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

TL;DR

This paper investigates the structure of weak$^*$ closures and derived sets in dual Banach spaces, revealing conditions under which these sets are dense, proper, or equal to their closures, depending on reflexivity properties.

Contribution

It characterizes the behavior of weak$^*$ derived sets in dual Banach spaces, distinguishing between non-quasi-reflexive, non-reflexive, and quasi-reflexive cases.

Findings

Existence of a subspace with dense weak$^*$ limits in non-quasi-reflexive spaces.

Construction of convex sets with different derived set closures in non-reflexive spaces.

Equality of derived set and closure in quasi-reflexive spaces for convex subsets.

Abstract

The main results of the paper: {\bf (1)} The dual Banach space $X^*$ contains a linear subspace $A\subset X^*$ such that the set $A^{(1)}$ of all limits of weak$^*$ convergent bounded nets in $A$ is a proper norm-dense subset of $X^*$ if and only if $X$ is a non-quasi-reflexive Banach space containing an infinite-dimensional subspace with separable dual. {\bf (2)} Let $X$ be a non-reflexive Banach space. Then there exists a convex subset $A\subset X^*$ such that $A^{(1)}\neq {\bar{A}\,}^*$ (the latter denotes the weak$^*$ closure of $A$). {\bf (3)} Let $X$ be a quasi-reflexive Banach space and $A\subset X^*$ be an absolutely convex subset. Then $A^{(1)}={\bar{A}\,}^*$.