The strong WCD property for Banach spaces

In this paper, we introduce 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 s a sequence ( A n ) of weak ∗ compact subsets of X ∗ ∗ such that if K ⊂ X is weakly compact, there is an ( n m ) ⊂ N such that K ⊂ ⋂ m = 1 ∞ A n m ⊂ X . In this case, ( A n ) is called a strongly determining sequence for X . We show that SWCG ⇒ 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 ℵ 0 -space. Using c 0 for an example, we show how weakly compact structure theorems may be used to construct strongly determining sequences.