Topology, isomorphic smoothness and polyhedrality in Banach spaces

TL;DR

This paper explores the role of topology in Banach space theory, focusing on $w^*$-locally relatively compact sets and their applications to isomorphic smoothness and polyhedrality.

Contribution

It develops the topological theory of $w^*$-locally relatively compact sets and applies it to problems in Banach space geometry.

Findings

Established properties of $w^*$-locally relatively compact sets.

Connected topological concepts with Banach space smoothness.

Provided new tools for analyzing polyhedrality in Banach spaces.

Abstract

In recent decades, topology has come to play an increasing role in some geometric aspects of Banach space theory. The class of so-called $w^*$-locally relatively compact sets was introduced recently by Fonf, Pallares, Troyanski and the author, and were found to be a useful topological tool in the theory of isomorphic smoothness and polyhedrality in Banach spaces. We develop the topological theory of these sets and present some Banach space applications.