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?

Thanks for a hint/answer.

  • 2
    $\begingroup$ $(E,\sigma(E,E'))$ is very often incomplete, e.g. for infinite dimensional Banach spaces. Rather exceptionally, it is complete for the Frechet space of all scalar sequences endowed with the product topology (which is already the weak topology). $\endgroup$ – Jochen Jan 11 at 11:29
  • $\begingroup$ Thanks for the answer. I already found an answer in a book by Hans Jarchow (Locally Convex spaces), in pages 147/148, where it is given a characterization for the completeness of a locally convex space when equipped with the weak topology. $\endgroup$ – serenus Jan 11 at 11:57

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