Compactness and countable compactness in weak topologies
Volume 112 / 1995
Studia Mathematica 112 (1995), 243-250
DOI: 10.4064/sm-112-3-243-250
Abstract
A bounded closed convex set K in a Banach space X is said to have quasi-normal structure if each bounded closed convex subset H of K for which diam(H) > 0 contains a point u for which ∥u-x∥ < diam(H) for each x ∈ H. It is shown that if the convex sets on the unit sphere in X satisfy this condition (which is much weaker than the assumption that convex sets on the unit sphere are separable), then relative to various weak topologies, the unit ball in X is compact whenever it is countably compact.
