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

${\displaystyle \Phi :X\rightarrow C_{b}(X)}$

defined by

${\displaystyle \Phi (x)(y)=d(x,y)-d(x_{0},y)\quad {\mbox{for all}}\quad x,y\in X}$

is an isometry.^[1]

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.^[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

${\displaystyle \Psi :X\rightarrow C_{b}(X)}$

defined by

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

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

In both of these embedding theorems, we may replace 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.

Formally speaking, this embedding was first introduced by Kuratowski,^[3] but a very close variation of this embedding appears already in the papers of Fréchet. Those papers make use of the embedding respectively to exhibit ${\displaystyle \ell ^{\infty }}$ as a "universal" separable metric space (it isn't itself separable, hence the scare quotes)^[4] and to construct a general metric on ${\displaystyle \mathbb {R} }$ by pulling back the metric on a simple Jordan curve in ${\displaystyle \ell ^{\infty }}$.^[5]

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