$C^k$-smooth approximations of LUR norms

TL;DR

This paper demonstrates that certain Banach spaces with smooth norms can be approximated by norms that are simultaneously smooth, locally uniformly rotund, and limits of smoother norms, enhancing the understanding of norm smoothness in functional analysis.

Contribution

It establishes that WCG Banach spaces with a $C^k$-smooth norm admit an equivalent norm that is $C^1$-smooth, LUR, and a uniform limit of $C^k$-smooth norms, extending to spaces of continuous functions on ordinals.

Findings

Existence of smooth, LUR, and limit-of-smoother norms in WCG Banach spaces.

Extension of results to spaces of continuous functions on ordinals with infinite smoothness.

Approximation of norms by smoother norms in the context of Banach space geometry.

Abstract

Let $X$ be a WCG Banach space admitting a $C^k$-Fr\' echet smooth norm. Then $X$ admits an equivalent norm which is simultaneously $C^1$-Fr\' echet smooth, LUR, and a uniform limit of $C^k$-Fr\' echet smooth norms. If $X=C([0,\alpha])$, where $\alpha$ is an ordinal, then the same conclusion holds true with $k=\infty$.