Weak${}^{\ast }$ closures and derived sets in dual Banach spaces

Abstract

The main results of the paper: {\bf (1)} The dual Banach space $X^{\ast }$ contains a linear subspace $A\subset X^{\ast }$ such that the set $A^{(1)}$ of all limits of weak${}^{\ast }$ convergent bounded nets in $A$ is a proper norm-dense subset of $X^{\ast }$ 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 $A\subset X^{\ast }$ such that $A^{(1)}\ne {\bar{A}}^{\ast }$ (the latter denotes the weak${}^{\ast }$ closure of $A$). {\bf (3)} Let $X$ be a quasi-reflexive Banach space and $A\subset X^{\ast }$ be an absolutely convex subset. Then $A^{(1)}={\bar{A}}^{\ast }$.