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 Fr├ęchet 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?

Thanks for a hint/answer.