ON KREIN-SMULIAN THEOREM FOR WEAKER TOPOLOGIES

Abstract. We investigate possible extensions of the classical Krein-Smulian theorem to various weak topologies. In particular, if X is a WCG Banach space and is any locally convex topology weaker than the norm-topology, then for every -compact norm-bounded set H, conv H is -compact. In arbitrary Banach spaces, the norm-fragmentability assumption on H is shown to be sucient for the last property to hold. A new proof to the following result is given: if a Banach space does not contain a copy of `1[0; 1], then the Krein-Smulian theorem holds for every topology induced by a norming set of functionals. We conclude that in such spaces a norm-bounded set is weakly compact if it is merely compact in topology induced by a boundary. On the other hand, the same statement is obtained for all C(K) and `1() spaces. 1