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mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
, a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
is called complete (or a Cauchy space) if every
Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
of points in has a
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
that is also in . 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 e.g. \sqrt is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the ''completion'' of a given space, as explained below.


Definition

Cauchy sequence A sequence x_1, x_2, x_3, \ldots in a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
(X, d) is called Cauchy if for every positive real number r > 0 there is a positive integer N such that for all positive integers m, n > N, d\left(x_m, x_n\right) < r. Complete space A metric space (X, d) is complete if any of the following equivalent conditions are satisfied: :#Every
Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
of points in X has a
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
that is also in X. :#Every Cauchy sequence in X converges in X (that is, to some point of X). :#Every decreasing sequence of non-empty closed subsets of X, with
diameters In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...
tending to 0, has a non-empty intersection: if F_n is closed and non-empty, F_ \subseteq F_n for every n, and \operatorname\left(F_n\right) \to 0, then there is a point x \in X common to all sets F_n.


Examples

The space Q of rational numbers, with the standard metric given by the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of the difference, is not complete. Consider for instance the sequence defined by x_1 = 1 and x_ = \frac + \frac. This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit x, then by solving x = \frac + \frac necessarily x^2 = 2, yet no rational number has this property. However, considered as a sequence of real numbers, it does converge to the
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
\sqrt. The
open interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Othe ...
, again with the absolute value metric, is not complete either. The sequence defined by is Cauchy, but does not have a limit in the given space. However the
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
interval is complete; for example the given sequence does have a limit in this interval and the limit is zero. 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 R''n'', with the usual distance metric. In contrast, infinite-dimensional normed vector spaces may or may not be complete; those that are complete are
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
s. The space C of continuous real-valued functions on a closed and bounded interval is a Banach space, and so a complete metric space, with respect to the supremum norm. However, the supremum norm does not give a norm on the space C of continuous functions on , for it may contain unbounded functions. Instead, with the topology of
compact convergence In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology. Definition Let (X, \mathcal) be a topologica ...
, C can be given the structure of a Fréchet space: a locally convex topological vector space whose topology can be induced by a complete translation-invariant metric. 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 of all sequences in S becomes a complete metric space if we define the distance between the sequences \left(x_n\right) and \left(y_n\right) to be \tfrac 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 Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
of a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
number of copies of the discrete space S. Riemannian manifolds which are complete are called
geodesic manifold In mathematics, a complete manifold (or geodesically complete manifold) is a ( pseudo-) Riemannian manifold for which, starting at any point , you can follow a "straight" line indefinitely along any direction. More formally, the exponential map a ...
s; completeness follows from the
Hopf–Rinow theorem Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow, who published it in 1931. Stefan Cohn-Vossen extended part of the Hopf–Rinow theorem ...
.


Some theorems

Every
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
metric space is complete, though complete spaces need not be compact. In fact, a metric space is compact if and only if it is complete and totally bounded. This is a generalization of the Heine–Borel theorem, which states that any closed and bounded subspace S of is compact and therefore complete. Let (X, d) be a complete metric space. If A \subseteq X is a closed set, then A is also complete. Let (X, d) be a metric space. If A \subseteq X is a complete subspace, then A is also closed. If X is a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
and M is a complete metric space, then the set B(X, M) of all
bounded function In mathematics, a function ''f'' defined on some set ''X'' with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number ''M'' such that :, f(x), \le M for all ''x'' in ''X''. A ...
s from to M is a complete metric space. Here we define the distance in B(X, M) in terms of the distance in M with the supremum norm d(f, g) \equiv \sup\ If X is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
and M is a complete metric space, then the set C_b(X, M) consisting of all
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
bounded functions f : X \to M is a closed subspace of B(X, M) and hence also complete. The
Baire category theorem The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the ...
says that every complete metric space is a
Baire space In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
. That is, the
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''U ...
of
countably many In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
nowhere dense In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywher ...
subsets of the space has empty interior. The
Banach fixed-point theorem In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certa ...
states that a contraction mapping on a complete metric space admits a fixed point. The fixed-point theorem is often used to prove the inverse function theorem on complete metric spaces such as Banach spaces.


Completion

For any metric space ''M'', it is possible to construct a complete metric space ''M′'' (which is also denoted as \overline), which contains ''M'' as a dense subspace. It has the following
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
: if ''N'' is any complete metric space and ''f'' is any
uniformly continuous function In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
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 Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
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 classes of Cauchy sequences in ''M''. For any two Cauchy sequences x_ = \left(x_n\right) and y_ = \left(y_n\right) in ''M'', we may define their distance as d\left(x_, y_\right) = \lim_n d\left(x_n, y_n\right) (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 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.
Cantor A cantor or chanter is a person who leads people in singing or sometimes in prayer. In formal Jewish worship, a cantor is a person who sings solo verses or passages to which the choir or congregation responds. In Judaism, a cantor sings and lead ...
'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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
that has the rational numbers as a subfield. This field is complete, admits a natural
total ordering In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexiv ...
, 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 In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ...
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 -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 In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
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

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 , which is not complete. In topology one considers ''
completely metrizable space In mathematics, a completely metrizable space (metrically topologically complete space) is a topological space (''X'', ''T'') for which there exists at least one metric ''d'' on ''X'' such that (''X'', ''d'') is a complete metric space and ''d'' in ...
s'', spaces for which there exists at least one complete metric inducing the given topology. Completely metrizable spaces can be characterized as those spaces that can be written as an intersection of countably many open subsets of some complete metric space. Since the conclusion of the
Baire category theorem The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the ...
is purely topological, it applies to these spaces as well. Completely metrizable spaces are often called ''topologically complete''. However, the latter term is somewhat arbitrary since metric is not the most general structure on a topological space for which one can talk about completeness (see the section Alternatives and generalizations). Indeed, some authors use the term ''topologically complete'' for a wider class of topological spaces, the
completely uniformizable space In mathematics, a topological space (''X'', ''T'') is called completely uniformizable (or Dieudonné complete) if there exists at least one complete uniformity that induces the topology ''T''. Some authors additionally require ''X'' to be Hausdor ...
s.Kelley, Problem 6.L, p. 208 A topological space homeomorphic to a separable complete metric space is called a Polish space.


Alternatives and generalizations

Since
Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
s 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) < \varepsilon, 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 An entourage () is an informal group or band of people who are closely associated with a (usually) famous, notorious, or otherwise notable individual. The word can also refer to: Arts and entertainment * L'entourage, French hip hop / rap collect ...
is a set of all pairs of points that are at no more than a particular "distance" from each other. 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 space In general topology and analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense. Cauchy spaces were introduced by H. H. Keller in 1968, as an axiomatic tool deriv ...
s; these too have a notion of completeness and completion just like uniform spaces.


See also

* * * * * *


Notes


References

* * Kreyszig, Erwin, ''Introductory functional analysis with applications'' (Wiley, New York, 1978). * Lang, Serge, "Real and Functional Analysis" * {{DEFAULTSORT:Complete Metric Space Metric geometry Topology Uniform spaces