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



Examples of TVS with no non-trivial open convex subsets
$W^{p,\infty}(\mathbb R^n)$ 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 


* 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)](http://www.mat.univie.ac.at/~michor/exotic-zoo.pdf)


