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I want to show that if a normed linear space has Schur's property (every weakly convergent sequence converges), then it is not guaranteed that every quotient space $X/Y$ will have that property, where $Y$ is a closed subspace of $X$.

I started with $X=\ell^1$. Here $X$ has Schur's property. Is it possible to find a closed subspace $Y$ of $X$ such that $X/Y\cong \ell^p (1<p<\infty)$ or $X/Y\cong c_0$? I got stuck at this point. Please help!

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    $\begingroup$ Yes. In fact any separable Banach space is a quotient of $\ell_1$. See this. $\endgroup$ – David Mitra May 16 at 5:34

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