Real isomorphic complex Banach spaces need not be complex isomorphic

Abstract:

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