Definitions

Complete metric space

Complete metric space

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 or alternatively if every Cauchy sequence in M converges in M.

Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because sqrt{2} is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges 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.

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 xn+1 := xn/2 + 1/xn. 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 sqrt{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 Qp 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 SN of all sequences in S becomes a complete metric space if we define the distance between the sequences (xn) and (yn) to be 1/N, where N is the smallest index for which xN is distinct from yN, 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.

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, by the Heine-Borel theorem.

A closed subspace of a complete space is complete. Conversely, a complete subspace of a metric space is closed.

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

d(f,g) := supleft{,d(f(x),g(x)) : xin X ,right}.

If X is a topological space and M is a complete metric space, then the set Cb(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.

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 (xn)n and (yn)n in M, we may define their distance as

d(x,y) = limn d(xn,yn).
(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. 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.

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 which has the rational numbers as a subfield. This field is complete, admits a natural total ordering, and is the unique totally ordered complete field (up to isomorphism). It is defined to be 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 p, the 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.

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 NN (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.

Alternatives and generalizations

Since Cauchy sequences can also be defined in general topological groups, an alternative to relying on a metric structure for defining completeness and constructing the completion of a space is to use a group structure. This is most often seen in the context of topological vector spaces, but requires only the existence of a continuous "subtraction" operation. In this setting, the distance between two points x and y is gauged not by a real number varepsilon via the metric d in the comparison d(x,y), but by an open neighbourhood N of 0 via subtraction in the comparison x-y in N.

A common generalisation of these definitions can be found in the context of a uniform space, where an entourage is a set of all pairs of points that are at no more than a particular "distance" from each other, and the uniform structure is the given collection of entourages for the space.

It is also possible to replace Cauchy sequences in the definition of completeness by Cauchy nets or Cauchy filters. If every Cauchy net (or equivalently every Cauchy filter) has a limit in X, then X is called complete. One can furthermore 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.

References

  • Kreyszig, Erwin, Introductory functional analysis with applications (Wiley, New York, 1978). ISBN 0-471-03729-X
  • Meise, Reinhold; Vogt, Dietmar; translated by Ramanujan, M.S. Introduction to functional analysis. Oxford: Clarendon Press; New York: Oxford University Press.

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