The Urysohn universal metric space is homeomorphic to a Hilbert space

The Urysohn universal metric space U is characterized up to isometry by the following properties: (1) U is complete and separable; (2) U contains an isometric copy of every separable metric space; (3) every isometry between two finite subsets of U can be extended to an isometry of U onto itself. We show that U is homeomorphic to the Hilbert space l2 (or to the countable power of the real line).