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 $\ell_1$.
answered May 26, 2018 at 17:25
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3$\begingroup$ And $c_0$ has the dual property--every quotient is isomorphic to a subspace. $\endgroup$ Commented May 26, 2018 at 17:29
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$\begingroup$ 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? $\endgroup$– MarkusCommented May 26, 2018 at 19:45
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1$\begingroup$ @Markus, take $\ell_1\oplus C[0,1]$. $\endgroup$ Commented May 26, 2018 at 19:50
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$\begingroup$ If $p\ne q$, can a quotient of $\ell_p$ be isomorphic to $\ell_q$ ? Any reference for that? $\endgroup$– AnupamCommented Jul 13 at 15:52
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1$\begingroup$ @Anupam, no because every subspace of $\ell_q$ has a further subspace isomorphic to $\ell_q$. Please see the Albiac--Kalton book. $\endgroup$ Commented Jul 13 at 20:36