Complete Space

In Mathématiques, a metric Espace $M$ is known as complete or complete space if all Suite of Cauchy of $M$ has a limit in $M$ (i.e. it converges in $M$). The property of complétude depends on the distance. It is thus important always to specify the distance which one takes when one speaks about complete space.

Intuitively, a space is complete if it “does not have a hole”, if it “does not have any missing point”. For example, the rational numbers do not form a complete space, since $\backslash \; sqrt\; \{2\}$ does not appear there whereas there exists a continuation of Cauchy of rational numbers having this limit. It is always possible “to fill the holes” thus bringing to completion with a given space.

The complétude can also be defined for uniform spaces, like the topological groups.

Examples

• Is space $\backslash \; mathbb\; Q$ of the rational numbers provided with the distance D (X, there) = |X - there| . This space is not complete. Indeed, let us consider the continuation defined by:

$x\_1\; =\; 1$ and $x\_\; \{n+1\}\; =\; \{x\_n\; \backslash \; over\; 2\}\; +\; \{1\; \backslash \; over\; x\_n\}$.

It is a continuation of Cauchy of rational numbers but it does not converge towards any limit belonging to $\backslash \; mathbb\; Q$. In fact it converges towards the irrational Nombre $\backslash \; sqrt\; \{2\}$.

• the open interval $]\; 0,1\; provided\; with\; the\; distance\; \text{'}\text{'}\; D\; (X,\; there)\; =\; |X\; -\; there|\text{'}\text{'}\; is\; not\; complete\; either.\; The\; continuation$ \backslash \; left\; (\{1\; \backslash \; over\; 2\},\; \{1\; \backslash \; over\; 3\},\; \{1\; \backslash \; over\; 4\},\; \{1\; \backslash \; over\; 5\}\; \backslash \; ldots\; \backslash \; right)$is\; a\; continuation\; of\; Cauchy\; but\; it\; does\; not\; have\; limit\; in\; the\; interval.$

• the unit] 0,1 provided with the distance $d\; (X,\; there)\; =\; |\; \backslash \; tan\; (\backslash \; pi\; X\; -\; \backslash \; pi/2)\; -\; \backslash \; tan\; (\backslash \; pi\; there\; -\; \backslash \; pi/2)\; |$ is complete.

• the real interval closed $$ provided with the usual distance is complete.

• space $\backslash \; R$ of the real numbers and spaces it $\backslash \; mathbb\; C$ of the complex numbers provided with the usual distance D (X, there) = |X - there| , is complete as well as the Euclidean Espace $\backslash \; R^n$ provided with the usual standard.

• the vector spaces normalized can be complete or not; those which are it are called spaces of Banach. All the normalized vector spaces of finished size are complete.

• space $\backslash \; mathbb\; Q\_p$ of the p-adic numbers provided with the p-adic distance is complete for all Prime number $p$. This space supplements $\backslash \; mathbb\; Q$ with metric p-adic just like the $\backslash \; R$ supplements $\backslash \; mathbb\; Q$ with the metric Euclidean one.

• If $S$ is a given unit, the unit $S^\; \{\backslash \; mathbb\; NR\}$ of the continuations of $S$ becomes a complete metric space if one defines the distance between the continuations $(x\_n)\; \_\; \{N\; \backslash \; in\; \backslash \; NR\}$ and $(y\_n)\; \_\; \{N\; \backslash \; in\; \backslash \; NR\}$ as being equal to $1\; \backslash \; over\; N$ where $N$ is the smallest index for which $x\_N\; \backslash \; y\_N$, or 0 if such an index does not exist.

Some theorems

• a metric space (E, d) is complete if and only if the intersection of any decreasing continuation of closed limited not vacuums $F\_n$ whose continuation of the diameters tends towards 0 has a nonempty intersection (theorem of complete encased).

• All metric Espace compact is complete. In fact, a metric space is compact if and only if it is complete and précompact.

• a subspace of a complete space is complete if and only if it is closed.

• If $X$ is a unit and $M$ a complete metric space, then the unit $B\; (X,\; M)$ of the limited functions of $X$ in $M$ is a complete metric space. One defines the distance in $B\; (X,\; M)$ in term of distance in $M$:

$d\; (F,\; G):\; =\; \backslash \; sup\; \backslash \; left\; \backslash \; \{\backslash ,\; D\; (F\; (X),\; G\; (X))\; :\; X\; \backslash \; in\; X\; \backslash ,\; \backslash \; right\; \backslash \}.$

• If $X$ is a topological Espace and $M$ a complete metric space, then the unit $C\_b\; (X,\; M)$ of the continuous functions limited $X$ in $M$ is a closed subspace of $B\; (X,\; M)$ and thus also complete.

• the theorem of Baire shows that any complete metric space is a Espace of Baire.

• Theorem of the fixed point: any contracting application $f$ of a complete metric space in itself admits single a Point fixes which is limiting of any in the following way defined continuation:

$x\_0\; \backslash ,$ unspecified

$x\_\; \{n+1\}\; =f\; (x\_n)\; \backslash ,$

• Any end product of metric spaces complete is complete for the induced distance.

• Is $(E,||.\; ||)$ a normalized vector space. The following properties are equivalent:

i) E is complete

II) any normally convergent series of elements of E is convergent.

