Mazur–Ulam theorem - Misplaced Pages

Surjective isometries are affine mappings

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

f\colon V\to W

is a surjective isometry, then $f$ 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 $u$ and $v$ in $V$ , and for any $t$ in $[0, 1]$ , write $r=\|u-v\|_{V}=\|f(u)-f(v)\|_{W}$ and denote the closed ball of radius R around v by ${\bar {B}}(v,R)$ . Then $tu+(1-t)v$ is the unique element of ${\bar {B}}(v,tr)\cap {\bar {B}}(u,(1-t)r)$ , so, since $f$ is injective, $f(tu+(1-t)v)$ is the unique element of $f{\bigl (}{\bar {B}}(v,tr)\cap {\bar {B}}(u,(1-t)r{\bigr )}=f{\bigl (}{\bar {B}}(v,tr){\bigr )}\cap f{\bigl (}{\bar {B}}(u,(1-t)r{\bigr )}={\bar {B}}{\bigl (}f(v),tr{\bigr )}\cap {\bar {B}}{\bigl (}f(u),(1-t)r{\bigr )},$ and therefore is equal to $tf(u)+(1-t)f(v)$ . Therefore $f$ 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 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.

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