Mazur–Ulam theorem

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

$f:V\to W$

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 $$r=\Vert u-v{\Vert }_V=\Vert f(u)-f(v){\Vert }_W$$ and denote the closed ball of radius R around v by $\overline{B}(v,R)$. Then $tu+(1-t)v$ is the unique element of $\overline{B}(v,tr)\cap \overline{B}(u,(1-t)r)$, so, since $f$ is injective, $f(tu+(1-t)v)$ is the unique element of $$f(\overline{B}(v,tr)\cap \overline{B}(u,(1-t)r)=f(\overline{B}(v,tr))\cap f(\overline{B}(u,(1-t)r)=\overline{B}(f(v),tr)\cap \overline{B}(f(u),(1-t)r),$$ 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.

See also

Aleksandrov–Rassias problem

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. https://gallica.bnf.fr/ark:/12148/bpt6k31473/f950.item. 
• Nica, Bogdan (2012). "The Mazur–Ulam theorem". Expositiones Mathematicae 30 (4): 397–398. 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.