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- Kuratowski_embedding abstract "In mathematics, the Kuratowski embedding allows one to view any metric space as a subset of some Banach space. It is named after Kazimierz Kuratowski.Specifically, if (X,d) is a metric space, x0 is a point in X, and Cb(X) denotes the Banach space of all bounded continuous real valued functions on X with the supremum norm, then the mapdefined byis an isometry.Note that this embedding depends on the chosen point x0 and is therefore not entirely canonical.The Kuratowski–Wojdysławski theorem states that every bounded metric space X is isometric to a closed subset of a convex subset of some Banach space. (N.B. the image of this embedding is closed in the convex subset, not necessarily in the Banach space.) Here we use the isometrydefined byThe convex set mentioned above is the convex hull of Ψ(X).In both of these embedding theorems, we may replace Cb(X) by the Banach space ℓ ∞(X) of all bounded functions X → R, again with the supremum norm, since Cb(X) is a closed linear subspace of ℓ ∞(X).These embedding results are useful because Banach spaces have a number of useful properties not shared by all metric spaces: they are vector spaces which allows one to add points and do elementary geometry involving lines and planes etc.; and they are complete. Given a function with codomain X, it is frequently desirable to extend this function to a larger domain, and this often requires simultaneously enlarging the codomain to a Banach space containing X.".
- Kuratowski_embedding wikiPageID "21014764".
- Kuratowski_embedding wikiPageRevisionID "543126314".
- Kuratowski_embedding hasPhotoCollection Kuratowski_embedding.
- Kuratowski_embedding subject Category:Functional_analysis.
- Kuratowski_embedding subject Category:Metric_geometry.
- Kuratowski_embedding comment "In mathematics, the Kuratowski embedding allows one to view any metric space as a subset of some Banach space.".
- Kuratowski_embedding label "Kuratowski embedding".
- Kuratowski_embedding label "Teorema di Fréchet-Kuratowski".
- Kuratowski_embedding label "Вложение Куратовского".
- Kuratowski_embedding sameAs Plongement_de_Kuratowski.
- Kuratowski_embedding sameAs Teorema_di_Fréchet-Kuratowski.
- Kuratowski_embedding sameAs m.05b5fdq.
- Kuratowski_embedding sameAs Q3984015.
- Kuratowski_embedding sameAs Q3984015.
- Kuratowski_embedding wasDerivedFrom Kuratowski_embedding?oldid=543126314.
- Kuratowski_embedding isPrimaryTopicOf Kuratowski_embedding.