On the Krein-Šmulian theorem for weaker topologies

Abstract

We investigate possible extensions of the classical Krein-\v{S}mulian theorem to various weak topologies. In particular, we show that if $X$ is a WCG Banach space and $\tau$ is any locally convex topology weaker than the norm-topology, then for every $\tau$-compact norm-bounded set $H$, $\overline{\operatorname{conv}}^{\,\tau}H$ is $\tau$-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 $\ell_1[0,1]$, then the Krein-\v{S}mulian theorem holds for every topology $\tau$ 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 $\ell_1(\Gamma)$ spaces.