TITLE: A Hausdorff abelian group with no character?
QUESTION [7 upvotes]: Pontryagin Duality for locally compact abliean groups gives plenty of continuous (unitary) characters $\chi : A \to \mathbb{R} / \mathbb{Z}$, but if we do not assume local compactness, can anything be said? In particular, is the following true?

Every abelian Hausdorff topological group has a nontrivial continuous (unitary) character

I believe this to be false, but cannot find a concrete example.

REPLY [8 votes]: If a topological vector space $X$ is not locally convex, then it usually has not non-zero linear continuous functionals, and this means that there are no non-trivial continuous characters on $X$. For example, you can take the space of all measurable functions on $[0,1]$ with the metrics
$$
d(x,y)=\int_0^1\frac{|x(t)-y(t)|}{1+|x(t)-y(t)|}d t.
$$