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Complete metric space

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Metric geometry

"Cauchy completion" redirects here. For the use in category theory, seeKaroubi envelope.

Inmathematical analysis, ametric spaceM is calledcomplete (or aCauchy space) if everyCauchy sequence of points inM has alimit that is also inM.

Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set ofrational numbers is not complete, because e.g. ${\sqrt {2}}$ 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 thecompletion of a given space, as explained below.

Definition

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Cauchy sequence

Asequence $x_{1},x_{2},x_{3},\ldots$ of elements from $X {\displaystyle X}$ of ametric space $(X,d)$ is calledCauchy if for every positivereal number $r>0$ there is a positiveinteger $N {\displaystyle N}$ such that for all positive integers $m,n>N,$ $d(x_{m},x_{n})<r.$

Complete space

A metric space $(X,d)$ iscomplete if any of the following equivalent conditions are satisfied:

Every Cauchy sequence in $X {\displaystyle X}$ converges in $X {\displaystyle X}$ (that is, has a limit that is also in $X {\displaystyle X}$ ).
Every decreasing sequence ofnon-empty closed subsets of $X, {\displaystyle X,}$ withdiameters tending to 0, has a non-emptyintersection: if $F_{n}$ is closed and non-empty, $F_{n+1}\subseteq F_{n}$ for every $n, {\displaystyle n,}$ and $\operatorname {diam} \left(F_{n}\right)\to 0,$ then there is a unique point $x\in X$ common to all sets $F_{n}.$

Examples

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The space $\mathbb {Q}$ of rational numbers, with the standardmetric given by theabsolute value of thedifference, is not complete. Consider for instance the sequence defined by

x_{1}=1\;

and

\;x_{n+1}={\frac {x_{n}}{2}}+{\frac {1}{x_{n}}}.

This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit $x, {\displaystyle x,}$ then by solving $x={\frac {x}{2}}+{\frac {1}{x}}$ necessarily $x^{2}=2,$ yet no rational number has this property. However, considered as a sequence ofreal numbers, it does converge to theirrational number ${\sqrt {2}}$ .

Theopen interval(0,1), again with the absolute difference metric, is not complete either. The sequence defined by $x_{n}={\tfrac {1}{n}}$ is Cauchy, but does not have a limit in the given space. However theclosed interval [0,1] is complete; for example the given sequence does have a limit in this interval, namely zero.

The space $\mathbb {R}$ of real numbers and the space $\mathbb {C}$ ofcomplex numbers (with the metric given by the absolute difference) are complete, and so isEuclidean space $\mathbb {R} ^{n}$ , with theusual distance metric. In contrast,infinite-dimensional normed vector spaces may or may not be complete; those that are complete areBanach spaces. The space C[a, b] ofcontinuous real-valued functions on a closed and bounded interval is a Banach space, and so a complete metric space, with respect to thesupremum norm. However, the supremum norm does not give a norm on the space C(a, b) of continuous functions on(a, b), for it may containunbounded functions. Instead, with thetopology ofcompact convergence, C(a, b) can be given the structure of aFréchet space: alocally convex topological vector space whose topology can be induced by a completetranslation-invariant metric.

The spaceQ_p ofp-adic numbers is complete for anyprime number $p . {\displaystyle p.}$ This space completesQ with thep-adic metric in the same way thatR completesQ with the usual metric.

If $S {\displaystyle S}$ is an arbitrary set, then the setS^N of all sequences in $S {\displaystyle S}$ becomes a complete metric space if we define the distance between the sequences $\left(x_{n}\right)$ and $\left(y_{n}\right)$ to be ${\tfrac {1}{N}}$ where $N {\displaystyle N}$ is the smallest index for which $x_{N}$ isdistinct from $y_{N}$ or $0 {\displaystyle 0}$ if there is no such index. This space ishomeomorphic to theproduct of acountable number of copies of thediscrete space $S . {\displaystyle S.}$

Riemannian manifolds which are complete are calledgeodesic manifolds; completeness follows from theHopf–Rinow theorem.

