# A weak Grothendieck compactness principle for Banach spaces with a symmetric basis

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 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# A weak Grothendieck compactness principle for Banach spaces with a symmetric basis

, Volume 18 (1) – Apr 18, 2013
13 pages

/lp/springer_journal/a-weak-grothendieck-compactness-principle-for-banach-spaces-with-a-70OD1BQwQa
Publisher
Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-013-0236-6
Publisher site
See Article on Publisher Site

### 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.

### Journal

PositivitySpringer Journals

Published: Apr 18, 2013

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