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 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 and in , and for any in , write and denote the closed ball of radius around by . Then is the unique element of , so, since is injective, is the unique element of and therefore is equal to . 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. See also Aleksandrov–Rassias problem References Normed spaces Theorems in functional analysis