Let $X$ be a set and $\operatorname{Bou}(X,\mathbb{R})$ be the set of bounded functions $f:X\to\mathbb{R}$ with norm $||f||=\operatorname{sup}\{|f(x)|:\;x\in X\}$ . Kuratowski's embedding theorem states that every metric space $(X,d)$ can be embedded isometrically into the Banach space $E=\operatorname{Bou}(X,\,\mathbb{R})$ .

Proof. One can assume that $X\ne \emptyset$ . Fix a point $a_0\in X$ and for every $a\in X$ define a function $f_a:X\to\mathbb{R}$ by$$f_a(x)=d(x,a)-d(x,a_0)$$ Then $|f_a(x)|\leq d(a,a_0)$ for every $x\in X$ so $f_a$ is bounded. By setting $\varphi :X\to E$ , $\varphi (a)=f_a$ , we have the mapping $\varphi : X\to E$ . It requires to prove that $\varphi$ is an isometry.

Let $a,\,b\in X$ . As $x\in X$ we have that$$|f_a(x)-f_b(x)|=|d(x,a)-d(x,b)|\leq d(a,b)$$ Therefore $||f_a-f_b||\leq d(a,b)$ . On the other hand$$|f_a(a)-f_b(a)|=|d(a,a)-d(a,a_0)-d(a,b)+d(a,a_0)|=d(a,b)$$ Therefore $||\varphi (a)-\varphi (b)||=||f_a-f_b||=d(a,b)$ .

Bibliography

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J. V¨AISÄLÄ: Topologia II. 2nd corrected issue, Limes ry., Helsinki, Finland (2005), ISBN 951-745-209-8