Weak^* closures and derived sets in dual Banach spaces


Abstract


The main results of the paper: \textbf{(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. \textbf{(2)} Let X be a non-reflexive Banach space. Then there exists a convex subset A\subset X^* such that A^{(1)}\neq {\overline{A}\,}^* (the latter denotes the weak^* closure of A). \textbf{(3)} Let X be a quasi-reflexive Banach space and A\subset X^* be an absolutely convex subset. Then A^{(1)}={\overline{A}\,}^*.

DOI Code: 10.1285/i15900932v31n1p129

Keywords: norming subspace ; quasi-reflexive Banach space ; total subspace ; weak$^*$ closure ; weak$^*$ derived set ; weak$^*$ sequential closure

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