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Properties (V) and (u) are not three-space properties

Published online by Cambridge University Press:  18 May 2009

Jesús M. F. Castillo  and
Manuel González
Jesús M. F. Castillo
Affiliation:
Departamento de Matemáticas, Universidad de Extremadura, Avda de Elvas s/n, 06071 Badajoz, España
Manuel González
Affiliation:
Departamento de Matemáticas, Universidad de Cantabria, Avda los Castros s/n, 39071 Santander, España
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In his fundamental papers [7,8], Pelczynski introduced properties (u), (V), and (V*) as tools as study the structure of Banach spaces. Let X be a Banach space. It is said that X has property (u) if, for every weak Cauchy sequence (xn) in X, there exists a weakly unconditionally Cauchy (wuC) series in X such that the sequence is weakly null. It is said that X has property (V) if, for every Banach space Z, every unconditionally converging operator from X into Z is weakly compact; equivalently, whenever K is a bounded subset of X* such that for every wuC series in X, then K is relatively weakly compact. A Banach space X is said to have property (V*) if whenever K is a bounded subset of X such that 0 for every wuC series in X*, then K is relatively weakly compact. Some well-known results which shall be needed later are contained in the following.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 36 , Issue 3 , September 1994 , pp. 297 - 299
Copyright
Copyright © Glasgow Mathematical Journal Trust 1994

References

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