Mazur–Ulam theorem
In mathematics, the Mazur–Ulam theorem states that if formula_1 and formula_2 are normed spaces over R and the mapping is a surjective isometry, then formula_4 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 formula_5 and formula_6 in formula_1, and for any formula_8 in formula_9, denoting formula_10, one has that formula_11 is the unique element of formula_12, so, being formula_4 injective, formula_14  is the unique element of formula_15, namely formula_16. Therefore formula_4 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.