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I saw several examples for bounded sequences in $L^1$ which do not admit a weak convergent subsequence. But what about $l^1$ the sequence space?

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The canonical Schauder basis $(e_n)_{n=1}^\infty$ does not admit any weakly convergent subsequence.

By the way, observe that weakly convergent sequences in $\ell_1$, by the Schur property, are the same as norm convergent ones.

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