Let $X$ be a normed vector space and $M$ a subset. I have proved that if $M$ is weakly sequentially closed then it is closed, but I am wondering whether the converse is true or not? Thanks
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No, the unit sphere in $\ell^2(\Bbb{N})$ is dense in the closed unit ball w.r.t. weak sequential convergence (and hence not closed w.r.t. weak sequential convergence), but closed w.r.t. the norm topology.
But if $M$ is convex and closed, then it is also weakly closed and hence also closed w.r.t. weak sequential convergence, see e.g. Mazur's Lemma and Convex set weakly closed if and only if strongly closed as well.
