Kuratowski embedding

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 C_b(X) denotes the Banach space of all bounded continuous real-valued functions on X with the supremum norm, then the map

:$\backslash Phi\; :\; X\; \backslash rarr\; C\_b(X)$

defined by

:$\backslash Phi(x)(y)\; =\; d(x,y)-d(x\_0,y)\; \backslash quad\backslash mbox\{for\; all\}\backslash quad\; x,y\backslash in\; X$

is an isometry.{{citation|title=Geometric embeddings of metric spaces|url=http://www.math.jyu.fi/research/reports/rep90.ps|author=Juha Heinonen|date=January 2003|accessdate=6 January 2009}}

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.{{citation|author=Karol Borsuk|title=Theory of retracts|year=1967|place=Warsaw}}. Theorem III.8.1 (N.B. the image of this embedding is closed in the convex subset, not necessarily in the Banach space.) Here we use the isometry

:$\backslash Psi\; :\; X\; \backslash rarr\; C\_b(X)$

defined by

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

The convex set mentioned above is the convex hull of Ψ(X).

In both of these embedding theorems, we may replace C_b(X) by the Banach space ℓ ^∞(X) of all bounded functions X → R, again with the supremum norm, since C_b(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.