A ξ-weak Grothendieck compactness principle

Abstract

AbstractFor 0 ≤ ξ ≤ ω1, we define the notion of ξ-weakly precompact and ξ-weakly compact sets in Banach spaces and prove that a set is ξ-weakly precompact if and only if its weak closure is ξ-weakly compact. We prove a quantified version of Grothendieck’s compactness principle and the characterisation of Schur spaces obtained in [7] and [9]. For 0 ≤ ξ ≤ ω1, we prove that a Banach space X has the ξ-Schur property if and only if every ξ-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, norm null) sequence. The ξ = 0 and ξ= ω1 cases of this theorem are the theorems of Grothendieck and [7], [9], respectively.