Let $X$ be a Banach space. Let $C$ be a convex, and normed-closed subset of $X$. It is well-known that $C$ becomes weakly closed subset of $X$. I want to know is there any well-know class of non convex sets which has this property?

i.e., a class of sets in $X$, not necessary convex on which, norm-closed implies weakly-closed.

  • $\begingroup$ Totally bounded sets have this property. $\endgroup$ – Sergei Akbarov Jun 20 at 6:20
  • $\begingroup$ @SergeiAkbarov Yeah.. but that's a trivial case , because in banach space closed totally bounded sets are compact. $\endgroup$ – Red shoes Jun 20 at 6:41
  • $\begingroup$ The proof for the convex case is by the Hahn-Banach theorem. For a new, substantially different, case: you will probably need a new, substantially different, proof. $\endgroup$ – Gerald Edgar Jun 20 at 9:48
  • $\begingroup$ Another trivial case: the union of finitely many closed convex sets. $\endgroup$ – Robert Israel Jun 20 at 15:53
  • $\begingroup$ @GeraldEdgar What if set be norm-locally convex? $\endgroup$ – Red shoes Jun 20 at 20:23

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