A Proof of the Mazur-Ulam Theorem

for all a, b ∈ E and 0 ≤ t ≤ 1. Equivalently, f is affine if the map T :E → F , defined by Tx = fx− f(0), is linear. An isometry need not be affine. To see this, let E be the real line R, let F be the plane with the norm ‖x‖ = max(|x1|, |x2|), and let φ:R → R be any function such that |φ(s)−φ(t)| ≤ |s−t| for all s, t ∈ R, for example, φ(t) = |t| or φ(t) = sin t. Setting f(s) = (s, φ(s)) we get an isometry f :E → F , which is usually not affine. An isometry f :E → F is affine if