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

Abstract

The main results of the paper: (1) The dual Banach space X * contains a linear subspace A ⊂ 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. (2) Let X be a non-reflexive Banach space. Then there exists a convex subset A ⊂ X * such that A (1) = A * (the latter denotes the weak * closure of A).(3) Let X be a quasi-reflexive Banach space and A ⊂ X * be an absolutely convex subset. Then A (1) = A * .