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.
Specifically, if is a metric space, x0 is a point in X, and Cb 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.
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. Here we use the isometry
defined by
The convex set mentioned above is the convex hull of Ψ.
In both of these embedding theorems, we may replace Cb by the Banach space ℓ ∞ of all bounded functions X → R, again with the supremum norm, since Cb is a closed linear subspace of ℓ ∞.
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 where he first introduces the notion of metric space.