Let be a Banach space, and two subsets of . Denote by and the distances to from the point and from the subset respectively. The Krein-Šmulian Theorem asserts that the closed convex hull of a weakly compact subset of a Banach space is weakly compact; in other words, every w-compact subset such that satisfies . We extend this result in the following way: if is a closed subspace of and is a wcompact subset of , then
Moreover, if is w-dense in , then . However, the equality holds in many cases, for instance, if , if has w-angelic dual unit ball (for example, if is WCG or WLD), if , if is fragmented by the norm of , etc. We also construct under a w-compact subset such that is w-dense in , and .
Cite this article
Antonio S. Granero, An extension of the Krein-Šmulian Theorem. Rev. Mat. Iberoam. 22 (2006), no. 1, pp. 93–110DOI 10.4171/RMI/450