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.