Weak$^*$ closures and derived sets for convex sets in dual Banach spaces

TL;DR

This paper explores the structure of convex sets in dual Banach spaces, establishing that for any nonreflexive space and countable ordinal, there exists a convex set with a specific derived set closure property, extending prior results.

Contribution

It constructs convex sets in dual Banach spaces with prescribed weak$^*$ derived set properties for any countable successor ordinal, generalizing earlier findings.

Findings

Existence of convex sets with prescribed derived set order in dual Banach spaces

Extension of weak$^*$ derived set theory to nonreflexive spaces

Generalization of previous results by Ostrovskii and Silber

Abstract

The paper is devoted to the convex-set counterpart of the theory of weak$^*$ derived sets initiated by Banach and Mazurkiewicz for subspaces. The main result is the following: For every nonreflexive Banach space $X$ and every countable successor ordinal $\alpha$, there exists a convex subset $A$ in $X^*$ such that $\alpha$ is the least ordinal for which the weak$^*$ derived set of order $\alpha$ coincides with the weak$^*$ closure of $A$. This result extends the previously known results on weak$^*$ derived sets by Ostrovskii (2011) and Silber (2021).