2026-06-28T10:11:36Zhttps://docta.ucm.es/rest/oai/requestoai:docta.ucm.es:20.500.14352/578942023-08-06T01:28:14Zcom_20.500.14352_14col_20.500.14352_15
Docta Complutense
author
Bombal Gordón, Fernando
2023-06-20T17:10:37Z
2023-06-20T17:10:37Z
1990-01-01
0017-0895
10.1017/S0017089500009113
https://hdl.handle.net/20.500.14352/57894
http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=5052868
http://journals.cambridge.org/action/login
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.
eng
On (V*) sets and Pelczynski's property (V*).
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