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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.
Positivity – Springer Journals
Published: Apr 18, 2013
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