A Reverse Theorem on the ·-w* Continuity of the Dual Map

Abstract EN

Given a Banach space X, x∈?X, and ?Xx=x*∈?X*:x*x=1, we define the set ?X*x of all x*∈?X* for which there exist two sequences xnn∈N⊆?X∖{x} and xn*n∈N⊆?X* such that xnn∈N converges to x, xn*n∈N has a subnet w*-convergent to x*, and xn*xn=1 for all n∈N.

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.