Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Real isomorphic complex Banach spaces need not be complex isomorphic


Author: J. Bourgain
Journal: Proc. Amer. Math. Soc. 96 (1986), 221-226
MSC: Primary 46B20
DOI: https://doi.org/10.1090/S0002-9939-1986-0818448-2
MathSciNet review: 818448
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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\Vert x \right\Vert _{\overline X }} = {\left\Vert {\bar x} \right\Vert _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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46B20

Retrieve articles in all journals with MSC: 46B20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0818448-2
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society