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