Note on Kadets Klee property and Asplund spaces

TL;DR

This paper proves that a weak Asplund Banach space with the $oldsymbol{ ext{w}^*}$-$oldsymbol{ ext{w}}$-Kadets Klee property is actually an Asplund space, clarifying the relationship between these properties.

Contribution

It establishes that the $oldsymbol{ ext{w}^*}$-$oldsymbol{ ext{w}}$-Kadets Klee property implies the Asplund property in weak Asplund Banach spaces, extending previous understanding.

Findings

Weak Asplund spaces with the $ ext{w}^*$-$ ext{w}$-Kadets Klee property are Asplund.

The result links geometric properties with differentiability in Banach spaces.

Provides conditions under which weak Asplund spaces are Asplund.

Abstract

A typical result in this note is that if $X$ is a Banach space which is a weak Asplund space and has the $\omega^*$-$\omega$-Kadets Klee property, then $X$ is already an Asplund space.