 Sequentially weakly closed set

A subset $K$ of a normed space is sequentially weakly closed when every weakly convergent sequence in $K$ has its limit in $K$. Every weakly closed set has this property. For a convex set in a Banach space, it is equivalent to closure in the norm topology: Mazur lemma produces norm convergent convex combinations from any weakly convergent sequence, and conversely norm convergence implies weak convergence. For a Hilbert space, the weak Banach–Saks theorem in a Hilbert space supplies these convex combinations as subsequence averages. No equivalence with weak closedness is asserted here for arbitrary nonconvex sets.