Completion

The completion of a metric space is a complete metric space containing an isometric dense copy of . It is obtained by adjoining limits of Cauchy sequences.

For example, the rational numbers, with the distance metric, are not complete because there exist Cauchy sequences that do not converge, e.g., 1, 1.4, 1.41, 1.414, ... does not converge because is not rational. The completion of the rationals is the real numbers. Note that the completion depends on the metric. For instance, for any prime , the rationals have a metric given by the p-adic norm, and then the completion of the rationals is the set of p-adic numbers. Another common example is the space of L²-functions on , identifying functions equal almost everywhere. It is the completion of the continuous functions on with the L²-norm (Axler 2026, pp. 201 and 205).

Technically, the completion consists of equivalence classes of Cauchy sequences, where iff . Its metric is

The limit exists and is independent of the representatives. The map sending to the class of its constant sequence embeds isometrically as a dense subspace. Every metric space has a completion, and any two completions are related by a unique isometry agreeing with their embeddings of the original space (Conrad, pp. 5-6).