Linear Structures in the Set of Norm-Attaining Functionals on a Banach Space

Abstract

In this note, we investigate the spaceability properties of the set NA(X) of all norm attaining functionals on a Banach space X. This study is intimately related with isometric duality theory. If X is a dual space, then NA(X) certainly contains its predual. It is also motivated by proximinality questions: indeed, an hyperplane H = kerx £ of X is proximinal in X if and only if x £ 2 NA(X). If a flnite codimensional subspace Y of X is proximinal in X, then Y ? † NA(X), and the converse also holds in some Banach spaces, but not always. We refer to [3, 19, 20, 28] for some recent progress on this question. When a Banach space X has an inflnite dimensional quotient which is isomorphic to a dual space, it is easy to construct an equivalent norm on X for which NA(X) is spaceable. We show in this paper that the converse holds for Banach spaces whose dual unit ball is