A ξ-weak Grothendieck compactness principle
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Abstract For 0 ≤ ξ ≤ ω1, we define the notion of ξ-weakly precompact and ξ-weakly compact sets in Banach spaces and prove that a set is ξ-weakly precompact if and only if its weak closure is ξ-weakly compact. We prove a quantified version of Grothendieck’s compactness principle and the characterisation of Schur spaces obtained in [7] and [9]. For 0 ≤ ξ ≤ ω1, we prove that a Banach space X has the ξ-Schur property if and only if every ξ-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, norm null) sequence. The ξ = 0 and ξ= ω1 cases of this theorem are the theorems of Grothendieck and [7], [9], respectively.
1977 ◽
Vol 29
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pp. 963-970
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2005 ◽
Vol 71
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pp. 425-433
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2002 ◽
Vol 65
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pp. 223-230
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2020 ◽
Vol 63
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pp. 475-496
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1972 ◽
pp. 235-274
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1976 ◽
Vol 80
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pp. 269-276
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2012 ◽
Vol 16
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pp. 227-232
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2014 ◽
Vol 90
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pp. 311-318
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