# Mazur–Ulam theorem

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In mathematics, the Mazur–Ulam theorem states that if ${\displaystyle V}$ and ${\displaystyle W}$ are normed spaces over R and the mapping

${\displaystyle f\colon V\to W}$

is a surjective isometry, then ${\displaystyle f}$ is affine.

It is named after Stanisław Mazur and Stanisław Ulam in response to an issue raised by Stefan Banach.

## References

• 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 transformationes isométriques d'espaces vectoriels normés". C. R. Acad. Sci. Paris. 194: 946–948.