Editing Complete metric space (section)

==Completion==
For any metric space ''M'', it is possible to construct a complete metric space ''M′'' (which is also denoted as <math>\overline{M}</math>), which contains ''M'' as a [[dense subspace]].  It has the following [[universal property]]: if ''N'' is any complete metric space and ''f'' is any [[uniformly continuous function]] from ''M'' to ''N'', then there exists a unique uniformly continuous function ''f′'' from ''M′'' to ''N'' that extends ''f''.  The space ''M''' is determined [[up to]] [[isometry]] by this property (among all complete metric spaces isometrically containing ''M''), and is called the ''completion'' of ''M''.

The completion of ''M'' can be constructed as a set of [[equivalence class]]es of Cauchy sequences in ''M''. For any two Cauchy sequences <math>x_{\bull} = \left(x_n\right)</math> and <math>y_{\bull} = \left(y_n\right)</math> in ''M'', we may define their distance as
<math display=block>d\left(x_{\bull}, y_{\bull}\right) = \lim_n d\left(x_n, y_n\right)</math>

(This limit exists because the real numbers are complete.) This is only a [[Pseudometric space|pseudometric]], not yet a metric, since two different Cauchy sequences may have the distance 0. But "having distance 0" is an [[equivalence relation]] on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion of ''M''.  The original space is embedded in this space via the identification of an element ''x'' of ''M''' with the equivalence class of sequences in ''M'' converging to ''x'' (i.e., the equivalence class containing the sequence with constant value ''x'').  This defines an isometry onto a dense subspace, as required. Notice, however, that this construction makes explicit use of the completeness of the real numbers, so completion of the rational numbers needs a slightly different treatment.

[[Georg Cantor|Cantor]]'s construction of the real numbers is similar to the above construction; the real numbers are the completion of the rational numbers using the ordinary absolute value to measure distances. The additional subtlety to contend with is that it is not logically permissible to use the completeness of the real numbers in their own construction. Nevertheless, equivalence classes of Cauchy sequences are defined as above, and the set of equivalence classes is easily shown to be a [[Field (mathematics)|field]] that has the rational numbers as a [[field extension|subfield]]. This field is complete, admits a natural [[total ordering]], and is the unique totally ordered complete field (up to [[isomorphism]]).  It is ''defined'' as the field of real numbers (see also [[Construction of the real numbers]] for more details). One way to visualize this identification with the real numbers as usually viewed is that the equivalence class consisting of those Cauchy sequences of rational numbers that "ought" to have a given real limit is identified with that real number. The truncations of the [[decimal expansion]] give just one choice of Cauchy sequence in the relevant equivalence class.

For a prime <math>p,</math> the [[p-adic number|{{math|''p''}}-adic numbers]] arise by completing the rational numbers with respect to a different metric.

If the earlier completion procedure is applied to a normed vector space, the result is a Banach space containing the original space as a dense subspace, and if it is applied to an [[inner product space]], the result is a [[Hilbert space]] containing the original space as a dense subspace.