Weak$^*$ derived sets of convex sets in duals of non-reflexive spaces

TL;DR

This paper studies the structure of weak$^*$ derived sets of convex subsets in duals of non-reflexive Banach spaces, revealing the existence of convex sets of various finite and transfinite orders.

Contribution

It demonstrates that duals of non-reflexive Banach spaces contain convex subsets of any finite order and of order ω+1, advancing understanding of their weak$^*$ limit structures.

Findings

Existence of convex subsets of any finite order in duals of non-reflexive spaces.

Existence of convex subsets of order ω+1 in such duals.

Characterization of weak$^*$ derived sets in non-reflexive Banach space duals.

Abstract

We investigate weak$^*$ derived sets, that is the sets of weak$^*$ limits of bounded nets, of convex subsets of duals of non-reflexive Banach spaces and their possible iterations. We prove that a dual space of any non-reflexive Banach space contains convex subsets of any finite order and a convex subset of order $\omega + 1$.