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A characterization of the Schur property by means of the Bohr topology

✍ Scribed by Salvador Hernández; Jorge Galindo; Sergio Macario

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
101 KB
Volume
97
Category
Article
ISSN
0166-8641

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✦ Synopsis

Let G be a MAPA group that is metrizable and satisfies Pontryagin duality; that is, it coincides with its topological bidual. We prove that the Bohr topology of G respects compactness if and only if every non-totally bounded subset contains an infinite discrete subset which is C * -embedded in the Bohr compactification of G. This result is used to characterize the Banach spaces which respect compactness, or, with a different terminology, have the Schur property (defined below). Among other equivalent properties, we prove that a Banach space E has the Schur property if and only if every bounded basic sequence contains an infinite subsequence equivalent to a l 1 -basis.

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