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 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.