International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 1, Pages 67-70
doi:10.1155/S0161171295000081

The strong WCD property for Banach spaces

Abstract

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 (An) of weak ∗ compact subsets of X∗∗ such that if K⊂X is weakly compact, there is an (nm)⊂N such that K⊂⋂m=1∞Anm⊂X. In this case, (An) 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 ℵ0-space. Using c0 for an example, we show how weakly compact structure theorems may be used to construct strongly determining sequences.