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On (V*) sets and Pelczynski's property (V*)

Published online by Cambridge University Press:  18 May 2009

Fernando Bombal
Fernando Bombal
Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid (Spain).
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The concept of (V*) set was introduced, as a dual companion of that of (V)-set, by Pelczynski in his important paper [14]. In the same paper, the so called properties (V) and (V*) are defined by the coincidence of the (V) or (V*) sets with the weakly relatively compact sets. Many important Banach space properties are (or can be) defined in the same way; that is, by the coincidence of two classes of bounded sets. In this paper, we are concerned with the study of the class of (V*) sets in a Banach space, and its relationship with other related classes. To this general study is devoted Section I. A (as far as we know) new Banach space property (we called it property weak (V*)) is defined, by imposing the coincidence of (V*) sets and weakly conditionally compact sets. In this way, property (V*) is decomposed into the conjunction of the weak (V*) property and the weak sequential completeness. In Section II, we specialize to the study of (V*) sets in Banach lattices. The main result in the section is that every order continuous Banach lattice has property weak (V*), which extends previous results of E. and P. Saab ([16]). Finally, Section III is devoted to the study of (V*) sets in spaces of Bochner integrable functions. We characterize a broad class of (V*) sets in L1(μ, E), obtaining similar results to those of Andrews [1], Bourgain [6] and Diestel [7] for other classes of subsets. Applications to the study of properties (V*) and weak (V*) are obtained. Extension of these results to vector valued Orlicz function spaces are also given.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 32 , Issue 1 , January 1990 , pp. 109 - 120
Copyright
Copyright © Glasgow Mathematical Journal Trust 1990

References

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