Stability of isometries on Banach spaces

Source abstract

Let X X and Y Y be Banach spaces. A mapping f : X → Y f:X \to Y is called an ε \varepsilon -isometry if | ‖ f ( x 0 ) − f ( x 1 ) ‖ − ‖ x 0 − x 1 ‖ | ⩽ ε |\left \| {f({x_0}) - f({x_1})} \right \| - \left \| {{x_0} - {x_1}} \right \|| \leqslant \varepsilon for all x 0 , x 1 ∈ X {x_0},{x_1} \in X . It is shown that there exist constants A A and B B such that if f : X → Y f:X \to Y is a surjective ε \varepsilon -isometry, then ‖ f ( ( x 0 + x 1 ) / 2 ) − ( f ( x 0 ) + f ( x 1 ) ) / 2 ‖ ⩽ A ( ε ‖ x 0 − x 1 ‖ ) 1 / 2 + B ε \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 ∈ X {x_0},{x_1} \in X . This, together with a result of Peter M. Gruber, is used to show that if f : X → Y f:X \to Y is a surjective ε \varepsilon -isometry, then there exists a surjective isometry I : X → Y I:X \to Y for which ‖ f ( x ) − I ( x ) ‖ ⩽ 5 ε \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.