Let $X$ a compact metric space; I have to identify the dual of the set of continuous functions on $X$, $C(X)^*$. By Riesz representation theorem we have that it can be identified with the space of regular Radon measures, normed by total variation(see for instance Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I).
question: What happens if I remove the compactness of $X$?