremum norm We can also show that for a continuous convex function f on D such that every subdifferential contains only atomic measures, f is Gâteaux differentiable on a \tau residual set.

(T)

Date received: April 4, 2000