On the Banach spaces with the property \((V^*)\) of Pelczynski
type: Item
instance of: scholarly article
title: On the Banach spaces with the property \((V^*)\) of Pelczynski
zbMATH Open document ID: 0677.46006
DOI: 10.1007/BF01766147
author: G. Emmanuele
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1988-01-01T00:00:00Z
review text: A bounded subset \(X\) of a Banach space \(E\) is called a \(V^*\)-set if \(\lim_{n}\sup_{x\in X}| x^*_ n(x)| =0\) whenever \(\sum x^*_ n\) is a weakly unconditionally convergent series in \(E^*\). If every \(V^*\)-set is relatively weakly compact then E is said to have property \((V^*)\). This property was first introduced and studied by \textit{A. Pelczynski} [Bull. Acad. Pol. Sci. 10, 641-648 (1962; Zbl 0107.325)]. It is shown that if \(E\) is a Banach lattice then \(E\) has property \((V^*)\) if and only if \(E\) is weakly sequentially complete. A Banach space \(E\) is shown to haveproperty \((V^*)\) if and only if either \(E\) is reflexive or \(E\) is weakly sequentially complete and has an unconditional Schauder decomposition \((E_ n)\) with \(E_ n\) having property \((V^*)\). Other results in the paper involve relationships between the \((V^*)\)- property, the Reciprocal Dunford Pettis property and the Radon Nikodym property.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B22
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 46B42
zbMATH DE Number: 4109350
zbMATH Keywords: \(V^*\)-set
zbMATH Keywords: weakly unconditionally convergent series
zbMATH Keywords: Banach lattice
zbMATH Keywords: property \((V^*)\)
zbMATH Keywords: weakly sequentially complete
zbMATH Keywords: unconditional Schauder decomposition
zbMATH Keywords: Reciprocal Dunford Pettis property
zbMATH Keywords: Radon Nikodym property
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01766147
OpenAlex ID: W4234509984
description: scientific article; zbMATH DE number 4109350