Theorem   Every isometry between normed vector spaces over 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 , as can be seen from the fact that complex conjugation is an isometry but is not affine over . (But complex conjugation is clearly affine over , and in general any normed vector space over can be considered as a normed vector space over , 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.)