On the Krein-Šmulian theorem for weaker topologies
On the Krein-Šmulian theorem for weaker topologies
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, …