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

Introduction

In the case that the space $X$ is compact then all continuous functions belongs to $C_{0}(X)$ as we will show in the next section. On the other hand if the space $X$ is not compact while we always have the inclusion $C(X)\supseteq C_{0}(X)$ there may be some continuous function that do not belong to $C_{0}(X)$. Some of them may even even be bounded and still not belong to $C_{0}(X)$, this motivates us to consider the dual space of $C_{0}(X)$ and the dual space of $C_{b}(X)$.

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. In the case $X$ si compact we can choose $K=X$ in the previous definition, since properties on the empty set are trivially true, we can conclude that $C(X)=C_{0}(X)$. 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 $C_{b}(X)$

We want now to investigate the properties of elements of the dual space of $C_{b}(X)$, not by looking at local properties but rather a very property condition on the limit at infinity of our elements of $C_{b}(X)$, it turns out that this allow us to extract also global information on the whole $C_{b}(X)$.

We say that a function $u\in C_{b}(X)$ admits a limit at infinity $u(\infty )$ if for any $\epsilon >0$ there exists a compact $K_{\epsilon }\subset X$ such that if $x\notin K_{\epsilon }$ implies $|u(x)-u(\infty )|\leq \epsilon$. We can see this operation as a linear function 'limit at infinity' thanks to Hahn-Banach we can build a continuous extension of it for all $C_{b}(X)$. This is another spectacular consequence of Axiom of Choice (Hahn-Banach theorem in this case) since the extension assign a limit to any continuous bounded function!

Kantorovich Duality for $C_{b}(X\times Y)$

As it can be found in Villani, the following version of Kantorovich duality holds: let $X$ and $Y$ locally compact Polish spaces, let $c$ be a lower semi-continuous non negative function on $X\times Y$ and let $\mu$ and $\nu$ be two Borel probability measures on $X\times Y$ respectively then:

$\inf _{\pi \in \Pi (\mu ,\nu )}\int _{X\times Y}c(x,y)d\pi (x,y)=\sup _{(\phi ,\psi )\in \Phi _{c}}\int _{X}\phi d\mu +\int _{X}\psi d\nu$,

Here $\Pi (\mu ,\nu )$ is the set of all probability measures $\pi$ that satisfies $\pi (A\times Y)=\mu (A)$ and $\pi (X\times B)=\nu (B)$ for any measurable set $A\subset X$ and any measurable set $B\subset Y$; $\Phi _{c}$ is the set of all measurable functions $(\phi ,\psi )\in L^{1}(d\mu )\times L^{1}(d\nu )$ that satisfies $\phi (x)+\psi (x)\leq c(x,y)$ for $d\mu$ almost all $x\in X$ and for $d\nu$ almost all $y\in Y$.

If we try to extend the proof of the compact case we run into a problem since the dual of $C_{b}(X\times Y)$ strictly contains ${\mathcal {M}}(X\times Y)$. If we restrict to the closed subspace $C_{0}(X\times Y)\subset C_{b}(X\times Y)$ than any elements $C_{b}(X\times Y)'$ acts continuously, and mentioned before, can be represented by a unique $\pi \in {\mathcal {M}}(X\times Y)$ such that

$<l,f>=\int _{X\times Y}f(x,y)d\pi ,{\text{ for every }}f\in C_{0}(X\times Y)$.

We can then write $l=\pi +R$ where $R$ is a continuous linear functional supported at infinity, i.e. $\forall f\in C_{0}(X\times Y)$ implies $<R,f>=0$.

For what discussed in the previous section the behavior of some $R$ may not be clear at first glance as the following result implies:

let $\mu$ and $\nu$ be two Borel probability measures on $X\times Y$ respectively, there is a continuous linear functional $l$ on $C_{b}(X\times Y)$, supported at infinity, such that the following holds

$\forall (\phi ,\psi )\in C_{0}(X)\times C_{0}(Y),<l,\phi +\psi >=\int _{X}\phi d\mu +\int _{Y}\psi d\nu$.

Because