  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 $A^{(1)}={\bar{A}\,}^*$.
