Approximate Isometries on Euclidean Spaces

This paper contains an exposition of two theorems on Banach spaces. Let X and Y be real Banach spaces and let f be a map from X to Y such that f(0)=0. The Mazur-Ulam Theorem says that if such a map is isometric (distance-preserving) and surjective, then it is linear. In general, it is necessary to assume that f is surjective. However, for a very large class of spaces this assumption is not necessary for the conclusion of the theorem. Let epsilon be a given positive number. An epsilon-isometry is a map from X to Y that preserves distances to within epsilon. It is known that if f is such a map between real Banach spaces, if f(0) = 0, and if f is surjective, then there exists a linear isometry g such that f and g are uniformly close. This was proved in 1945 by Hyers and Ulam for the special case of Hilbert spaces, and then extended to Banach spaces over the years by several authors. Here the assumption that f be surjective is necessary even when X and Y are Euclidean spaces.