Smooth and polyhedral approximation in Banach spaces

TL;DR

This paper demonstrates that certain Banach space norms can be uniformly approximated by smooth and polyhedral norms, including on spaces like c0, and provides conditions for polyhedral norm existence in weakly compactly generated spaces.

Contribution

It introduces methods for approximating norms with smooth and polyhedral norms on Banach spaces and extends known results to broader classes such as c0 and weakly compactly generated spaces.

Findings

Norms on specific Banach spaces can be approximated by smooth and polyhedral norms.

Any equivalent norm on c0(Γ) can be approximated uniformly by smooth and polyhedral norms.

A necessary condition for polyhedral norms on weakly compactly generated Banach spaces is established.

Abstract

We show that norms on certain Banach spaces $X$ can be approximated uniformly, and with arbitrary precision, on bounded subsets of $X$ by $C^{\infty}$ smooth norms and polyhedral norms. In particular, we show that this holds for any equivalent norm on $c_0(\Gamma)$, where $\Gamma$ is an arbitrary set. We also give a necessary condition for the existence of a polyhedral norm on a weakly compactly generated Banach space, which extends a well-known result of Fonf.