It is known that the continuous finite-dimensional irreducible unitary representations separate the points of Hausdorff compact topological groups (this is, in part, the Peter-Weyl theorem). These, though, are not the only such groups - $\mathbb R^n$ is so, too. My question, therefore, is:
is it possible to characterize the topological groups the points of which can be separated by their continuous finite-dimensional representations?