Mazur-Ulam theorem


Every isometry ( between normed vector spacesPlanetmathPlanetmath over R is an affine transformationMathworldPlanetmathPlanetmath.

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]


  • 1 S. Mazur and S. Ulam, Sur les transformationsMathworldPlanetmath 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 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