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

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).