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
Φ : X → C b ( X ) {\displaystyle \Phi :X\rightarrow C_{b}(X)}
defined by
Φ ( x ) ( y ) = d ( x , y ) − d ( x 0 , y ) for all x , y ∈ X {\displaystyle \Phi (x)(y)=d(x,y)-d(x_{0},y)\quad {\mbox{for all}}\quad x,y\in X}
is an isometry. 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. (N.B. the image of this embedding is closed in the convex subset, not necessarily in the Banach space.) Here we use the isometry
Ψ : X → C b ( X ) {\displaystyle \Psi :X\rightarrow C_{b}(X)}
defined by
Ψ ( x ) ( y ) = d ( x , y ) for all x , y ∈ X {\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 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, 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 ℓ ∞ {\displaystyle \ell ^{\infty }} as a "universal" separable metric space (it isn't itself separable, hence the scare quotes) and to construct a general metric on R {\displaystyle \mathbb {R} } by pulling back the metric on a simple Jordan curve in ℓ ∞ {\displaystyle \ell ^{\infty }} .
See also Tight span, an embedding of any metric space into an injective metric space defined similarly to the Kuratowski embedding
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