Kuratowski’s embedding theorem

Let X be a set and Bou(X,) be the set of bounded functions f:X with norm  ||f||=sup{|f(x)|:xX}. Kuratowski’s embeddingMathworldPlanetmathPlanetmath theorem states that every metric space (X,d) can be embedded isometrically into the Banach spaceMathworldPlanetmathE=Bou(X,).

Proof.  One can assume that X. Fix a point a0X and for every aX define a function fa:X by


Then |fa(x)|d(a,a0) for every xX so fa is boundedPlanetmathPlanetmathPlanetmathPlanetmath. By setting  φ:XE,  φ(a)=fa, we have the mapping φ:XE. It requires to prove that φ is an isometry.

Let a,bX. As xX we have that


Therefore ||fa-fb||d(a,b). On the other hand


Therefore ||φ(a)-φ(b)||=||fa-fb||=d(a,b).


  • 1 J. VÃÂisÃÂlÃÂ: Topologia II.  2nd corrected issue, Limes ry., Helsinki, Finland (2005), ISBN 951-745-209-8
Title Kuratowski’s embedding theorem
Canonical name KuratowskisEmbeddingTheorem
Date of creation 2013-03-22 18:24:48
Last modified on 2013-03-22 18:24:48
Owner puuhikki (9774)
Last modified by puuhikki (9774)
Numerical id 10
Author puuhikki (9774)
Entry type Theorem
Classification msc 54-00