Some theorems

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Everycompact metric space is complete, though complete spaces need not be compact. In fact, a metric space is compactif and only if it is complete andtotally bounded. This is a generalization of theHeine–Borel theorem, which states that any closed and bounded subspace $S {\displaystyle S}$ ofRⁿ is compact and therefore complete.^[1]

Let $(X,d)$ be a complete metric space. If $A\subseteq X$ is a closed set, then $A {\displaystyle A}$ is also complete. Let $(X,d)$ be a metric space. If $A\subseteq X$ is a complete subspace, then $A {\displaystyle A}$ is also closed.

Theorem—Let $(X,d)$ be a complete metric space, and let $(A,d)$ be a subspace of $X {\displaystyle X}$ . Then $A {\displaystyle A}$ is complete if and only if $A {\displaystyle A}$ is a closed subset of $X {\displaystyle X}$ .

Proof

Assume $A {\displaystyle A}$ is a closed subset of $X {\displaystyle X}$ . If $(a_{n})$ is a Cauchy sequence in $(A,d)$ , then it is also a Cauchy sequence in $X {\displaystyle X}$ . Since $X {\displaystyle X}$ is complete, the sequence $(a_{n})$ converges to a point $a_{0}$ in $X {\displaystyle X}$ . Since $A {\displaystyle A}$ is a closed subset of $X {\displaystyle X}$ , the point $a_{0}$ is in $A {\displaystyle A}$ . This shows that every Cauchy sequence in $A {\displaystyle A}$ converges in $A {\displaystyle A}$ , and hence, $(A,d)$ is complete.

Conversely, assume that $A {\displaystyle A}$ is not a closed subset of $X {\displaystyle X}$ . Then there is a sequence $(a_{n})$ in $A {\displaystyle A}$ that converges to a point $a_{0}$ in $X {\displaystyle X}$ , but $a_{0}$ is not in $A {\displaystyle A}$ . Since the Cauchy sequence $(a_{n})$ in $(A,d)$ does not converge in $A {\displaystyle A}$ , thus $(A,d)$ is not complete.

If $X {\displaystyle X}$ is aset and $M {\displaystyle M}$ is a complete metric space, then the set $B(X,M)$ of all bounded functionsf fromX to $M {\displaystyle M}$ is a complete metric space. Here we define the distance in $B(X,M)$ in terms of the distance in $M {\displaystyle M}$ with thesupremum norm $d(f,g)\equiv \sup\{d[f(x),g(x)]:x\in X\}$

If $X {\displaystyle X}$ is atopological space and $M {\displaystyle M}$ is a complete metric space, then the set $C_{b}(X,M)$ consisting of allcontinuous bounded functions $f:X\to M$ is a closed subspace of $B(X,M)$ and hence also complete.

TheBaire category theorem says that every complete metric space is aBaire space. That is, theunion of countably manynowhere dense subsets of the space has emptyinterior.

TheBanach fixed-point theorem states that acontraction mapping on a complete metric space admits afixed point. The fixed-point theorem is often used toprove theinverse function theorem on complete metric spaces such as Banach spaces.

Theorem^[2] (C. Ursescu)—Let $X {\displaystyle X}$ be a complete metric space and let $S_{1},S_{2},\ldots$ be a sequence of subsets of $X . {\displaystyle X.}$

If each $S_{i}$ is closed in $X {\displaystyle X}$ then ${\textstyle \operatorname {cl} \left(\bigcup _{i\in \mathbb {N} }\operatorname {int} S_{i}\right)=\operatorname {cl} \operatorname {int} \left(\bigcup _{i\in \mathbb {N} }S_{i}\right).}$
If each $S_{i}$ isopen in $X {\displaystyle X}$ then ${\textstyle \operatorname {int} \left(\bigcap _{i\in \mathbb {N} }\operatorname {cl} S_{i}\right)=\operatorname {int} \operatorname {cl} \left(\bigcap _{i\in \mathbb {N} }S_{i}\right).}$

Completion

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For any metric spaceM, it is possible to construct a complete metric spaceM′ (which is also denoted as ${\overline {M}}$ ), which containsM as adense subspace. It has the followinguniversal property: ifN is any complete metric space andf is anyuniformly continuous function fromM toN, then there exists a unique uniformly continuous functionf′ fromM′ toN that extendsf. The spaceM' is determinedup to isometry by this property (among all complete metric spaces isometrically containingM), and is called thecompletion ofM.

