On \(c_ 0\) sequences in Banach spaces (Q807930)

A Banach space has property (S) if every normalized weakly null sequence contains a subsequence equivalent to the canonical basis of \((c_ 0)\). It is shown that equivalence constants can be choosen independent of the original sequence. It is also shown that property (S) implies property (a) introduced by \textit{A. Pelczynski} [Bull. Acad. Polon. Sci., Ser. Sci. Math. Astron Phys. 6, 251-253 (1958; Zbl 0082.108)]. This settles - in the negative - a conjecture by \textit{J. Hagler} [Stud. Math. 60, 289-307 (1977; Zbl 0387.46015)] that certain separable Banach spaces with property (a) have separable duals.