Stability of isometries

Source abstract

A map T : E → F T:E \to F ( E , F Banach spaces) is called an ε \varepsilon -isometry if | ‖ T ( x ) − T ( y ) ‖ − ‖ x − y ‖ | ⩽ ε \left |\,{\left \| {T(x)-T(y)} \right \|-\left \|{x -y}\right \|}\,\right |\, \leqslant \varepsilon whenever x , y ∈ E x,\,y \in E . Hyers and Ulam raised the problem whether there exists a constant κ \kappa , depending only on E , F , such that for every surjective ε \varepsilon -isometry T : E → F T:E \to F there exists an isometry I : E → F I:E \to F with ‖ T ( x ) − I ( x ) ‖ ⩽ κ ε {\left \| {T(x) - I(x)} \right \|}\leqslant \kappa \varepsilon for every x ∈ E x \in E . It is shown that, whenever this problem has a solution for E , F , one can assume κ ⩽ 5 \kappa \leqslant 5 . In particular this holds true in the finite dimensional case.