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 ?
$\begingroup$
$\endgroup$
4
-
4$\begingroup$ 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 ChoiCommented Apr 13, 2018 at 10:44
-
4$\begingroup$ 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. $\endgroup$– Tomasz KaniaCommented Apr 13, 2018 at 11:40
-
$\begingroup$ (just to clarify: Jochen and Willie's comment address the same duplicate) $\endgroup$– YCorCommented 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 $\ell_1$. $\endgroup$– Bill JohnsonCommented Apr 13, 2018 at 22:43
Add a comment
|