On Weakly Compact Subsets of a Banach Space

Abstract

In the last few years use of the Banach space weak topology has proven exceedingly fruitful in the hands of Bourbaki (4), Smulian (24), Kakutani (14), Ghantmacher and Smulian (9), Sirvint (23), Dunford and Pettis (8), and others. These investigations have been concerned either with sequentially compact subsets or with bicompact subsets. In this paper an attempt has been made to bridge the gap between these two extremes by considering Ma-compact subsets where K, is any cardinal number. One could define a subset to be t.-compact if every covering of power K, of the set's weak closure contained a finite subcovering. Bourgin has done this, treating the problem from a topological point of view (5). We have preferred an analysis approach, which, we believe, permits a much more direct and essentially simpler development.