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. The statement obviously holds for the empty space.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 map defined by is an isometry. The above construction can be seen as embedding a pointed metric space into a Banach space. defined by The convex set mentioned above is the convex hull of Ψ(X).
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| - Einbettungssatz von Arens-Eells (de)
- Plongement de Kuratowski (fr)
- Kuratowski embedding (en)
- Teorema di Fréchet-Kuratowski (it)
- Вложение Куратовского (ru)
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| - Der Einbettungssatz von Arens-Eells (englisch Arens-Eells embedding theorem) ist ein mathematischer Lehrsatz, welcher im Übergangsfeld zwischen den mathematischen Teilgebieten Analysis, Funktionalanalysis und Topologie einzuordnen ist. Er geht zurück auf die beiden Mathematiker Richard Friederich Arens und James Eells und behandelt die Frage der Einbettbarkeit beliebiger metrischer Räume in komplexe normierte Räume und insbesondere in komplexe Banachräume. (de)
- En mathématiques, le plongement de Kuratowski permet d'identifier tout espace métrique à une partie d'un espace de Banach (de façon non canonique). (fr)
- Вложение Куратовского — определённое изометрическое вложение метрического пространства в банахово пространство непрерывных ограниченных функций на нём. (ru)
- 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. The statement obviously holds for the empty space.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 map defined by is an isometry. The above construction can be seen as embedding a pointed metric space into a Banach space. defined by The convex set mentioned above is the convex hull of Ψ(X). (en)
- In matematica, il teorema di Kuratowski-Wojdysławski o teorema di Fréchet-Kuratowski, che prende il nome da Kazimierz Kuratowski e Maurice René Fréchet, stabilisce che ogni spazio metrico può essere incluso in un particolare spazio di Banach. Questa inclusione permette di vedere ogni spazio metrico come sottoinsieme di uno spazio di Banach, consentendo così di sfruttare le proprietà degli spazi di Banach che non sono condivise da tutti gli spazi metrici (come la completezza). (it)
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| - Der Einbettungssatz von Arens-Eells (englisch Arens-Eells embedding theorem) ist ein mathematischer Lehrsatz, welcher im Übergangsfeld zwischen den mathematischen Teilgebieten Analysis, Funktionalanalysis und Topologie einzuordnen ist. Er geht zurück auf die beiden Mathematiker Richard Friederich Arens und James Eells und behandelt die Frage der Einbettbarkeit beliebiger metrischer Räume in komplexe normierte Räume und insbesondere in komplexe Banachräume. (de)
- 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. The statement obviously holds for the empty space.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 map defined by is an isometry. The above construction can be seen as embedding a pointed metric space into a Banach space. 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 isometry defined by The 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. (en)
- En mathématiques, le plongement de Kuratowski permet d'identifier tout espace métrique à une partie d'un espace de Banach (de façon non canonique). (fr)
- In matematica, il teorema di Kuratowski-Wojdysławski o teorema di Fréchet-Kuratowski, che prende il nome da Kazimierz Kuratowski e Maurice René Fréchet, stabilisce che ogni spazio metrico può essere incluso in un particolare spazio di Banach. Questa inclusione permette di vedere ogni spazio metrico come sottoinsieme di uno spazio di Banach, consentendo così di sfruttare le proprietà degli spazi di Banach che non sono condivise da tutti gli spazi metrici (come la completezza). Introdotta da Kuratowski, una variante molto simile si ritrovava già in una pubblicazione di Fréchet dove viene introdotta per la prima volta l'idea di spazio metrico. (it)
- Вложение Куратовского — определённое изометрическое вложение метрического пространства в банахово пространство непрерывных ограниченных функций на нём. (ru)
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