Stability of Isometries on Banach Spaces

Let X and Y be Banach spaces. A mapping f:X → Y is called an ε-isometry if | \| f(x₀) - f(x₁) \| - \| x₀ - x₁ \|| ε for all x₀,x₁ ∈ X. It is shown that there exist constants A and B such that if f:X → Y is a surjective ε-isometry, then \| f((x₀ + x₁)/2) - (f(x₀) + f(x₁))/2 \| A(ε \| x₀ - x₁ \|)1/2 + Bε for all x₀,x₁ ∈ X. This, together with a result of Peter M. Gruber, is used to show that if f:X → Y is a surjective ε-isometry, then there exists a surjective isometry I:X → Y for which \| f(x) - I(x) \| 5ε, thus answering a question of Hyers and Ulam about the stability of isometries on Banach spaces.