Mazur–Ulam theorem
In mathematics, the Mazur–Ulam theorem states that if and are normed spaces over R and the mapping
is a surjective isometry, then is affine.
It is named after Stanisław Mazur and Stanisław Ulam in response to an issue 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 and in, and for any in, denoting, one has that is the unique element of, so, being injective, is the unique element of, namely. Therefore 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.
