Strictly convex norms and topology

TL;DR

This paper introduces a new topological property (*) and explores its implications for Banach spaces and compact spaces, characterizing when they admit equivalent strictly convex norms and relating to known topological classes.

Contribution

It defines the property (*) and uses it to characterize Banach spaces and scattered compact spaces with strictly convex norms, linking topology and functional analysis.

Findings

Characterization of Banach spaces with strictly convex norms using (*)

Identification of scattered compact spaces with (*) that admit such norms

Examples including Kunen's compact space and under CH, a non-Gruenhage space with (*)

Abstract

We introduce a new topological property called (*) and the corresponding class of topological spaces, which includes spaces with $G_\delta$-diagonals and Gruenhage spaces. Using (*), we characterise those Banach spaces which admit equivalent strictly convex norms, and give an internal topological characterisation of those scattered compact spaces $K$, for which the dual Banach space $C(K)^*$ admits an equivalent strictly convex dual norm. We establish some relationships between (*) and other topological concepts, and the position of several well-known examples in this context. For instance, we show that $C(\mathcal{K})^*$ admits an equivalent strictly convex dual norm, where $\mathcal{K}$ is Kunen's compact space. Also, under the continuum hypothesis CH, we give an example of a compact scattered non-Gruenhage space having (*).