On Weak Sequential Completeness of Spaces Where Weakly Compact Sets are Super Weakly Compact

Abstract We show that every Banach space in which weakly compact sets are super weakly compact is automatically weakly sequentially complete answering a question from [11]. In the proof we show how to build a weakly compact set which is not super weakly compact from an arbitrary nontrivial weakly Cauchy sequence using the notion of a summing subsequence of Rosenthal or Singer.