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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 map

\( \Phi : X \rarr C_b(X) \)

defined by

\( \Phi(x)(y) = d(x,y)-d(x_0,y) \quad\mbox{for all}\quad x,y\in X \)

is an isometry.[1]

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.[2] (N.B. the image of this embedding is closed in the convex subset, not necessarily in the Banach space.) Here we use the isometry

\( \Psi : X \rarr C_b(X) \)

defined by

\( \Psi(x)(y) = d(x,y) \quad\mbox{for all}\quad x,y\in X \)

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  ∞(X) of all bounded functions XR, again with the supremum norm, since Cb(X) is a closed linear subspace of  ∞(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.


Formally speaking, this embedding was first introduced by Kuratowski[3], but a very close variation of this embedding appears already in the paper of Fréchet[4] where he first introduces the notion of metric space.
See also

Tight span, an embedding of any metric space into an injective metric space defined similarly to the Kuratowski embedding


^ Juha Heinonen (January 2003), Geometric embeddings of metric spaces, retrieved 6 January 2009
^ Karol Borsuk (1967), Theory of retracts, Warsaw. Theorem III.8.1
^ Kuratowski, C. Quelques problèmes concernant les espaces métriques non-separables. Fund. Math. 25 (1935), 534-545.
^ Fréchet M. Sur quelques points du calcul fonctionnel. — Rendiconti del Circolo Matematico di Palermo. — 1906. — 22. — pp. 1—74.

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