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.

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