 Fréchet embedding

Fix a point $x_0$ in a metric space $X$. The Fréchet embedding sends $x$ to the bounded function $a\mapsto d(x,a)-d(x_0,a)$ in ${\ell }^{\infty }(X)$. The triangle inequality bounds each coordinate by $d(x,x_0)$ and gives $\Vert f_x-f_y{\Vert }_{\infty }\le d(x,y)$. Taking $a=x$ gives equality, so the map is an isometric embedding. For a separable metric space a countable dense set of anchors suffices. Finite collections of distances to selected subsets instead give one-Lipschitz coordinates that need not be an exact Fréchet embedding.