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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, ''x''0 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 :$\Phi : X \rarr C_b\left(X\right)$ defined by :$\Phi\left(x\right)\left(y\right) = d\left(x,y\right)-d\left(x_0,y\right) \quad\mbox\quad x,y\in X$ is an isometry. Note that this embedding depends on the chosen point ''x''0 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 isometry :$\Psi : X \rarr C_b\left(X\right)$ defined by :$\Psi\left(x\right)\left(y\right) = d\left(x,y\right) \quad\mbox\quad x,y\in X$ 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 ''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''.

History

Formally speaking, this embedding was first introduced by Kuratowski, but a very close variation of this embedding appears already in the paper of Fréchet''Fréchet M.'' (1906) "Sur quelques points du calcul fonctionnel", Rendiconti del Circolo Matematico di Palermo 22: 1–74. where he first introduces the notion of metric space.