It is shown that complex Banach spaces may be isomorphic as real spaces and not as complex spaces. If
X
X
is a complex Banach space, denote
X
¯
\overline X
the Banach space with same elements and norm as
X
X
but scalar multiplication defined by
z
⋅
x
=
z
¯
⋅
x
z \cdot x = \bar z \cdot x
for
z
∈
C
,
x
∈
X
z \in {\mathbf {C}},x \in X
. If
X
X
is a space of complex sequences,
X
¯
\overline X
identifies with the space of coordinate-wise conjugate sequences and its norm is given by
‖
x
‖
X
¯
=
‖
x
¯
‖
X
{\left \| x \right \|_{\overline X }} = {\left \| {\bar x} \right \|_X}
, where
x
¯
=
(
z
¯
1
,
z
¯
2
,
…
)
\bar x = ({\bar z_1},{\bar z_2}, \ldots )
for
x
=
(
z
1
,
z
2
,
…
)
x = ({z_1},{z_2}, \ldots )
. Obviously
X
X
and
X
¯
\overline X
are isometric as real spaces. In this note, we prove that
X
X
and
X
¯
\overline X
may not be linearly isomorphic (in the complex sense). The method consists in constructing certain finite dimensional spaces by random techniques.