It is shown that the double sequence
{
λ
m
n
}
\{ {\lambda _{mn}}\}
with
λ
m
n
=
1
{\lambda _{mn}} = 1
if
n
⩽
m
n \leqslant m
and
0
0
otherwise is an
L
p
{L^p}
multiplier for the Walsh system in two dimensions only if
p
=
2
p = 2
. This result is then used to show that the one-dimensional trigonometric system and the Walsh system are nonequivalent bases of the Banach space
L
p
[
0
,
1
]
{L^p}[0,\;1]
, and hence have different
L
p
{L^p}
multipliers,
1
>
p
>
∞
,
p
≠
2
1 > p > \infty ,\;p \ne 2
.