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

Introduction

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

Background and Statement

Let $C_{0}(X)=\{f\in C(X){\text{ and }}\forall \epsilon >0{\text{ }}\exists {\text{ a compact set}}K\subset X{\text{ s.t. }}\mid f(x)\mid <\epsilon {\text{ }}\forall x\in X\setminus K\}$ equipped with the sup norm. In other words this is the space of continuous functions vanishing at infinity. When $X$ is compact we can choose $K=X$ in the previous definition, and 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. Again when $X$ is compact we have not introduced a new space since every continuous function on a compact metric space is bounded, to see this assume on the contrary that there is a sequence $\{x_{n}\}_{n\in \mathbb {N} }$ such that $|f(x_{n})|\rightarrow \infty$ as $n\rightarrow \infty$. By compactness there is a sub-sequence $\{x_{n_{k}}\}_{i\in \mathbb {N} }$ converging to a point ${\hat {x}}\in X$. Therefore by continuity of $f$ we have $|f(x_{n_{k}})|\rightarrow |f({\hat {x}})|<\infty$, and this is our desired contradiction. We conclude that $C(X)=C_{b}(X)=C_{0}(X)$.

The rest of this discussion will consider the case where $X$ is not compact. Rather than equality of the three spaces, we have the inclusions: $C(X)\supset C_{b}(X)\supset C_{0}(X)$

The case of $C_{0}(X)'$

The representation of the dual space of $C_{0}(X)$ is a described by the following well known result in Functional Analysis (Riesz Representation Theorem 6.19 in Rudin ):

Let $X$ be a locally compact Hausdorff space For any bounded linear function $\phi$, i.e. an element of the dual space $C_{0}(X)'$, there is a unique complex Borel measure $\mu$ such that the following holds:

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

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

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

To describe the dual space of $C_{b}(X)$, we will focus on the behavior of functions at infinity, as in Exercise 1.23 of . We first need a preliminary result: $C_{0}(X)$ is a closed (vector) subspace of $C_{b}(X)$. In other words, $C_{0}(X)$ contains all its limit points. Let $f_{n}(x)$ be a convergent sequence in $C_{0}(X)'$, where $f_{n}(x)$ are continuous functions vanishing at infinity and let $f(x)ne$ their limit then f is continuous since the uniform norm we are using provides uniform convergence. It remains to show that the limit $f(x)$ vanishes at infinity: let $n\in \mathbb {N}$ be such that $\|f-f_{n}\|_{\infty }<\epsilon /2$.Now since each $f_{n}(x)$ vanishes at infinity, we can find $K_{n}\subset X$ such that $|f_{n}(x)|<\epsilon /2$ for any $x\in X\setminus K_{n}$. Then we can conclude by the triangle inequality that

$|f(x)|\leq |f(x)-f_{n}(x)|+|f_{n}(x)|<\epsilon$.

That is, $f(x)\in C_{0}(X)$. This proves $C_{0}(X)$ is a closed subspace of $C_{b}(X)$. We may now carefully specify the local property at infinity for $C_{b}(X)$.

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 set $K_{\epsilon }\subset X$ such that $x\notin K_{\epsilon }$ 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 of $C_{b}(X)$. This is another spectacular consequence of Axiom of Choice (Hahn-Banach theorem [1] in this case). Intuitively we can partition the space $C_{b}(X)$ into equivalence classes of the equivalence relation of having the same limit at infinity. Then by to the axiom of choice we choose a representative for each class. The problem with this argument, however, is that we don't know yet that every function in $C_{b}(X)$ admits such a limit. But this will not stop us from falling down the rabbit hole: note that every function in $C_{0}(X)$ admits such a limit, let $l:C_{0}(X)\rightarrow \mathbb {R}$

$u\rightarrow u(\infty )$,

Since the functions vanish at infinity this operation of assigning the limit at infinity is clearly a linear map. It's not hard to see that $l\in C_{0}(X)'$, i.e. a bounded linear operator on $C_{0}(X)$. We showed before that $C_{0}(X)$ is a closed (vector) subspace of $C_{b}(X)$ therefore we can extend $l$ to all of $C_{b}(X)$ using the formulation of the Hahn Banach Theorem for normed spaces. Let $L$ be such extension, $L\in C_{b}(X)'$ and $L=l$ on $C_{0}(X)$. Note that this functional is supported at infinity, in the sense that for any $u\in C_{0}(X)$, we have $<L,u>=0$.

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

As it can be found in Villani, Proposition 1.22 also ([2]), the following version of Kantorovich duality holds: let $X$ and $Y$ be 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 satisfy $\pi (A\times Y)=\mu (A)$ 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 satisfy $\phi (x)+\psi (x)\leq c(x,y)$ for $d\mu$ for almost all $x\in X$ and for $d\nu$ almost all $y\in Y$.

As mentioned in Villani Section 1.3 pg. 39, 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 $C_{0}(X\times Y)\subset C_{b}(X\times Y)$ then any element in $C_{b}(X\times Y)'$ which acts continuously, as 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 $<R,f>=0$.

From what is discussed in the previous section, the behavior of some $R$ may not be clear at first glance as the following result shows in exercise 1.23 of .

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:

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

To prove this we want to apply what we have seen in the previous section. Lets consider the function $u(x,y)\in C_{0}(X\times Y)$: for fixed $x$ we can see this function as a function of $y$ i,e, let ${\hat {u}}(y)=u(x,y)$. Noticing that ${\hat {u}}\in C_{0}(Y)$, so we can assign a limit at infinity, $l$, to ${\hat {u}}$ and then extend it to all $\phi \in C_{b}(Y)$ following the construction of $L$ in the precious section. Similarly we will consider for any fixed $y$ the function $u(x,y)\in C_{0}(X\times Y)$ as a function of $x$. Note that such an extension is supported at infinity! This will allow us to first write the two functions:

$u_{1}(x,\infty )=l({\hat {u}}(y)),u_{2}(\infty ,y)=l({\hat {u}}(x))$.

Since we are only considering functions that vanish at infinity we can conclude that our $u_{1}\in C_{b}(X)$ and $u_{2}\in C_{b}(Y)$ satisfy the following:

$<l,u_{1}+u_{2}>=\int _{X}u_{1}(x,\infty )d\mu (x)+\int _{Y}u_{2}(\infty ,y)d\nu (y)$.

Here $l$, with a slightly abuse of notation, is the simultaneous assignment of the limit at infinity in $C_{b}(X)$ and $C_{b}(Y)$. The simultaneous extension $L$, again with a little abuse of notation, will be a bounded linear functional on $C_{b}(X\times Y)$and it will satisfy $\bigstar$ when restricted to $C_{0}(X)\times C_{0}(Y)$. By construction, shown in the previous section, $L$ is supported at infinity which means that when restricted to $C_{0}(X\times Y)$, it acts like the $0$ map. This means that we can't conclude easily that $L$ can be represented as an element of $\Pi (\mu ,\nu )$.

It turns out that in our hypothesis we can have the decomposition $L=\pi +R$ where $R$ is a continuous linear functional supported at infinity and then $L$ can be indeed represented as an element of $\Pi (\mu ,\nu )$; we will write: $L\in \Pi (\mu ,\nu )$. The key idea to prove this is to use the identity $<L,1>=1$. The detailed proof can be found again in Villani lemma 1.25 .