The completion ofM can be constructed as a set ofequivalence classes of Cauchy sequences inM. For any two Cauchy sequences $x_{\bullet }=\left(x_{n}\right)$ and $y_{\bullet }=\left(y_{n}\right)$ inM, we may define their distance as $d\left(x_{\bullet },y_{\bullet }\right)=\lim _{n}d\left(x_{n},y_{n}\right)$

(This limit exists because the real numbers are complete.) This is only apseudometric, not yet a metric, since two different Cauchy sequences may have the distance 0. But "having distance 0" is anequivalence relation on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion ofM. The original space is embedded in this space via the identification of an elementx ofM' with the equivalence class of sequences inM converging tox (i.e., the equivalence class containing the sequence with constant valuex). 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 afield that has the rational numbers as asubfield. This field is complete, admits a naturaltotal ordering, and is the unique totally ordered complete field (up toisomorphism). It isdefined as the field of real numbers (see alsoConstruction 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 thedecimal expansion give just one choice of Cauchy sequence in the relevant equivalence class.

For a prime $p, {\displaystyle p,}$ thep-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 aninner product space, the result is aHilbert space containing the original space as a dense subspace.

Topologically complete spaces

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Completeness is a property of themetric and not of thetopology, meaning that a complete metric space can behomeomorphic 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.

Intopology one considerscompletely 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 that can be written as an intersection of countably many open subsets of some complete metric space. Since the conclusion of theBaire category theorem is purely topological, it applies to these spaces as well.

Completely metrizable spaces are often calledtopologically 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 sectionAlternatives and generalizations). Indeed, some authors use the termtopologically complete for a wider class of topological spaces, thecompletely uniformizable spaces.^[3]

A topological space homeomorphic to aseparable complete metric space is called aPolish space.

Alternatives and generalizations

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Main article:Uniform space § Completeness

SinceCauchy sequences can also be defined in generaltopological 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 oftopological vector spaces, but requires only the existence of a continuous "subtraction" operation. In this setting, the distance between two points $x {\displaystyle x}$ and $y {\displaystyle y}$ is gauged not by a real number $\varepsilon$ via the metric $d {\displaystyle d}$ in the comparison $d(x,y)<\varepsilon ,$ but by anopen neighbourhood $N {\displaystyle N}$ of $0 {\displaystyle 0}$ via subtraction in the comparison $x-y\in N.$

A common generalization of these definitions can be found in the context of auniform space, where anentourage 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 Cauchysequences in the definition of completeness by Cauchynets or Cauchyfilters. If every Cauchy net (or equivalently every Cauchy filter) has a limit in $X, {\displaystyle X,}$ then $X {\displaystyle X}$ 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 isCauchy spaces; these too have a notion of completeness and completion just like uniform spaces.

Notes

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^Sutherland, Wilson A. (1975).Introduction to Metric and Topological Spaces. Clarendon Press.ISBN 978-0-19-853161-6.
^Zalinescu, C. (2002).Convex analysis in general vector spaces. River Edge, N.J. London: World Scientific. p. 33.ISBN 981-238-067-1.OCLC 285163112.
^Kelley, Problem 6.L, p. 208

References

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Kelley, John L. (1975).General Topology. Springer.ISBN 0-387-90125-6.
Kreyszig, Erwin,Introductory functional analysis with applications (Wiley, New York, 1978).ISBN 0-471-03729-X
Lang, Serge, "Real and Functional Analysis"ISBN 0-387-94001-4
Meise, Reinhold; Vogt, Dietmar (1997).Introduction to functional analysis. Ramanujan, M.S. (trans.). Oxford: Clarendon Press; New York: Oxford University Press.ISBN 0-19-851485-9.

Metric spaces (Category)

Basic concepts

Main results

Maps

Types of
metric spaces

Sets

Examples

Manifolds	Euclidean distance Riemannian
Functional analysis andMeasure theory	Chebyshev distance Inner product space Lévy metric Lévy–Prokhorov metric Metrizable topological vector space Normed space Taxicab geometry Wasserstein metric
General topology	Discrete space Intrinsic metric Laakso space Product metric