Isometry - Linear Isometry

Given two normed vector spaces V and W, a linear isometry is a linear map f : V → W that preserves the norms:

for all v in V. Linear isometries are distance-preserving maps in the above sense. They are global isometries if and only if they are surjective.

By the Mazur-Ulam theorem, any isometry of normed vector spaces over R is affine.