On Weakly Compact Subsets of Banach Spaces

Introduction. The two sections of this note are independent, but they are related by the fact that both use the results of [5 ] to obtain information on the properties of weakly compact sets in Banach spaces. In the first section we prove some results on a class of compact sets which is believed to include all weakly compact subsets of Banach spaces. We are interested in the properties of the nonmetrizable sets of this form. (Our results become trivial in the metrizable case.) We show in particular that such sets have a dense subset consisting of Gt points. Weakly compact sets are known to possess many properties which are similar to those of metrizable spaces (Eberlein's Theorem, for example). Our result exhibits a new property of this kind. In the second section we show that in a separable reflexive space, every weakly compact set is the intersection of finite unions of cells.