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In $\ell^1(\mathbb N)$, weak convergence implies strong convergence. Is there a classification of infinite-dimensional Banach spaces for which such a property holds true ?

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    $\begingroup$ If you are only asking about convergence of sequences, then perhaps Schur's property is what you are after? en.wikipedia.org/wiki/Schur%27s_property $\endgroup$
    – Yemon Choi
    Commented Apr 13, 2018 at 10:44
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    $\begingroup$ To be more precise, the spaces where weak convergence implies strong convergence are precisely the finitnie-dimensinonal ones. For $\ell_1$ the key point is that we consider only sequences, not more general nets. $\endgroup$ Commented Apr 13, 2018 at 11:40
  • $\begingroup$ (just to clarify: Jochen and Willie's comment address the same duplicate) $\endgroup$
    – YCor
    Commented Apr 13, 2018 at 16:16
  • $\begingroup$ Rosenthal's $\ell_!$ theorem implies that a space has the Schur property if and only if every bounded non totally bounded sequence has a subsequence that is equivalent to the unit vector basis of $\ell_1$. $\endgroup$ Commented Apr 13, 2018 at 22:43

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