A universal differentiability set in Banach spaces with separable dual

TL;DR

This paper proves that in Banach spaces with separable duals, there exists a special subset where every Lipschitz function is Fréchet differentiable somewhere, extending the understanding of differentiability in infinite-dimensional spaces.

Contribution

It constructs a universal differentiability set of Hausdorff dimension 1 in Banach spaces with separable duals, showing all Lipschitz functions are differentiable somewhere in this set.

Findings

Existence of a universal differentiability set in such Banach spaces.

The set is totally disconnected, closed, bounded, and has Hausdorff dimension 1.

Every Lipschitz function on the space is Fréchet differentiable at some point in the set.

Abstract

We show that any non-zero Banach space with a separable dual contains a totally disconnected, closed and bounded subset S of Hausdorff dimension 1 such that every Lipschitz function on the space is Fr\'echet differentiable somewhere in S.