Question

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.

Answer 1

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!)

Answer 2

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.)

Answer 3

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.