Mazur–Ulam theorem

In mathematics, the Mazur–Ulam theorem states that if

are normed spaces over R and the mapping is a surjective isometry, then

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

, write

and denote the closed ball of radius R around v by

is the unique element of

{\displaystyle {\bar {B}}(v,tr)\cap {\bar {B}}(u,(1-t)r)}

is injective,

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)}

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.

Aleksandrov–Rassias problem