 Krein–Smulian theorem

The closed convex hull of a weakly compact subset of a Banach space is weakly compact.

The Krein–Smulian theorem is a result in functional analysis that provides conditions under which a weakly compact set in a Banach space is also weak*-compact in the dual space. Specifically, it gives a characterization of weakly compact convex subsets of a dual space in terms of their weak*-closed subsets.