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 $\u2102$, as can be seen from the fact that complex conjugation is an isometry $\u2102\to \u2102$ but is not affine over $\u2102$. (But complex conjugation is clearly affine over $\mathbb{R}$, and in general any normed vector space over $\u2102$ 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 |