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It is well known that when $X$ is a compact space (or locally compact space), the dual space of $C(X)=\{f |f: X\rightarrow X \text{ is continuous and bounded} \}$ 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.

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4  
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

3 Answers 3

up vote 18 down vote accepted

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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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.

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Interesting! This is also related to a question that I asked recently (mathoverflow.net/questions/105147). Could you give some more precise references, perhaps here or over at the other question? –  Igor Khavkine Nov 1 '12 at 12:48
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There is a fairly systematic treatment in the book "Saks spaces and applications to functional analysis". –  jbc Nov 1 '12 at 12:55

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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