Abstract
A Banach space has property (S) if every normalized weakly null sequence contains a subsequences equivalent to the unit vector basis ofc0. We show that the equivalence constant can be chosen “uniformly”, i.e., independent of the choice of the normalized weakly null sequence. Furthermore we show that a Banach space with property (S) has property (u). This solves in the negative the conjecture that a separable Banach space with property (u) not containingl1 has a separable dual.
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