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Suppose $X$ is a Banach space not isomorphic to a Hilbert space. Can we always find a subspace of $X$ that is not isomorphic to a quotient of $X$?

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Every separable Banach space is a quotient of $\ell_1$, so in particular every subspace of $\ell_1$ is a quotient of $\ell_1$.

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    $\begingroup$ And $c_0$ has the dual property--every quotient is isomorphic to a subspace. $\endgroup$ May 26, 2018 at 17:29
  • $\begingroup$ Is there an example, other than a Hilbert space, in which both are true? Every subspace isomorphic with a quotient, and every quotient isomorphic with a subspace? $\endgroup$
    – Markus
    May 26, 2018 at 19:45
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    $\begingroup$ @Markus, take $\ell_1\oplus C[0,1]$. $\endgroup$ May 26, 2018 at 19:50

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