# on completeness of a complete locally convex space w.r.t. the weak topology

by serenus   Last Updated January 11, 2019 11:20 AM

Let $$E$$ be a complete locally convex space, let $$E'$$ denote its topological dual, and let $$\sigma(E,E')$$ denote the weak topology on $$E$$. Is it true that the space $$E$$ is complete when equipped with the topology $$\sigma(E,E')$$, that is, is the locally convex space $$(E,\sigma(E,E'))$$ complete, too? And, what is the situation if in particular $$E$$ is a Fréchet space?

A related question would be the following: If $$E$$ is a locally convex space, is the $$\sigma(E,E')$$-closure of $$(E,\sigma(E,E'))$$ in $$E$$ necessarily $$E$$, that is, $$\overline{(E,\sigma(E,E'))}^{\sigma(E,E')}=E$$ ? And, what is the situation if $$E$$ is, in addition, complete?

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