# complete metric space

(redirected from*Banach completion*)

## complete metric space

[kəm′plēt ¦me·trik ′spās]## Complete Metric Space

a metric space in which the Cauchy convergence criterion is satisfied.

A sequence of points *x*_{1}, *x*_{2}, …, *x*_{n}, … on a line, in a plane, or in space is said to be a Cauchy, or fundamental, sequence if for sufficiently large numbers *n* and *m* the distance between the points *x*_{n} and *x*_{m} becomes arbitrarily small. According to the Cauchy criterion, such a sequence of points has a limit if, and only if, it is a Cauchy sequence. For many sets of mathematical objects, such as functions or operators, a concept of distance can be introduced that has properties analogous to the properties of ordinary distance. The set is then said to be a metric space. The concept of the limit of a sequence of points in the metric space can be defined in the usual way. If the Cauchy criterion holds, the space is called complete.

Examples of complete metric spaces are Euclidean spaces and many other linear spaces—for example, the space of continuous functions on the interval *[a, b]* with distance

and Hubert space. A closed subset of a complete metric space is itself a complete metric space. If a metric space is not complete, it can be made complete in a way analogous to the way the set of rational numbers can be augmented by the irrationals to form the set of real numbers.

The concept of completeness can be extended to nonmetric topological spaces in which neighborhoods of different points can be compared. Examples of such spaces are topological groups and rings.