A weak Grothendieck compactness principle for Banach spaces with a symmetric basis |
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Authors: | P. N. Dowling D. Freeman C. J. Lennard E. Odell B. Randrianantoanina B. Turett |
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Affiliation: | 1. Department of Mathematics, Miami University, Oxford, OH, 45056, USA 2. Department of Mathematics and Computer Science, St. Louis University, St. Louis, MO, 63103, USA 3. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA 4. Department of Mathematics, The University of Texas at Austin, Austin, TX, 78712, USA 5. Department of Mathematics, Oakland University, Rochester, MI, 48309, USA
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Abstract: | 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 Dowling et al. (J Funct Anal 263(5):1378–1381, 2012), 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 $ell ^1$ and do not contain a subspace isomorphic to $c_0$ 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. |
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