On the Banach spaces with the property \((V^*)\) of Pelczynski
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.
weakly unconditionally convergent series
scientific article; zbMATH DE number 4109350
scientific article