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, ''x''₀ 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

:$\Phi : X \rarr C_b(X)$

defined by

:$\Phi(x)(y) = d(x,y)-d(x_0,y) \quad\mbox\quad x,y\in X$

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

:$\Psi : X \rarr C_b(X)$

defined by

:$\Psi(x)(y) = d(x,y) \quad\mbox\quad x,y\in X$

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,
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 $\ell^\infty$ as a "universal" separable metric space (it isn't itself separable, hence the scare quotes) and to construct a general metric on $\mathbb$ by pulling back the metric on a simple Jordan curve in $\ell^\infty$.

See also

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

References

{{DEFAULTSORT:Kuratowski Embedding
Functional analysis
Metric geometry