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

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Introduction

In the case that the space is compact then all continuous functions belong to as we will show in the next section. On the other hand if the space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is not compact, we always have the inclusion , but there may be some continuous functions that do not belong to . Some of them may even even be bounded and still not belong to , which motivates us to consider the dual space of and the dual space of .

Background and Statement

Let equipped with the sup norm. In other words this is the space of continuous functions vanishing at infinity. When is compact we can choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K=X} in the previous definition, and since properties on the empty set are trivially true, we can conclude that . Let be the space of bounded continuous functions on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X } together with the sup norm. Again when 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 such that as . By compactness there is a sub-sequence converging to a point . Therefore by continuity of we have , and this is our desired contradiction. We conclude that .

The rest of this discussion will consider the case where is not compact. Rather than equality of the three spaces, we have the inclusions:

The case of

The representation of the dual space of is a described by the following well known result in Functional Analysis (Riesz Representation Theorem 6.19 in Rudin [1]):

Let be a locally compact Hausdorff space For any bounded linear function , i.e. an element of the dual space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{0}(X)' } , there is a unique complex Borel measure such that the following holds:

.

This allows us to identify with , the space of complex Borel measures. Moreover we can endow with the total variation norm: .

The case of

To describe the dual space of , we will focus on the behavior of functions at infinity, as in Exercise 1.23 of [2]. We first need a preliminary result: is a closed (vector) subspace of . In other words, contains all its limit points. Let be a convergent sequence in , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_n(x) } are continuous functions vanishing at infinity and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 vanishes at infinity: let be such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|f - f_n\|_\infty <\epsilon/2 } .Now since each vanishes at infinity, we can find such that for any . Then we can conclude by the triangle inequality that

.

That is, . This proves is a closed subspace of . We may now carefully specify the local property at infinity for .

We say that a function admits a limit at infinity, , if for any there exists a compact set such that implies . 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 . This is another spectacular consequence of Axiom of Choice (Hahn-Banach theorem [1] in this case). Intuitively we can partition the space 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 admits such a limit. But this will not stop us from falling down the rabbit hole: note that every function in admits such a limit, let

,

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 , i.e. a bounded linear operator on . We showed before that is a closed (vector) subspace of therefore we can extend to all of using the formulation of the Hahn Banach Theorem for normed spaces. Let be such extension, and on . Note that this functional is supported at infinity, in the sense that for any , we have .

Kantorovich Duality for

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

,

Here is the set of all probability measures that satisfy and for any measurable set and any measurable set ; is the set of all measurable functions that satisfy for for almost all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X } and for almost all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 strictly contains . If we restrict to the closed subspace then any element in which acts continuously, as mentioned before, can be represented by a unique Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi \in \mathcal{M}(X\times Y)} such that

.

We can then write where is a continuous linear functional supported at infinity, i.e. implies .

From what is discussed in the previous section, the behavior of some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R } may not be clear at first glance as the following result shows in exercise 1.23 of [5].

Let and be two Borel probability measures on respectively There is a continuous linear functional on , supported at infinity, such that the following holds:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 : for fixed we can see this function as a function of i,e, let . Noticing that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{u} \in C_0(Y) } , so we can assign a limit at infinity, , to and then extend it to all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi \in C_b(Y) } following the construction of in the precious section. Similarly we will consider for any fixed the function as a function of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } . Note that such an extension is supported at infinity! This will allow us to first write the two functions:

.

Since we are only considering functions that vanish at infinity we can conclude that our Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_1 \in C_b(X) } and satisfy the following:

.

Here Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l } , with a slightly abuse of notation, is the simultaneous assignment of the limit at infinity in and . The simultaneous extension , again with a little abuse of notation, will be a bounded linear functional on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{b}(X\times Y)} and it will satisfy when restricted to . By construction, shown in the previous section, is supported at infinity which means that when restricted to , it acts like the map. This means that we can't conclude easily that can be represented as an element of .

It turns out that in our hypothesis we can have the decomposition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = \pi +R } where is a continuous linear functional supported at infinity and then can be indeed represented as an element of ; we will write: . The key idea to prove this is to use the identity . 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.