Consider $\ell_1$ as a dual of $c_0$. It is well known that there must exist sequences of elements in $\ell_1$ which converge in the weak*-topology (wrt $c_0$) but not weakly. Can one give an example (likely using ultrafilters / Banach limits) of such a sequence?
2026-05-04 09:01:29.1777885289
Sequences in $\ell_1$ which converge in the weak*-topology (wrt $c_0$) but not weakly
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The sequence of natural unit vectors: $e_n$ has $1$ in the $n$th place, zero elsewhere. Show it converges to zero weak* but not weakly. Knowing the dual of $\ell_1$ should help.