Suppose that $X\subseteq Y \subseteq Z$ are Banach spaces such that $X$ is complemented in $Y$ and the duals $X^*, Z^*$ are isomorphic. Must the dual $Y^*$ be isomorphic to $X^*$?
Squeeze theorem for duals of Banach spaces
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functional-analysis
banach-spaces
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0I don't think so. Here's how I would go about constructing an example. Start with $E$ such that $E \times E \cong E$. Choose $F \lt E$ non-complemented. Put $X = E$, $Y = E \times F$ and $Z = E \times E$. – 2011-09-02
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0Thank you. However in the particular case I am interested in, $Z$ is not isomorphic to its square. – 2011-09-02