most recent 30 from http://mathoverflow.net 2013-05-21T10:35:50Z http://mathoverflow.net/feeds/question/71687 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71687/metrizable-dual-space Romanov 2011-07-30T21:42:06Z 2011-07-30T22:08:48Z

<p>I've got the following questions concerning the theory of locally convex spaces :</p> <p>Let $X$ be a locally convex metrizable space, what is the necessary and sufficient condition to have its dual $X^*$ metrizable? </p> <p>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?</p> <p>Thank you in advance for the answer.</p>

http://mathoverflow.net/questions/71687/metrizable-dual-space/71688#71688 Todd Trimble 2011-07-30T22:08:48Z 2011-07-30T22:08:48Z

<p>The <a href="http://ncatlab.org/nlab/show/Fr%C3%A9chet+space#properties_5" rel="nofollow">nLab</a> cites a theorem that the dual of a Fr&eacute;chet space $X$ is Fr&eacute;chet if and only if $X$ is a Banach space. (Reference: paragraph 29.1 (7) in Gottfried Koethe, <i>Topological Vector Spaces I</i>.) Even if $X$ is non-complete, the dual of $X$ is isomorphic to the dual of its completion, so $X^\ast$ cannot be Fr&eacute;chet if $X$ is a non-normable locally convex metrizable TVS. </p>