A Weak Grothendieck Compactness Principle for Banach spaces with a Symmetric Basis

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 [1], an analogue of the Grothendieck compactness principle for the weak topology was used to characterize Banach spaces with the Schur property. Using a different analogue of the Grothendieck compactness principle for the weak topology, a characterization of the Banach spaces with a symmetric basis that are not isomorphic to ` and do not contain a subspace isomorphic to c0 is given. As a corollary, it is shown that, in the Lorentz space d(w, 1), every weakly compact set is contained in the closed convex hull of the rearrangement invariant hull of a norm null sequence. The research of D. Freeman and E. Odell was partially supported by the National Science Foundation P.N. Dowling Department of Mathematics, Miami University, Oxford, OH 45056, Tel.: +513-529-5831, Fax: +513-529-1493 E-mail: [email protected] D. Freeman Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, E-mail: [email protected] C.J. Lennard Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, E-mail: [email protected] E. Odell Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, E-mail: [email protected] B. Randrianantoanina Department of Mathematics, Miami University, Oxford, OH 45056, E-mail: [email protected] B. Turett Department of Mathematics, Oakland University, Rochester, MI 48309, E-mail: [email protected] 2 P.N. Dowling et al.