Differentiability of Banach Spaces via Constructible Sets

TL;DR

This paper proves that certain separability conditions in Banach spaces ensure the existence of an equivalent Fréchet differentiable norm, advancing the understanding of smoothness in infinite-dimensional spaces.

Contribution

It establishes new conditions under which Banach spaces admit an equivalent Fréchet differentiable norm, linking separability properties to smoothness.

Findings

Banach spaces with weak*-separable dual balls admit differentiable norms

Weak*-closed convex subsets being separable imply differentiability

Constructibility of convex sets in dual spaces leads to smooth norms

Abstract

the main goal of this paper is to prove that any Banach space X, that every dual ball in X** is weak* -separable, or every weak* -closed convex subset in X** is weak* -separable, or every norm-closed convex set in X* is constructible, admits an equivalent Frechet differentiable norm.