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.
@article{bwmeta1.element.bwnjournal-article-smv112i3p243bwm,
author = {W. Kirk},
title = {Compactness and countable compactness in weak topologies},
journal = {Studia Mathematica},
volume = {113},
year = {1995},
pages = {243-250},
zbl = {0824.46020},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv112i3p243bwm}
}
Kirk, W. Compactness and countable compactness in weak topologies. Studia Mathematica, Tome 113 (1995) pp. 243-250. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv112i3p243bwm/
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