On Pelczynski's property u for Banach spaces (Q1113045)

A Banach space X is said to have property u if corresponding to each weak Cauchy sequence \((x_ n)\) there exists a sequence \((y_ n)\) such that \(\sum y_ n\) is weakly unconditionally Cauchy and \((\sum x_ n- \sum^{n}_{1}y_ i)\) converges weakly to 0; we then write \(X\in u\). Using an earlier result [\textit{J. Howard} and \textit{K. Melendez}, Bull. Aust. Math. Soc. 7, 183-190 (1972; Zbl 0244.47011)] the author observes that if X is wealy sequentially complete then \(X\in u\). The following results are obtained: (a) \(\ell_{\infty}\not\in u.\) (b) If \(X\in u\) is a conjugate space then X is wealy sequentially complete.