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]

References

1

S. Mazur and S. Ulam, Sur les transformations isométriques d'espaces vectoriels normés, C. R. Acad. Sci., Paris 194 (1932), 946-948.

2

Jussi Väisälä, A proof of the Mazur-Ulam theorem, Amer. Math. Mon. 110, #7 (2003), 633-635. (A preprint is available on Väisälä's website.)