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.

History

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]

See also

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

References

1. Juha Heinonen (January 2003), Geometric embeddings of metric spaces, retrieved 6 January 2009
2. Karol Borsuk (1967), Theory of retracts, Warsaw{{citation}}: CS1 maint: location missing publisher (link). Theorem III.8.1
3. Kuratowski, C. (1935) "Quelques problèmes concernant les espaces métriques non-separables" (Some problems concerning non-separable metric spaces), Fundamenta Mathematicae 25: pp. 534–545.
4. Fréchet, Maurice (1 June 1910). "Les dimensions d'un ensemble abstrait". Mathematische Annalen. 68 (2): 161–163. doi:10.1007/BF01474158. ISSN 0025-5831. Retrieved 17 March 2024.
5. Frechet, Maurice (1925). "L'Expression la Plus Generale de la "Distance" Sur Une Droite". American Journal of Mathematics. 47 (1): 4–6. doi:10.2307/2370698. ISSN 0002-9327. Retrieved 17 March 2024.