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{{r|Complete metric space}}

{{r|Complete metric space}}

{{r|Complete metric space}}

{{r|Complete metric space}}

{{r|Complete metric space}}

{{r|Complete metric space}}

{{r|Complete metric space}}

...the '''Banach's fixed-point theorem''' states that a contracting map in a complete metric space has a unique fixed-point. Given a complete metric space (''X'',''ρ''), i.e. a metric space in which every [[Cauchy sequence]] {''x

...sconnected]]. The rational numbers do not form a [[completeness (topology)|complete metric space]]; the [[real numbers]] are the completion of <math>\mathbb{Q}</math>.

Every Euclidean space is also a complete metric space. Moreover, all geometric notions immanent to a Euclidean space can be chara

...ch that <math>c_1\rho(x,y)\le\rho(f(x),f(y))\le c_2\rho(x,y)</math> from a complete metric space (''X'',''ρ''), the dimension is preserved, because the inverse map is also

...es, the corresponding qualitative theory of the eigenfunctions and their [[Complete metric space|completeness]] in a suitable [[function space]] became known as '''Sturm–

...nce]]s. This notion has led to the fundamental mathematical concept of a [[complete metric space]]. The [[Cauchy condition]] for the convergence of [[series (mathematics)|s