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
Canonical name	MazurUlamTheorem
Date of creation	2013-03-22 16:22:50
Last modified on	2013-03-22 16:22:50
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	12
Author	yark (2760)
Entry type	Theorem
Classification	msc 46B04