A Reverse Theorem on the ·-w* Continuity of the Dual Map. (10th March 2015) - de Kock, Mienie; García-Pacheco, Francisco Javier
Abstract : Given a Banach space X, x ∈ 𝖲 X, and 𝖩 X x = x * ∈ 𝖲 X * : x * x = 1, we define the set 𝖩 X * x of all x * ∈ 𝖲 X * for which there exist two sequences x n n ∈ N ⊆ 𝖲 X ∖ { x } and x n * n ∈ N ⊆ 𝖲 X * such that x n n ∈ N converges to x, x n * n ∈ N has a subnet w * -convergent to x *, and x n * x n = 1 for all n ∈ N . We prove that if X is separable and reflexive and X * enjoys the Radon-Riesz property, then 𝖩 X * x is contained in the boundary of 𝖩 X x relative to 𝖲 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 𝖩 X x relative to 𝖲 X * is contained in 𝖩 X * x .