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.