A Weak Grothendieck Compactness Principle

The Grothendieck compactness principle states that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. In this article, an analogue of the Grothendieck compactness principle is considered when the norm topology of a Banach space is replaced by its weak topology. It is shown that every weakly compact subset of a Banach space is contained in the closed convex hull of a weakly null sequence if and only if the Banach space has the Schur property. In [3, p. 112], Alexander Grothendieck proved that every norm compact subset of a Banach space X is contained in the closed convex hull of a norm null sequence. Grothendieck remarks that this result is implicitly contained in an article of J. Dieudonné and L. Schwartz [1, proof of Th. 5]. Despite Grothendieck’s remark, we refer to this result as the Grothendieck compactness principle. It is known that an analogue of Grothendieck’s compactness principle does not hold for all Banach spaces if the norm topology is replaced by the weak topology. Lindenstrauss and Phelps [5, Corollary 1.2] proved that the closed unit ball in an infinite-dimensional reflexive Banach space cannot be contained in the closed convex hull of a weakly null sequence. Also, in [7], it is noted that the closed unit ball of `, considered as a subset of c0, is not contained in the closed convex hull of a weakly null sequence in c0. The question that provides the impetus for the current article is: For which Banach spaces do analogues of Grothendieck’s compactness principle hold if the norm topology of a Banach space is replaced by its weak topology? That is, for which Banach spaces is it true that every weakly compact set in the Banach space is contained in the closed convex hull of a weakly null sequence? If X is a Banach space with the Schur property, that is, a space in which weak convergence and norm convergence of sequences coincide, then every weakly compact set is norm compact; and therefore, by Grothendieck’s result, in spaces with the Schur property, every weakly compact set is contained the closed convex hull of a weakly (in fact, norm) null sequence. It turns out that no other spaces have this property. Theorem 1. Every weakly compact subset of a Banach space X is contained in the closed convex hull of a weakly null sequence if and only if X has the Schur property. Before proving the theorem, let us recall a definition and a few facts. A Schauder basis {ei}i=1 for a Banach space is bimonotone if, for every n,m ∈ N with n < m, the 1 2 P.N. DOWLING, D. FREEMAN, C.J. LENNARD, E. ODELL, B. RANDRIANANTOANINA, AND B. TURETT projection P[n,m) defined by P[n,m) ( ∑∞ i=1 aiei) = ∑m−1 i=n aiei satisfies ‖P[n,m)‖ = 1. It is worth noting that, if {ei} is a basis for a Banach space X, then X can be renormed to have an equivalent bimonotone norm: ||| ∑∞ i=1 aiei||| = supn ∥∥∥∥P[q2j−1,q2j) ( M ∑