Abstract
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.
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