Weak$^*$ closures and derived sets for convex sets in dual Banach spaces
Volume 268 / 2023
Studia Mathematica 268 (2023), 291-310
MSC: Primary 46B10; Secondary 46B20.
DOI: 10.4064/sm211211-25-6
Published online: 6 October 2022
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 $\mathcal {X}$ and every countable successor ordinal $\alpha $, there exists a convex subset $A$ in $\mathcal {X}^*$ 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).
