Abstract
A (not necessarily linear) mapping
Φ from a Banach space
X to a Banach space
Y is said to be a 2
-local isometry if for any pair
x
,
y
of elements of
X, there is a surjective linear isometry
T
:
X
→
Y
such that
T
x
=
Φ
x
and
T
y
=
Φ
y
. We show that under certain conditions on locally compact Hausdorff spaces
Q,
K and a Banach space
E, every 2-local isometry on
C
0
(
Q
,
E
)
to
C
0
(
K
,
E
)
is linear and surjective. We also show that every 2-local isometry on
ℓ
p
is linear and surjective for
1
⩽
p
<
∞
,
p
≠
2
, but this fails for the Hilbert space
ℓ
2
.