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 $$[0,1]$$, 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.