Examples of TVS with no non-trivial open convex subsets 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?
 A: $W^{p,\infty}(\mathbb R^n)$ for $0<p<1$, which is the space of all smooth functions such that each partial derivative  is in $L^p$. Instead of $\mathbb R^n$ one can also take a Riemannian manifold of bounded geometry, where now each iterated covariant derivative has to be in $L^p$ (of sections of the appropriate tensor bundle).
One can also go to Denjoy-Carleman ultra differentiable functions of this type, like 
$W^{\{M\},p}(\mathbb R^n)$ or $W^{(M),p}(\mathbb R^n)$ for $p<1$, as described in 


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*Andreas Kriegl, Peter W. Michor, Armin Rainer: An exotic zoo of diffeomorphism groups on ℝn. Ann. Glob. Anal. Geom. 47, 2 (2015), 179-222. (pdf)
A: You may be interested in the paper Nearly exotic topologies on normed spaces (1971) where N. T. Peck shows that any infinite-dimensional normed space can be re-topologized in such a (weaker) way that the new topology admits no nontrivial continuous linear functionals. (Hence also, if I understand this answer correctly, no nontrivial open convex subsets.)