Supplemented of a metric space

For any metric space $M$, it is possible to build a complete metric space $M\text{'}$ (also noted $\backslash \; M$ tilde or $\backslash \; hat\; M$) which contains $M$ like dense subspace . It has the following property: if $N$ is an unspecified complete metric space and $f$ is a function uniformly continuous of $M$ towards $N$, then there exists a single function uniformly continuous $f\text{'}$ of $M\text{'}$ towards $N$ which prolongs $f$. $M\text{'}$ is called supplemented of $M$.

Supplemented $M$ can be built like the whole of the classes of equivalence of the series Cauchy of $M$. For two series Cauchy $(u\_n)\; \_\; \{N\; \backslash \; in\; \backslash \; NR\}$ and $(v\_n)\; \_\; \{N\; \backslash \; in\; \backslash \; NR\}$ of $M$, one defines the relation then:

$U\; \backslash \; mathcal\; R\; V\; \backslash \; Leftrightarrow\; \backslash \; lim\_\; \{N\; \backslash \; to\; \backslash \; infty\}\; D\; (u\_n,\; v\_n)\; =\; 0$

where $d$ is the distance on the $M$ unit. This relation is well a Relation of equivalence. One notes then $\backslash \; tilde\; M$ his Ensemble quotient.

It is then a question of providing $\backslash \; M$ tilde with a distance which will make it complete. On the whole of the series Cauchy, one defines the application F which, with two series Cauchy $U\; =\; (u\_n)$ and $V\; =\; (v\_n)$, associates reality $f\; (U,\; V)\; =\; \backslash \; lim\_\; \{N\; \backslash \; to\; \backslash \; infty\}\; D\; (u\_n,\; v\_n)$. This relation is well an application because, the continuations U and V being of Cauchy, one can prove that the continuation $(D\; (u\_n,\; v\_n))$ is a continuation of Cauchy of $\backslash \; R\_+$, therefore a convergent continuation (because $\backslash \; R\_+$, provided with the usual distance, is complete). This application checks all the properties of a distance except one: F (U, V) = 0 does not imply inevitably only U = V .

On the other hand, of this application, one can induce an application on the unit quotient $\backslash \; M$ tilde, application which, with the classes of U and V, noted $\backslash \; dowry\; U$ and $\backslash \; dowry\; V$, associates $d\; (\backslash \; dowry\; U,\; \backslash \; dowry\; V)\; =\; F\; (U,\; V)$. It is shown that this definition is independent of the selected representatives and defines a distance well on $\backslash \; M$ tilde.

Original space is plunged in new space by identification of an element $x$ of $M$ to the class of equivalence which contains the constant continuation of value $x$.

It is shown whereas space $\backslash \; M$ tilde, provided with the distance D , is complete and that M is dense in $\backslash \; M$ tilde.

The construction of the real numbers is a particular case; the whole of the real numbers is supplemented of the whole of the rational numbers, the usual absolute value being used like distance. By using other concepts of distance on the rational numbers, one obtains other units, the p-adic numbers.

If this procedure is applied to a vector Space normalized, one obtains a Espace of Banach containing original space like dense subspace. Applied to a Space préhilbertien, one obtains a Espace of Hilbert.

Topologically complete space

The complétude is a topological property metric, but not , which means that a complete metric space can be homeomorphic with a space which is not it. The whole of the real numbers, for example, is complete and homeomorphic with the interval $]\; 0,1\; provided\; with\; the\; topology\; induced\; by\; the\; usual\; topology\; of$ \backslash \; mathbb\; R$qui\; is\; not\; complete.$

In topology, a space is regarded as topologically complete if there exists metric complete inducing the topology of this space. Such a space is also called Polish Espace.

Such a space is a particular case of Espace of Baire.

Example: it is the case of the unit] 0,1 which is not complete with the usual distance, but which becomes it with the distance $d\; (X,\; there)\; =\; |\; \backslash \; tan\; (\backslash \; pi\; X\; -\; \backslash \; pi/2)\; -\; \backslash \; tan\; (\backslash \; pi\; there\; -\; \backslash \; pi/2)\; |$.

Another meaning of the term

The epithet " complet" is sometimes used in the following direction: a Ensemble ordered is known as complete if very part admits a higher Borne (with the convention which any element raises the empty set and thus that $sup\; (\backslash \; emptyset)\; =min\; (E)$). This is equivalent (see below) so that very part has a lower Borne (with the convention opposed for the empty set: $inf\; (\backslash \; emptyset)\; =max\; (E)$). For example, all segment is a complete ordered unit. On the other hand, $\backslash \; mathbb\; \{R\}$ is complete for the usual distance but not as an ordered unit. To avoid any confusion Bourbaki had proposed the term completed , which was not essential. Thus, $\backslash \; mathbb\; \{R\}$ is not completed but $\backslash \; overline\; \{\backslash \; mathbb\; \{R\}\}\; =\; \backslash \; mathbb\; \{R\}\; \backslash \; cup\; \backslash \; \{-\; \backslash \; infty,\; +\; \backslash \; infty\; \backslash \}$ is, from where its name of completed real Droite . Another example is the unit P (E) parts of a unit E with for order inclusion: the upper limit is the meeting and the limit the lower intersection.

In addition, such a unit is a particular case of complete lattice. One thus has the Théorème of Knaster-Tarski: any increasing application of an ordered unit completed in itself has a fixed point.

See too

• Supplemented of a metric space on the site Les-Mathématiques.net
• complete Space on the site Les-Mathématiques.net
• Lattice

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