# Dual of a topological vector space. Is it nontrivial?

In the case of normed spaces we know their duals are nonempty using a quick application of the Hahn Banach Theorem.

If we step back to the larger class of locally convex spaces, an enthralling sequence of separation results yields another nontrivial dual.

What about general topological vector spaces? Is it possible to have a nontrivial topological vector space over which every linear functional is discontinuous?

The case of $L^p$ spaces for $0\lt p\lt 1$ endowed with the distance $$d(f,g):=\int |f(x)-g(x)|^p\mathrm dx$$ gives a normed space whose unique continuous linear functional is the null one. Indeed, each $f$ can be written as $\frac 1n\sum_{i=1}^nf_i$, where $d(f_i,0)$ is small enough (at least in the case where the measure space is the unit interval).
• Are you taking any particular value of $p$ here? – roo Sep 30 '13 at 20:33
• Sorry I meant $0\lt p\lt 1$, otherwise what I wrote is not true. – Davide Giraudo Sep 30 '13 at 20:34