Transitivity of proximinality and norm attaining functionals

Abstract

We study the question of when the set of norm attaining functionals on a Banach space is a linear space. We show that this property is preserved by factor reflexive proximinal subspaces in R(1)^~ spaces and generally by taking quotients by proximinal subspaces. We show, for K(l₂) and c0-direct sums of families of reflexive spaces, the transitivity of proximinality for factor reflexive subspaces. We also investigate the linear structure of the set of norm attaining functionals on hyperplanes of c0 and show that, for some particular hyperplanes of c0, linearity and orthogonal linearity coincide for the set of norm attaining functionals.