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In that post, I give an example of a TVS for which the topological dual is equal to $0$. But in the example, there is no open convex subset different from the empty set or the space itself.

Do you have an example of a TVS with a null dual topological space but having a non trivial open convex subset?

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There can't be such an example.

Given an open convex subset $U$ not containing the origin of a (Hausdorff) TVS $E$, there is by the geometric version of Hahn-Banach as given in Schaefer's book on topological vector spaces, a closed hyperplane $H$ disjoint from $U$. The quotient projection $E \to E/H$ is a non-zero continuous linear functional.

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    $\begingroup$ You sure have the most appropriate username to answer this question. $\endgroup$
    – KConrad
    May 17, 2015 at 18:01
  • $\begingroup$ And I'm really proud having Hahn-Banach answering my question! Thanks hahnbanach. $\endgroup$ May 19, 2015 at 6:29

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