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Say we have a metric space $(M,d)$, and we want to complete it in the following sense:

Definition: A completion of $(M,d)$ is a complete metric space $(\widetilde{M},d')$ together with a Lipschitz funcion $i:M\rightarrow\widetilde{M}$ such that for every other complete metric space $(N,\rho)$ together with a Lipschitz function $f:M\rightarrow N$, there exists an unique Lipschitz function $F:\widetilde{M}\rightarrow N$ such that $F\circ i=f$.

This definition is adapted from the definition for uniform spaces (wikipedia). I changed the condition that the functions are uniformly continuous to Lipschitz so any two completions of a metric space would be "equivalent" as metric spaces, and not just be "uniformly equivalent". One could also suppose the functions are isometries, for example. (I think the better "morfisms" in the category of metric spaces are Lipschitz functions.)

The usual completion of $M$ is defined to be (a quotient of) the set $\widetilde{M}$ of Cauchy sequences on $M$ with the (pseudo)metric $d'\left((x_n)_n,(y_n)_n\right)=\lim d(x_n,y_n)$ and the inclusion $i:x\in M\mapsto (x)_n\in\widetilde{M}$ (it's the same as the one given in wikipedia).

However, suppose we are working with a normed (vector) space, let's say $(X,\Vert\cdot\Vert)$. The completion of $X$ (making the proper adaptations in the definition: that the functions are linear, etc...) can be very easily defined as the closure of $ev(X)$ as a subspace of $X''$, where $Y'$ denotes the dual of a normed space $Y$ with the operator norm, and $ev:X\rightarrow X''$ is the evaluation function: $ev(x)(f)=f(x)$ for every $x\in X$ and $f\in X'$.

My question is: would we be able to make a similar construction for general metric spaces, that is, to find a nice definition of dual of a metric space for which the dual of it would be a complete metric space? If so, we could try to just define the completion of $M$ as the closure of $ev(M)$ in $dual(dual(M))$. In other words, I would like to find a kind of (nice) (algebraic, analytic, etc..) structure so the category of sets with that structure is dual to the category of metric spaces.

The first possible "dual" of $(M,d)$ that comes to my mind is $C_b(M)$, the set of continuous bounded functions from $M$ to $\mathbb{R}$ with the $\infty$-norm. Problem is that this is a commutative, unital $C^*$-Algebra, and we know the natural dual of a commutative $C^*$-Algebra is a compact hausdorff topological space, not a metric space.

(also, I guess a dual notion of metric space couls have many applications other than just making completions)

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  • $\begingroup$ What is a "limited" function from $M$ to $\mathbb R$? I've never seen this term used before. Cool idea by the way. $\endgroup$ Jul 3, 2013 at 15:12
  • $\begingroup$ @PatrickDaSilva I'm almost sure it means "bounded". $\endgroup$ Jul 3, 2013 at 15:51
  • $\begingroup$ @Daniel : Me too, but the word is still a bit weird. $\endgroup$ Jul 3, 2013 at 16:21
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    $\begingroup$ @Luiz : In a similar way that not all Banach spaces are reflexive, you might not have a theory of a dual of a metric space with a bijection in the bidual. In the case of Banach spaces you always have an injection, but we had to restrict our attention to complete normed vector spaces. Perhaps some assumptions are required to get something nice out of $C_c(M)$, just like that I'm thinking about "$M$ is a locally compact completely regular topological space", so that you can try building an injection from $M$ to $C_c(M)$. It's just an idea. $\endgroup$ Jul 3, 2013 at 17:55
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    $\begingroup$ Interesting question, but I am pretty sure that the answer is "No, there is no such construction." (unfortunately) $\endgroup$ Jul 3, 2013 at 21:41

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This might not be the construction that you are expecting in this question. But, let me explain you a possible natural definition for "dual of a metric space". Since all the duality results are secretly rooted in category theory, one should categorify the notion of metric spaces to find the "correct" dual of it. First I should confess you that I am learning some of the notions (in category theory) mentioned in this answer and, therefore would like to keep things elementary as possible.

In this paper, William Lawvere found a way to interpret of metric spaces (or rather generalized spaces) as categories enriched over the monoidal poset category $([0,\infty),\ge,+)$. For more details look at this ncatlab page and this nice YouTube video. But in this setting, Cauchy completion is bit involve and, can be thought of as "the enriched Yoneda embedding" :). Now that we are working with categories instead of metric spaces, we have a natural dual (opposite category) and, it provide us a dual for the initial metric space.

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