Mazur-Ulam theorem

Theorem.

Every isometry (http://planetmath.org/Isometry) between normed vector spaces over $\mathrm{R}$is an affine transformation.

Note that we consider isometries to be surjective by definition.The result is not in general true for non-surjective isometric mappings.

The result does not extend to normed vector spaces over $ℂ$,as can be seen from the fact that complex conjugation is an isometry $ℂ\to ℂ$but is not affine over $ℂ$.(But complex conjugation is clearly affine over $ℝ$,and in general any normed vector space over $ℂ$can be considered as a normed vector space over $ℝ$,to which the theorem can be applied.)

This theorem was first proved by Mazur and Ulam.[1]A simpler proof has been given by Jussi Väisälä.[2]