On the Krein-Šmulian theorem for weaker topologies

Abstract

We investigate possible extensions of the classical Krein-Šmulian theorem to various weak topologies. In particular, we show that if X is a WCG Banach space and τ is any locally convex topology weaker than the norm-topology, then for every τ-compact normbounded set H, conv τ H is τ-compact. In arbitrary Banach spaces, the norm-fragmentability assumption on H is shown to be sufficient 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-Šmulian theorem holds for every topology τ 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 the topology induced by a boundary. On the other hand, the same statement is obtained for all C(K) and l 1(Γ) spaces.