Suppose $X$ is an infinite-dimensional Banach algebra (hence not locally compact).
Does there exist any sort of Riesz representation theorem that says something about elements of $C(X)^*$?
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I would guess "the" natural topology on this space should be that of uniform convergence on compact subsets. Since a function on $X$ is continuous iff it its restrictions on compact subsets are continuous, $C(X)$ is actually the projective limit of $C(K), K \subset X$, $K$ compact. Which means that its dual should be the inductive limit of spaces of measures on $K$, that is, the space of measures with compact support.