Mazur-Ulam theorem
Encyclopedia
In mathematics, the Mazur–Ulam theorem states that if 
and 
are normed spaces over R and the map
ping
is a surjective isometry
, then
is affine
.
and are normed spaces over R and the mapMap (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
ping
is a surjective isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
, then
is affineAffine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
.