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Mazur–Ulam theorem

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In mathematics, the Mazur–Ulam theorem states that if and are normed spaces over R and the mapping

is a surjective isometry, then is affine. It was proved by Stanisław Mazur and Stanisław Ulam in response to a question raised by Stefan Banach.

For strictly convex spaces the result is true, and easy, even for isometries which are not necessarily surjective. In this case, for any and in , and for any in , write and denote the closed ball of radius R around v by . Then is the unique element of , so, since is injective, is the unique element of and therefore is equal to . Therefore is an affine map. This argument fails in the general case, because in a normed space which is not strictly convex two tangent balls may meet in some flat convex region of their boundary, not just a single point.

See also

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Aleksandrov–Rassias problem

References

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  • Richard J. Fleming; James E. Jamison (2003). Isometries on Banach Spaces: Function Spaces. CRC Press. p. 6. ISBN 1-58488-040-6.
  • Stanisław Mazur; Stanisław Ulam (1932). "Sur les transformations isométriques d'espaces vectoriels normés". C. R. Acad. Sci. Paris. 194: 946–948.
  • Nica, Bogdan (2012). "The Mazur–Ulam theorem". Expositiones Mathematicae. 30 (4): 397–398. arXiv:1306.2380. doi:10.1016/j.exmath.2012.08.010.
  • Jussi Väisälä (2003). "A Proof of the Mazur–Ulam Theorem". The American Mathematical Monthly. 110 (7): 633–635. doi:10.1080/00029890.2003.11920004. JSTOR 3647749. S2CID 43171421.