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.



Metrizable dual space
The [nLab](http://ncatlab.org/nlab/show/Fr%C3%A9chet+space#properties_5) 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. 


