A topological characterization of dual strict convexity in Asplund spaces

TL;DR

This paper characterizes dual strict convexity in Asplund spaces through a topological property of the dual unit sphere, linking geometric and topological aspects of Banach space theory.

Contribution

It establishes an equivalence between the existence of an equivalent strictly convex dual norm and a specific topological property of the dual sphere in Asplund spaces.

Findings

Dual strict convexity characterized by property (*)

Topological property (*) introduced for dual spheres

Equivalence between norm geometry and topology in Asplund spaces

Abstract

Let $X$ be an Asplund space. We show that the existence of an equivalent norm on $X$ having a strictly convex dual norm is equivalent to the dual unit sphere $S_{X^*}$ (equivalently $X^*$) possessing a non-linear topological property called ($*$), which was introduced by J. Orihuela, S. Troyanski and the author.