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 Cb(X) denotes the Banach space of all bounded continuous real-valued functions on X with the supremum norm, then the map
- [math]\displaystyle{ \Phi : X \rarr C_b(X) }[/math]
defined by
- [math]\displaystyle{ \Phi(x)(y) = d(x,y)-d(x_0,y) \quad\mbox{for all}\quad x,y\in X }[/math]
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
- [math]\displaystyle{ \Psi : X \rarr C_b(X) }[/math]
defined by
- [math]\displaystyle{ \Psi(x)(y) = d(x,y) \quad\mbox{for all}\quad x,y\in X }[/math]
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,[3] but a very close variation of this embedding appears already in the paper of Fréchet[4] where he first introduces the notion of metric space.
See also
- Tight span, an embedding of any metric space into an injective metric space defined similarly to the Kuratowski embedding
References
- ↑ Juha Heinonen (January 2003), Geometric embeddings of metric spaces, http://www.math.jyu.fi/research/reports/rep90.ps, retrieved 6 January 2009
- ↑ Karol Borsuk (1967), Theory of retracts, Warsaw. Theorem III.8.1
- ↑ Kuratowski, C. (1935) "Quelques problèmes concernant les espaces métriques non-separables" (Some problems concerning non-separable metric spaces), Fundamenta Mathematicae 25: pp. 534–545.
- ↑ Fréchet M. (1906) "Sur quelques points du calcul fonctionnel", Rendiconti del Circolo Matematico di Palermo 22: 1–74.
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