Difference between revisions of "Complete metric space"
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| − | A metric space in which each [[Cauchy criteria|Cauchy sequence]] converges. A complete metric space is a particular case of a [[ | + | A [[metric space]] in which each [[Cauchy criteria|Cauchy sequence]] converges. A complete metric space is a particular case of a [[complete uniform space]]. A closed subset $A$ of a complete metric $(X,d)$ space is itself a complete metric space (with the distance which is the restriction of $d$ to $A$). The converse is true in a general metric space: if $(X,d)$ is a metric space, not necessarily complete, and $A\subset X$ is such that $(A,d)$ is complete, then $A$ is necessarily a closed subset. |
| − | Given any metric space $(X,d)$ there exists a unique [[ | + | Given any metric space $(X,d)$ there exists a unique [[completion]] of $X$, that is a triple $(Y,\rho,i)$ such that: |
*$(Y, \rho)$ is a complete metric space; | *$(Y, \rho)$ is a complete metric space; | ||
*$i: X \to Y$ is an [[Isometric mapping|isometric embedding]], namely a map such that $d(x,y) = \rho (i(x), i(y))$ for any pair of points $x,y\in X$; | *$i: X \to Y$ is an [[Isometric mapping|isometric embedding]], namely a map such that $d(x,y) = \rho (i(x), i(y))$ for any pair of points $x,y\in X$; | ||
*$i(X)$ is dense in $Y$. | *$i(X)$ is dense in $Y$. | ||
Often people refer to the metric space $(Y, \rho)$ as the completion. Both the space and the isometric embedding are unique up to isometries. | Often people refer to the metric space $(Y, \rho)$ as the completion. Both the space and the isometric embedding are unique up to isometries. | ||
Revision as of 20:35, 9 November 2014
2020 Mathematics Subject Classification: Primary: 54E50 [MSN][ZBL]
A metric space in which each Cauchy sequence converges. A complete metric space is a particular case of a complete uniform space. A closed subset $A$ of a complete metric $(X,d)$ space is itself a complete metric space (with the distance which is the restriction of $d$ to $A$). The converse is true in a general metric space: if $(X,d)$ is a metric space, not necessarily complete, and $A\subset X$ is such that $(A,d)$ is complete, then $A$ is necessarily a closed subset.
Given any metric space $(X,d)$ there exists a unique completion of $X$, that is a triple $(Y,\rho,i)$ such that:
- $(Y, \rho)$ is a complete metric space;
- $i: X \to Y$ is an isometric embedding, namely a map such that $d(x,y) = \rho (i(x), i(y))$ for any pair of points $x,y\in X$;
- $i(X)$ is dense in $Y$.
Often people refer to the metric space $(Y, \rho)$ as the completion. Both the space and the isometric embedding are unique up to isometries.
Complete metric space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_metric_space&oldid=30896