Mazur–Ulam theorem

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. For strictly convex spaces the result is true, and easy, even for isometries which are not necessarily surjective. In this case, for any ${\displaystyle u}$ and ${\displaystyle v}$ in ${\displaystyle V}$, and for any ${\displaystyle t}$ in ${\displaystyle [0,1]}$, denoting ${\displaystyle r:=\|u-v\|_{V}=\|f(u)-f(v)\|_{W}}$, one has that ${\displaystyle tu+(1-t)v}$ is the unique element of ${\displaystyle {\bar {B}}(v,tr)\cap {\bar {B}}(u,(1-t)r)}$, so, being ${\displaystyle f}$ injective, ${\displaystyle f(tu+(1-t)v)}$ is the unique element of ${\displaystyle f{\big (}{\bar {B}}(v,tr)\cap {\bar {B}}(u,(1-t)r{\big )}=f{\big (}{\bar {B}}(v,tr){\big )}\cap f{\big (}{\bar {B}}(u,(1-t)r{\big )}={\bar {B}}{\big (}f(v),tr{\big )}\cap {\bar {B}}{\big (}f(u),(1-t)r{\big )}}$, namely ${\displaystyle tf(u)+(1-t)f(v)}$. Therefore ${\displaystyle f}$ 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.

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.
• Jussi Väisälä (2003). "A Proof of the Mazur-Ulam Theorem". The American Mathematical Monthly. 110 (7): 633–635.

External links

• Nica, Bogdan (2013). "A proof of the Mazur–Ulam theorem assuming f is bijective". arXiv:1306.2380. Cite journal requires |journal= (help)
• Väisälä, Jussi. "A proof of the Mazur–Ulam theorem" (PDF). Archived from the original (PDF) on 16 May 2018.