Stability of isometries on Banach spaces

Abstract:

Let $X$ and $Y$ be Banach spaces. A mapping $f:X \to Y$ is called an $\varepsilon$-isometry if $|\left \| {f({x_0}) - f({x_1})} \right \| - \left \| {{x_0} - {x_1}} \right \|| \leqslant \varepsilon$ for all ${x_0},{x_1} \in X$. It is shown that there exist constants $A$ and $B$ such that if $f:X \to Y$ is a surjective $\varepsilon$-isometry, then $\left \| {f(({x_0} + {x_1})/2) - (f({x_0}) + f({x_1}))/2} \right \| \leqslant A{(\varepsilon \left \| {{x_0} - {x_1}} \right \|)^{1/2}} + B\varepsilon$ for all ${x_0},{x_1} \in X$. This, together with a result of Peter M. Gruber, is used to show that if $f:X \to Y$ is a surjective $\varepsilon$-isometry, then there exists a surjective isometry $I:X \to Y$ for which $\left \| {f(x) - I(x)} \right \| \leqslant 5\varepsilon$, thus answering a question of Hyers and Ulam about the stability of isometries on Banach spaces.