Dual space of C 0(x) vs C b(x)

The dual of $C_{0}(X)$ and the dual of $C_{b}(X)$ in the case $X$ is not compact.

Background and Statement

Let $C_{0}(X)=\{f\in C(X){\text{ and }}\forall \epsilon >0{\text{ }}\exists {\text{ a compact set}}K\subset X{\text{ s.t. }}\mid f(x)\mid <\epsilon {\text{ }}\forall x\in X\setminus K\}$, in other words this is the space of continuous function vanishing at infinity, and let $C_{b}(X)$ be the space of bounded continuous functions on $X$ together with the sup norm. With this norm $C_{0}(X)$ is a closed subspace of $C_{b}(X)$. Note that these two spaces coincides when $X$ is compact. The representation of the dual space of $C_{0}(X)$ is a well described by the following well known result in Functional Analysis (Riesz Representation Theorem 6.19 in Rudin):

Let $X$ be locally compact Hausdorff space, $\phi \in C_{0}(X)'$ there is a unique complex Borel measure $\mu$ such that the following holds:

$\phi f=\int _{X}fd\mu ,{\text{ for every }}f\in C_{0}(X)$

Moreover we can endow $C_{0}(X)'$ with the total variation norm: $\|\phi \|=|\mu |(X)$. This allows us to identify $C_{0}(X)'$ with ${\mathcal {M}}(X)$, space of complex Borel measures.

The case of $C_{b}(X)$

The idea is to produce, using Hahn-Banach elements of $C_{b}(X)'$ that only look at the behavior of functions of $C_{b}(X)$ at infinity (or on the boundary).