the dual space of C(X) (X is noncompact metric space)

It is well known that when X is compact space (or locally compact space), $C(X)=\{f:f X\rightarrow X \text{is continous,and bounded} \}$ ,the dual space of $C(X) C(X)^{*}$is correspond to $M(X)$ space of Radon measure with bounded variation.

However according to my knowledge. there are few books discuss case that when X is noncompact, for example complete vsperable metric space.

even for the simplest example, when taking X is R, means real line, what does $(C(X))^{*}$ mean.

Any advice and reference will be appreaciated.

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I think, you should specify what do you mean by $C(X)$, when $X$ is non-compact. For example, in the case of $X={\mathbb R}$ you can mean by by $C(\mathbb R)$ the space of all continuous functions on $\mathbb R$ endowed with the compact-open topology, and after that the dual space $C(\mathbb R)^*$ becomes exactly the space of all measures with compact support. –  Sergei Akbarov Nov 1 '12 at 18:35
@Sergei: I completely agree. This should have been said right from the start. –  Alain Valette Nov 1 '12 at 19:24

What you state in the first paragraph is the Riesz Representation Theorem (see http://en.wikipedia.org/wiki/Riesz_representation_theorem#The_representation_theorem_for_linear_functionals_on_Cc.28X.29) This is valid for all locally compact Hausdorff spaces; so in particular for $\mathbb R$ (ah, I guess, if you look at $C_0(\mathbb R)^*$).

If $X$ is any topological space, then of course we can talk of $C^b(X)$ (the bounded continuous functions on $X$). This is still a commutative C$^*$-algebra, and so is isomorphic to $C(K)$, where $K$ is some compact Hausdorff space. The process of moving from $X$ to $K$ is functorial; purely at the topological level it corresponds to constructing the Stone-Cech compactification (see http://en.wikipedia.org/wiki/Stone_cech_compactification ) Point evaluation at $x\in X$ induces a character on $C^b(X) = C(K)$ and hence a point $k$ of $K$; we thus get a (continuous) map $X\rightarrow K$. This is injective if $X$ is completely regular; but it can fail to be injective (basically, we might lack enough continuous functions to separate points of $X$).

Back to your question: $C^b(X)^* = C(K)^* = M(K)$. For $\mathbb R$, we find that $K$ is nothing but $\beta\mathbb R$ the Stone-Cech compactification (quite a large space!)

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A nice reference, taken from my answer to another question:

V. S. Varadarajan, MEASURES ON TOPOLOGICAL SPACES, AMS Transl. 48 (1965) 161--228.

Measures on topological spaces as dual to continuous functions on the space, or to bounded continuous functions on the space. (Also, beware of an error in the appendix.)

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