TITLE: the dual space of C(X) (X is noncompact metric space)
QUESTION [21 upvotes]: It is well known that when $X$ is a compact space (or locally compact space), the dual space of $C(X)=\{f |f:
X\rightarrow \mathbb{C}  \text{ is continuous and bounded} \}$  is $M(X)$, the space of Radon measures with bounded variation.
However, according to my knowledge, there are few books which discuss the case when X is noncompact, for example a complete separable metric space.
Even for the simplest example, when taking  $X=\mathbb{R}$, what does $(C(X))^{*}$ mean?
Any advice and reference will be much appreciated.

REPLY [11 votes]: The problem of obtaining a  useful generalisation of the Riesz representation theorem for non-compact spaces was addressed in the 50's by R.C. Buck, amongst others.  It was clear that it was necessary to leave the context of Banach spaces for a nice theory.  Buck introduced the so-called strict topology on the space of bounded, continuous functions on a locally compact space and showed that the dual is the space of bounded Radon measurs on the underlying space.  This was generalised to the case of completely regular spaces in the 60's using  the theory of mixed topologies or Saks spaces which had been developed by the Polish school.  The most succinct definition of the resulting topology on the above space is that it is the finest locally convex topology which agrees with compact convergence on bounded sets.  There is a relatively complete theory---in particular, the Riesz representation theorem holds in its natural form.