Out of idle curiosity, I was mulling the idea of the completion of a metric space.
In a nutshell, one starts with a metric space $(M, d)$, defines an equivalence relation $\sim$ on the set of Cauchy sequences by $(x_n) \sim (y_n)$ if and only if $\lim_{n \to \infty} d(x_n, y_n) = 0$. We let $\overline{M}$ be the set of equivalence classes, and define $\overline{d}$, a metric on $\overline{M}$, by $$\overline{d}((x_n), (y_n)) = \lim_{n \to \infty} d(x_n, y_n).$$ Naturally, one needs to prove that the above limit exists, and the metric is well-defined, but this all works out. We get the unique complete metric space, up to isometry, containing $M$ as a dense subspace.
Now, what if, instead of making an equivalence relation on Cauchy sequences, we instead use bounded sequences? For bounded sequences $(x_n), (y_n)$ in $M$, we say $(x_n) \sim (y_n)$ if and only if $\operatorname*{limsup}_{n\to \infty} d(x_n, y_n) \to 0$. We then define $\widetilde{M}$ to be the set of equivalence classes, and let $$\widetilde{d}((x_n), (y_n)) = \operatorname*{limsup}_{n \to \infty} d(x_n, y_n).$$ It's not difficult to see $\widetilde{d}$ is a well-defined metric on $\widetilde{M}$.
Question: Has this construction been studied before? Does it, or something equivalent to it, have a name? Where could I find more information about it?
Some observations:
- $\widetilde{M}$ contains $\overline{M}$ as a (complete) subspace.
- $\widetilde{M}$ is itself complete.
- If $X$ is a normed linear space, then $\widetilde{X}$ is a linear space as well: the sequence space $\ell^\infty(X)$, quotiented by the closed subspace of null-convergent sequences of $X$. If $d$ is the metric derived from the norm $\|\cdot\|_X$, then the metric $\widetilde{d}$ is derived from the quotient norm $$\|(x_n)\|_{\widetilde{X}} = \operatorname*{limsup}_{n \to \infty} \|x_n\|_X.$$
This is wildly more complicated than the Cauchy completion. Here is what it looks like for $M = \mathbb{R}$; first, the vector space of bounded sequences $\mathbb{N} \to \mathbb{R}$ can be identified with the commutative $C^{\ast}$-algebra of continuous functions $\beta \mathbb{N} \to \mathbb{R}$ where $\beta \mathbb{N}$ is the Stone-Cech compactification of $\mathbb{N}$, which can be identified with the space of ultrafilters on $\mathbb{N}$. $\widetilde{ \mathbb{R} }$ is obtained by quotienting by the subspace of sequences converging to $0$, which can be identified with the closed ideal of functions on $\beta \mathbb{N}$ vanishing on the complement $\beta \mathbb{N} \setminus \mathbb{N}$.
The quotient by this closed ideal then produces another $C^{\ast}$-algebra, which is the $C^{\ast}$-algebra of continuous functions on $\beta \mathbb{N} \setminus \mathbb{N}$, the space of non-principal ultrafilters on $\mathbb{N}$. This is a very complicated space. Somewhat more explicitly, the bijection from $\widetilde{\mathbb{R}}$ to this algebra of continuous functions sends a bounded sequence to the function which sends a non-principal ultrafilter to the ultralimit of the sequence with respect to the ultrafilter. (Maybe this continues to be a bijection for more general $M$, at least if $M$ is complete, but I don't know.)