Almost weak Asplund spaces

Abstract

Continuous convex functions have long been known to be generically differentiable on Euclidean spaces. However, in 1968 Asplund decided to investigate and classify those Banach spaces that possess this Euclidean space property. Specifically, Asplund investigated those Banach spaces on which every continuous convex function is Gâteaux (Fréchet) differentiable at the points of a residual subset of their domain. Such spaces are now known as weak Asplund (Asplund) spaces. While the study of Asplund spaces has flourished into a beautifully detailed story admitting several characterisations, the corresponding theory for weak Asplund spaces has been rather thin. In particular, there are no known characterisations for this class of spaces. In one attempt to provide such a characterisation, the class of Gâteaux differentiability spaces, that is, those spaces on which every continuous convex function is densely Gâteaux differentiable, was introduced. While it follows from the definition that every weak Asplund space is a Gâteaux differentiability space, it has been a long standing open problem, first asked by Larman and Phelps in 1979, as to whether every Gâteaux differentiability space is a weak Asplund space. The main goal of this thesis is to construct a Gâteaux differentiability space that is not weak Asplund, thus answering the question of Larman and Phelps from 1979. We achieve this counter-example by considering Banach spaces of the form (C(K),∥·∥∞), where K is a compact Hausdorff space.