nLab
Mazur-Ulam theorem

The Mazur–Ulam theorem (after Stanislaw Mazur? and Stanislaw Ulam?) states that if EE and FF are normed vector spaces over \mathbb{R} and f:EFf\colon E \to F is an isometric isomorphism, then ff is an affine map. For a proof, see either of the references below by Akhil Mathew and Jussi Väisälä. It is based on the idea that a midpoint between two points can be characterized in purely norm-theoretic terms. This is more or less trivial if the norm is strictly convex (compare the discussion at isometry); in the general case, one cleverly exploits properties of the reflection map x2zxx \mapsto 2 z - x where zz is a midpoint in question, to arrive at the conclusion.

References

A. Mathew, Post on his blog Climbing Mount Bourbaki, November 19, 2009 (link)

J. Väisälä, A proof of the Mazur–Ulam theorem (link)

Revised on November 12, 2011 08:40:16 by Toby Bartels (139.55.238.24)