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Lifting results for sequences in Banach spaces

Published online by Cambridge University Press:  24 October 2008

M. Gonzalez  and
V. M. Onieva
M. Gonzalez
Affiliation:
University of Santander, Spain
V. M. Onieva
Affiliation:
University of Zaragoza, Spain
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Extract

Several important classes of Banach spaces are characterized by means of convergence properties of sequences. For example, if X is a Banach space, then X belongs to the class Nl1 of spaces without copies of l1, the class R of reflexive spaces or the class F of finite-dimensional spaces if and only if each bounded sequence has respectively a weakly Cauchy (w-Cauchy), weakly convergent (w-convergent) or convergent subsequence. Similarly X is in the class WSC of weakly sequentially complete spaces, or the class SCH of spaces with the Schur property if and only if each w-Cauchy sequence is w-convergent, or convergent, respectively; note that X ∈ SCH if and only if each w-convergent sequence of X is convergent (see [12], p. 47).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 1 , January 1989 , pp. 117 - 121
Copyright
Copyright © Cambridge Philosophical Society 1989

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