## Metrizable dual space

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?

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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.

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 Thank you very much. – Romanov Jul 30 2011 at 22:43