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.



the dual space of C(X) (X is noncompact metric space)
What you state in the first paragraph is the Riesz Representation Theorem (see [wikipedia](https://en.wikipedia.org/wiki/Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem)). 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 [wikipedia](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!)


