The other standard formulation says that a convex subset of a Banach-space dual is weak-star closed if its intersection with every norm-closed ball is weak-star closed.
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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.