Cardinality of Λ Determines the Geometry of \({B_{{\ell _\infty }\left( \Lambda \right)}}\) and \({B_{{\ell _\infty }\left( \Lambda \right)*}}\)

Abstract

We study the geometry of the unit ball of ℓ_∞(Λ) and of the dual space, proving, among other things, that Λ is countable if and only if 1 is an exposed point of \({B_{{\ell _\infty }\left( \Lambda \right)}}\). On the other hand, we prove that Λ is finite if and only if the δ_λ are the only functionals taking the value 1 at a canonical element and vanishing at all other canonical elements. We also show that the restrictions of evaluation functionals to a 2-dimensional subspace are not necessarily extreme points of the dual of that subspace. Finally, we prove that if Λ is uncountable, then the face of \({B_{{\ell _\infty }\left( \Lambda \right)*}}\) consisting of norm 1 functionals attaining their norm at the constant function 1 has empty interior relative to \({S_{{\ell _\infty }\left( \Lambda \right)*}}\).