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The strong WCD property for Banach spaces. (English) Zbl 0830.46013
Summary: We introduce a weakly compact version of the weakly countably determined (WCD) property, the strong WCD (SWCD) property. A Banach space \(X\) is said to be SWCD if there is a sequence \((A_n)\) of weak* compact subsets of \(X^{**}\) such that if \(K\subset X\) is weakly compact, there is an \((n_m) \subset \mathbb{N}\) such that \(K\subset \bigcap_{m=1}^\infty A_{n_m} \subset X\). In this case, \((A_n)\) is called a strongly determining sequence for \(X\). We show that \(\text{SWCG} \Rightarrow \text{SWCD}\) and that the converse does not hold in general. In fact, \(X\) is a separable SWCD space if and only if (\(X\), weak) is an \(\aleph_0\)-space. Using \(c_0\) for an example, we show how weakly compact structure theorems may be used to construct strongly determining sequences.
MSC:
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
46B10 Duality and reflexivity in normed linear and Banach spaces
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