A Reverse Theorem on the ·-w* Continuity of the Dual Map - Mienie de Kock; Francisco Javier García-Pacheco (Journal of Function Spaces)
Given a Banach space X, x∈𝖲X, and 𝖩Xx=x*∈𝖲X*:x*x=1, we define the set 𝖩X*x of all x*∈𝖲X* for which there exist two sequences xnn∈N⊆𝖲X∖{x} and xn*n∈N⊆𝖲X* 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 𝖩X*x is contained in the boundary of 𝖩Xx relative to 𝖲X*. We also show that if X is infinite dimensional and separable, then there exists an equivalent norm on X such that the interior of 𝖩Xx relative to 𝖲X* is contained in 𝖩X*x.