I give here the classical example of the space $E = L^p([0,1])$ which has no open convex subsets apart from $\emptyset$ and $E$. Consequently, there is no non-trivial continuous linear form on $E$.

Apart from spaces of functions or sequences, do you have examples of Topological Vector Spaces (TVS) with no open convex subsets other than $\emptyset$ and the space itself?

Also, do you have an example of a TVS with no non-trivial continuous linear form but which contains an open convex subset different from $\emptyset$ and the space itself?