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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 $E$.

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?

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  • $\begingroup$ For what it's worth, this construction of TVS with dual space $\{0\}$ works for $L^p(X)$ where $X$ is any measure space with no atoms and $0 < p < 1$. In this way the sequence spaces and functions on $[0,1]$ that you describe are special instances of a more general construction. See math.uconn.edu/~kconrad/blurbs/analysis/lpspace.pdf. $\endgroup$ – KConrad May 16 '15 at 19:01
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    $\begingroup$ "Apart from spaces of functions or sequences" Doesn't that exclude basically all naturally occurring spaces? I mean the whole point of functional analysis and the study of TVS is to understand certain function spaces better. $\endgroup$ – Johannes Hahn May 16 '15 at 20:07
  • $\begingroup$ @Ricardo l'll be interested in having a big list of examples for my first question. I'll raise a separate topic for the second question. $\endgroup$ – mathcounterexamples.net May 16 '15 at 20:36
  • $\begingroup$ @Johannes Hahn. You're probably right. I just mean that I'm also interested by "strange objects". $\endgroup$ – mathcounterexamples.net May 19 '15 at 6:27
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$W^{p,\infty}(\mathbb R^n)$ 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)
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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.)

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