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

## Introduction

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

## Background and Statement

Let ${\displaystyle C_{0}(X)=\{f\in C(X){\text{ and }}\forall \epsilon >0{\text{ }}\exists {\text{ a compact set}}K\subset X{\text{ s.t. }}\mid f(x)\mid <\epsilon {\text{ }}\forall x\in X\setminus K\}}$ equipped with the sup norm. In other words this is the space of continuous functions vanishing at infinity. When ${\displaystyle X}$ is compact we can choose $\displaystyle K=X$ in the previous definition, and since properties on the empty set are trivially true, we can conclude that ${\displaystyle C(X)=C_{0}(X)}$. Let ${\displaystyle C_{b}(X)}$ be the space of bounded continuous functions on $\displaystyle X$ together with the sup norm. Again when ${\displaystyle X}$ 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 ${\displaystyle \{x_{n}\}_{n\in \mathbb {N} }}$ such that ${\displaystyle |f(x_{n})|\rightarrow \infty }$ as ${\displaystyle n\rightarrow \infty }$. By compactness there is a sub-sequence ${\displaystyle \{x_{n_{k}}\}_{i\in \mathbb {N} }}$ converging to a point ${\displaystyle {\hat {x}}\in X}$. Therefore by continuity of ${\displaystyle f}$ we have ${\displaystyle |f(x_{n_{k}})|\rightarrow |f({\hat {x}})|<\infty }$, and this is our desired contradiction. We conclude that ${\displaystyle C(X)=C_{b}(X)=C_{0}(X)}$.

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

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

The representation of the dual space of ${\displaystyle C_{0}(X)}$ is a described by the following well known result in Functional Analysis (Riesz Representation Theorem 6.19 in Rudin [1]):

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

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

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

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

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

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

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

We say that a function ${\displaystyle u\in C_{b}(X)}$ admits a limit at infinity, ${\displaystyle u(\infty )}$, if for any ${\displaystyle \epsilon >0}$ there exists a compact set ${\displaystyle K_{\epsilon }\subset X}$ such that ${\displaystyle x\notin K_{\epsilon }}$ implies ${\displaystyle |u(x)-u(\infty )|\leq \epsilon }$. 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 ${\displaystyle C_{b}(X)}$. This is another spectacular consequence of Axiom of Choice (Hahn-Banach theorem [1] in this case). Intuitively we can partition the space ${\displaystyle C_{b}(X)}$ 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 ${\displaystyle C_{b}(X)}$ admits such a limit. But this will not stop us from falling down the rabbit hole: note that every function in ${\displaystyle C_{0}(X)}$ admits such a limit, let ${\displaystyle l:C_{0}(X)\rightarrow \mathbb {R} }$

${\displaystyle u\rightarrow u(\infty )}$,

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 ${\displaystyle l\in C_{0}(X)'}$, i.e. a bounded linear operator on ${\displaystyle C_{0}(X)}$. We showed before that ${\displaystyle C_{0}(X)}$ is a closed (vector) subspace of ${\displaystyle C_{b}(X)}$ therefore we can extend ${\displaystyle l}$ to all of ${\displaystyle C_{b}(X)}$ using the formulation of the Hahn Banach Theorem for normed spaces. Let ${\displaystyle L}$ be such extension, ${\displaystyle L\in C_{b}(X)'}$ and ${\displaystyle L=l}$ on ${\displaystyle C_{0}(X)}$. Note that this functional is supported at infinity, in the sense that for any ${\displaystyle u\in C_{0}(X)}$, we have ${\displaystyle =0}$.

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

As it can be found in Villani, Proposition 1.22 [3] also ([2]), the following version of Kantorovich duality holds: let ${\displaystyle X}$ and ${\displaystyle Y}$ be locally compact Polish spaces, let ${\displaystyle c}$ be a lower semi-continuous non negative function on ${\displaystyle X\times Y}$ and let ${\displaystyle \mu }$ and ${\displaystyle \nu }$ be two Borel probability measures on ${\displaystyle X\times Y}$ respectively, then,

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

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

As mentioned in Villani Section 1.3 pg. 39[4], if we try to extend the proof of the compact case we run into a problem since the dual of ${\displaystyle C_{b}(X\times Y)}$ strictly contains ${\displaystyle {\mathcal {M}}(X\times Y)}$. If we restrict to the closed subspace ${\displaystyle C_{0}(X\times Y)\subset C_{b}(X\times Y)}$ then any element in ${\displaystyle C_{b}(X\times Y)'}$ which acts continuously, as mentioned before, can be represented by a unique $\displaystyle \pi \in \mathcal{M}(X\times Y)$ such that

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

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

From what is discussed in the previous section, the behavior of some $\displaystyle R$ may not be clear at first glance as the following result shows in exercise 1.23 of [5].

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

$\displaystyle \forall (\phi,\psi) \in C_{0}(X) \times C_{0}(Y), = \int_{X}\phi d\mu + \int_{Y}\psi d\nu. \text{ } \bigstar$ .

To prove this we want to apply what we have seen in the previous section. Lets consider the function ${\displaystyle u(x,y)\in C_{0}(X\times Y)}$: for fixed ${\displaystyle x}$ we can see this function as a function of ${\displaystyle y}$ i,e, let ${\displaystyle {\hat {u}}(y)=u(x,y)}$. Noticing that $\displaystyle \hat{u} \in C_0(Y)$ , so we can assign a limit at infinity, ${\displaystyle l}$, to ${\displaystyle {\hat {u}}}$ and then extend it to all $\displaystyle \phi \in C_b(Y)$ following the construction of ${\displaystyle L}$ in the precious section. Similarly we will consider for any fixed ${\displaystyle y}$ the function ${\displaystyle u(x,y)\in C_{0}(X\times Y)}$ as a function of $\displaystyle x$ . Note that such an extension is supported at infinity! This will allow us to first write the two functions:

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

Since we are only considering functions that vanish at infinity we can conclude that our $\displaystyle u_1 \in C_b(X)$ and ${\displaystyle u_{2}\in C_{b}(Y)}$ satisfy the following:

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

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

It turns out that in our hypothesis we can have the decomposition $\displaystyle L = \pi +R$ where ${\displaystyle R}$ is a continuous linear functional supported at infinity and then ${\displaystyle L}$ can be indeed represented as an element of ${\displaystyle \Pi (\mu ,\nu )}$; we will write: ${\displaystyle L\in \Pi (\mu ,\nu )}$. The key idea to prove this is to use the identity ${\displaystyle =1}$. The detailed proof can be found again in Villani lemma 1.25 [6].

1. Rudin, Walter. Real and Complex Analysis, 1966.
2. Villani, Cedric. Topics In Optimal Transportation. American Mathematical Soc., 2003.
3. Villani, Cedric. Topics In Optimal Transportation. American Mathematical Soc., 2003.
4. Villani, Cedric. Topics In Optimal Transportation. American Mathematical Soc., 2003.
5. Villani, Cedric. Topics In Optimal Transportation. American Mathematical Soc., 2003.
6. Villani, Cedric. Topics In Optimal Transportation. American Mathematical Soc., 2003.