# Mazur-Ulam theorem

###### Theorem.

Every isometry (http://planetmath.org/Isometry) between normed vector spaces over $\mathbb{R}$ 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 $\mathbb{C}$, as can be seen from the fact that complex conjugation is an isometry $\mathbb{C}\to\mathbb{C}$ but is not affine over $\mathbb{C}$. (But complex conjugation is clearly affine over $\mathbb{R}$, and in general any normed vector space over $\mathbb{C}$ can be considered as a normed vector space over $\mathbb{R}$, 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 http://www.helsinki.fi/%7Ejvaisala/mazurulam.pdfavailable on Väisälä’s website.)
Title Mazur-Ulam theorem MazurUlamTheorem 2013-03-22 16:22:50 2013-03-22 16:22:50 yark (2760) yark (2760) 12 yark (2760) Theorem msc 46B04