On the Weak Compactness in Banach Lattices

The Grothendieck’s criterion of weak compactness in space C(S) has been extended byC. P. Niculescu for weakly sequentially complete Banach lattices. This paper first provides some sufficient andnecessary conditions for weakly sequentially complete Banach lattices, with the result that the criterionC.P.Niculescu obtained of the weak compactness has actually characterized the weakly sequentially Banachlattices. At the same time we extend C. P. Niculescu’s important results. Then we solve negatively Problem 1, ll which was posed by C. P. Niculescu, and a sufficient condition forthis is given which generalizes Pelczynski’s result.