Complete metric space

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For Cauchy completion in category theory, see Karoubi envelope.

In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M.

Intuitively, a space is complete if it "doesn't have any holes", if there aren't any "points missing". For instance, the rational numbers are not complete, because √2 is "missing" even though you can construct a Cauchy sequence of rational numbers that converge to it. (See the examples below.) It is always possible to "fill all the holes", leading to the completion of a given space, as will be explained below.

Contents 1 Examples 2 Some theorems 3 Completion 4 Topologically complete spaces 5 Generalizations 6 Reference

[edit] Examples

The space Q of rational numbers, with the standard metric given by the absolute value, is not complete. Consider for instance the sequence defined by x₁ := 1 and x_n+1 := x_n/2 + 1/x_n. This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: Such a limit x of the sequence would have the property that x² = 2, but no rational numbers have that property. But considered as a sequence of real numbers R it converges towards the irrational number √2, the square root of two.

The open interval (0,1), again with the absolute value metric, is not complete either. The sequence (1/2, 1/3, 1/4, 1/5, ...) is Cauchy, but does not have a limit in the space. However the closed interval [0,1] is complete; the sequence above has the limit 0 in this interval.

The space R of real numbers and the space C of complex numbers (with the metric given by the absolute value) are complete, and so is Euclidean space Rⁿ. Other normed vector spaces may or may not be complete; those which are, are the Banach spaces.

The space Q_p of p-adic numbers is complete for any prime number p. This space completes Q with the p-adic metric in the same way that R completes Q with the usual metric.

If S is an arbitrary set, then the set S^N of all sequences in S becomes a complete metric space if we define the distance between the sequences (x_n) and (y_n) to be 1/N, where N is the smallest index for which x_N is distinct from y_N, or 0 if there is no such index. This space is homeomorphic to the product of a countable number of copies of the discrete space S.

The topological vector space R^ω of sequences of real numbers which have finitely many nonzero terms (the topology of this space can alternatively be defined as the limit topology of the Rⁿ or as the coproduct of infinitely many copies of R) is not complete (even though its underlying field is). The completion of this space is the product ΠR of infinitely many copies of R. If instead we endow the space with the l^p norm, its completion is the space l^p(N).

[edit] Some theorems

Every compact metric space is complete. In fact, a metric space is compact if and only if it is complete and totally bounded.

A closed subspace of a complete space is complete. [1] Conversely, a complete subspace of a metric space is closed. [2] (In other words, there may be a closed subspace of a metric space that is not complete.)

If X is a set and M is a complete metric space, then the set B(X,M) of all bounded functions f from X to M is a complete metric space. Here we define the distance in B(X,M) in terms of the distance in M as

If X is a topological space and M is a complete metric space, then the set C_b(X,M) consisting of all continuous bounded functions f from X to M is a closed subspace of B(X,M) and hence also complete.

The Baire category theorem says that every complete metric space is a Baire space. That is, the interior of a union of countably many nowhere dense subsets of the space is empty.

[edit] Completion

For any metric space M, one can construct a complete metric space M' (which is also denoted as M with a bar over it), 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 which extends f. The space M' is determined up to isometry by this property, and is called the completion of M.

The completion of M can be constructed as a set of equivalence classes of Cauchy sequences in M. For any two Cauchy sequences (x_n)_n and (y_n)_n in M, we may define their distance as

d(x,y) = lim_n d(x_n,y_n).

(This limit exists because the real numbers are complete.) This is only a 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 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.

Cantor's construction of the real numbers is a special case of this; the real numbers are the completion of the rational numbers using the ordinary absolute value to measure distances. By using different notions of distance on the rationals, one obtains different incomplete metric spaces whose completions are the p-adic numbers.

If this completion procedure is applied to a normed vector space, one obtains a Banach space containing the original space as a dense subspace, and if it is applied to an inner product space, one obtains a Hilbert space containing the original space as a dense subspace.

[edit] Topologically complete spaces

Note that completeness is a property of the metric and not of the topology, meaning that a complete metric space can be homeomorphic to a non-complete one. An example is given by the real numbers, which are complete but homeomorphic to the open interval (0,1), which is not complete. Another example is given by the irrational numbers, which are not complete as a subspace of the real numbers but are homeomorphic to N^N (a special case of an example in Examples above).

In topology one considers topologically complete (or completely metrizable) spaces, spaces for which there exists at least one complete metric inducing the given topology. Completely metrizable spaces can be characterized as those spaces which can be written as an intersection of countably many open subsets of some complete metric space. Since the conclusion of the Baire category theorem is purely topological, it applies to these spaces as well.

[edit] Generalizations

It is also possible to define the concept of completeness for uniform spaces using Cauchy nets instead of Cauchy sequences. If every Cauchy net has a limit in X, then X is called complete. One can also construct a completion for an arbitrary uniform space similar to the completion of metric spaces. The most general situation in which Cauchy nets apply is Cauchy spaces; these too have a notion of completeness and completion just like uniform spaces.

A topological space may be completely uniformisable without being completely metrisable; it is then still not topologically complete.

[edit] Reference

• Kreyszig, Erwin, Introductory functional analysis with applications (Wiley, New York, 1978). ISBN 0-471-03729-X