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 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