= Sequentially weakly closed set
= Weakly sequentially closed set
{synonym}
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>.
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