Metrizable dual space - MathOverflow 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 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 Answer by Todd Trimble for Metrizable dual space 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>