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.