Mazur–Ulam theorem

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 ,1/math>, write
$r=\, u-v\, _V=\, f(u)-f(v)\, _W$
and denote the closed ball of radius around 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.

See also

Aleksandrov–Rassias problem

References

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Normed spaces
Theorems in functional analysis

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