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 \(\{x_n\}_{n \in \N}\) such that \(|f(x_n)| \rightarrow \infty\) as \(n \rightarrow \infty\). By compactness there is a sub-sequence \(\{x_{n_k}\}_{i \in \N}\) 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:

:\(<\phi, f > = \int_{X} f d \mu, \text{ for every } f \in C_{0}(X)\). 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 \(n \in \N\) 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 \(K_{\epsilon} \sub X\) 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 [https://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem] 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 \(l : C_{0}(X) \rightarrow \R\)

:\(u \rightarrow u(\infty)\), 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 Y) $

As it can be found in Villani, Proposition 1.22 also ([http://34.106.105.83/wiki/Kantorovich_Problem]), 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 .