A reverse theorem on the · - w∗ Continuity of the dual map

Given a banach space x,xx,and xx=x∗x∗:x∗x=1,we define the set x∗x of all x∗x∗ for which there exist two sequences xnnnx{x} and xn∗nnx∗ such that xnnn converges to x,xn∗nn has a subnet w∗-convergent to x∗,and xn∗xn=1 for all nn. we prove that if x is separable and reflexive and x∗ enjoys the radon-riesz property,then x∗x is contained in the boundary of xx relative to x∗. we also show that if x is infinite dimensional and separable,then there exists an equivalent norm on x such that the interior of xx relative to x∗ is contained in x∗x. © 2015 mienie de kock and francisco javier garcía-pacheco.