If converges weakly to in a Banach space, there are convex combinations of each tail that converge in norm to .
The weak closure and norm closure of a convex subset of a real or complex normed space coincide. If a point lies outside the norm closure, the Hahn-Banach separation theorem supplies a weakly continuous linear functional that separates it from the convex set.
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Mazur's Lemma is a result in functional analysis, specifically in the context of convex analysis and the study of weak compactness in Banach spaces. The lemma is named after the Polish mathematician Stanisław Mazur. The central idea of Mazur's Lemma concerns weakly convergent sequences and the nature of the weak closure of convex sets.