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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
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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.


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

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