Real Isomorphic Complex Banach Spaces Need not be Complex Isomorphic

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 X the Banach space with same elements and norm as X but scalar multiplication defined by z · x = z · x for z ∈ C,x ∈ X. If X is a space of complex sequences, X identifies with the space of coordinate-wise conjugate sequences and its norm is given by \| x \|X = \| x \|X, where x = ( z₁, z₂, … ) for x = (z₁,z₂, … ). Obviously X and X are isometric as real spaces. In this note, we prove that X and X may not be linearly isomorphic (in the complex sense). The method consists in constructing certain finite dimensional spaces by random techniques.