Cauchy completion
E825425
Cauchy completion is a construction in metric space theory that embeds a given space into a complete metric space by formally adding limits of all its Cauchy sequences.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
construction in metric space theory
ⓘ
mathematical construction ⓘ |
| adds | formal limits of all Cauchy sequences ⓘ |
| alsoKnownAs | metric completion ⓘ |
| appliesTo | pseudometric spaces ⓘ |
| basedOn | Cauchy sequences ⓘ |
| categoryTheoreticView | reflective subcategory embedding of metric spaces into complete metric spaces ⓘ |
| characterization |
every Cauchy sequence in the completion converges
ⓘ
every point of the completion is a limit of a Cauchy sequence from the original space ⓘ |
| condition | a metric space is complete if and only if it is isometric to its Cauchy completion ⓘ |
| constructionMethod |
equivalence classes of Cauchy sequences
ⓘ
quotient of the set of Cauchy sequences by the equivalence relation of vanishing distance ⓘ |
| containsIsometricCopyOf | original metric space ⓘ |
| defines | distance between equivalence classes via limit of distances of representatives ⓘ |
| embeddingType | isometric embedding ⓘ |
| ensures |
completeness of the resulting metric space
ⓘ
every metric space admits a Cauchy completion ⓘ original points correspond to constant Cauchy sequences ⓘ |
| equivalenceRelation | two Cauchy sequences are equivalent if their distance tends to zero ⓘ |
| example |
completion of Q with the usual metric is R
ⓘ
completion of continuous functions with respect to L2 norm gives an L2 space ⓘ completion of polynomials under suitable norm yields function spaces like C[0,1] or Lp spaces ⓘ |
| field |
analysis
ⓘ
metric space theory ⓘ topology ⓘ |
| generalizationOf | completion of the rational numbers to the real numbers ⓘ |
| input | metric space ⓘ |
| output | complete metric space ⓘ |
| preserves |
dense image of the original space in its completion
ⓘ
isometries up to unique isometry of completions ⓘ |
| property |
functorial up to isometry
ⓘ
original space is dense in its Cauchy completion ⓘ |
| purpose | to embed a metric space into a complete metric space ⓘ |
| relatedConcept |
Banach space
ⓘ
Cauchy sequence ⓘ complete metric space ⓘ completion of a uniform space ⓘ uniform continuity ⓘ |
| uniqueness | unique up to unique isometry ⓘ |
| universalProperty |
every uniformly continuous map from the original space to a complete metric space extends uniquely
ⓘ
initial object among complete metric spaces receiving an isometric embedding of the original space ⓘ |
| usedIn |
category-theoretic treatments of metric spaces
ⓘ
functional analysis ⓘ measure theory ⓘ metric geometry ⓘ probability theory ⓘ |
Referenced by (1)
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