In [that](http://www.mathcounterexamples.net/a-topological-vector-space-with-no-non-trivial-continuous-linear-form/) 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? 



TVS with null topological dual space
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](https://books.google.com/books?id=9kXY742pABoC&pg=PA46) 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.


