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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 $E$. ...

What is an example of a non-paracompact topological vector space? I'm aware of this question, but I don't care if my tvs is locally convex. In fact the wilder the better. The only criterion is that ...