A Reverse Theorem on the ·-w* Continuity of the Dual Map. (10th March 2015)

• Record Type: Journal Article
• Authors: de Kock, Mienie; García-Pacheco, Francisco Javier
• Other Names: Hudzik Henryk Academic Editor.
• Is Part Of: Journal of function spaces. Volume 2015(2015)
• Publication Date: 2015-03-10
• Subjects: Function spaces -- Periodicals; 515.7305
• DOI: 10.1155/2015/864173
• Languages: English
• ISSNs: 2314-8896

Abstract

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 .