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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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