MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I've got the following questions concerning the theory of locally convex spaces :

Let $X$ be a locally convex metrizable space, what is the necessary and sufficient condition to have its dual $X^*$ metrizable?

Is it possible that $X^*$ is the F-space when $X$ is a locally convex non-complete metrizable space which is not a normed space?

Thank you in advance for the answer.

share|cite|improve this question
See also… – Andrey Rekalo Jul 30 '11 at 23:03
up vote 3 down vote accepted

The nLab cites a theorem that the dual of a Fréchet space $X$ is Fréchet if and only if $X$ is a Banach space. (Reference: paragraph 29.1 (7) in Gottfried Koethe, Topological Vector Spaces I.) Even if $X$ is non-complete, the dual of $X$ is isomorphic to the dual of its completion, so $X^\ast$ cannot be Fréchet if $X$ is a non-normable locally convex metrizable TVS.

share|cite|improve this answer
Thank you very much. – Romanov Jul 30 '11 at 22:43

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.