On nonlinear perturbations of isometries (Q1906493)

Let \(X\) and \(Y\) be real Banach spaces. A mapping \(f: X\to Y\) is called an \(\varepsilon\)-isometry if \(|\;|f(x)- f(y)|- |x-y |\;|\leq \varepsilon\) holds for all \(x,y\in X\). The problem of uniform approximation of such maps by true isometries was raised by Hyers and Ulam in 1945. It has been already known that for every surjective \(\varepsilon\)-isometry \(f\) there exists an isometry \(U\) of \(X\) onto \(Y\) such that \(|f(x)- U(x)|\leq 5\varepsilon\) for all \(x\in X\). We prove that this upper bound \(5\varepsilon\) can be replaced by \(2\varepsilon\) and that the new inequality is sharp, thus giving the final solution to the problem of Hyers and Ulam. We also solve the problem of how far a surjective mapping preserving distance one in both directions can be from an affine isometry.