Approximation of norms on Banach spaces

TL;DR

This paper extends the approximation of equivalent norms by smooth and polyhedral norms from classical spaces to more general Banach spaces with uncountable symmetric bases, and constructs specific compact spaces for norm approximation.

Contribution

It generalizes previous results on norm approximation to broader classes of Banach spaces and constructs scattered compact spaces with prescribed properties for norm approximation.

Findings

Approximation of norms on Banach spaces with uncountable symmetric bases.

Extension of norm approximation results to Lorentz, Nakano, and Orlicz spaces.

Existence of scattered compact spaces with high Cantor-Bendixson height for norm approximation.

Abstract

Relatively recently it was proved that if $\Gamma$ is an arbitrary set, then any equivalent norm on $c_0(\Gamma)$ can be approximated uniformly on bounded sets by polyhedral norms and $C^\infty$ smooth norms, with arbitrary precision. We extend this result to more classes of spaces having uncountable symmetric bases, such as preduals of the `discrete' Lorentz spaces $d(w,1,\Gamma)$, and certain symmetric Nakano spaces and Orlicz spaces. We also show that, given an arbitrary ordinal number $\alpha$, there exists a scattered compact space $K$ having Cantor-Bendixson height at least $\alpha$, such that every equivalent norm on $C(K)$ can be approximated as above.