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?
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.
