I wonder if $l_{\infty}$ is the quotient of $l_{1}(\aleph_{1})$.
If so, how to prove it?
2026-03-28 22:36:55.1774737415
$l_{\infty}$ is the quotient of $l_{1}(\aleph_{1})$?
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The statement is equivalent to the Continuum hypothesis. Indeed, taking a quotient cannot increase the density character, and the density character of $l_\infty$ is $c$. This gives one implication.
If $\aleph_1=c$, then pick a dense subset of cardinality $\aleph_1$ in the unit ball of $l_\infty$. Map the standard basis vectors of $l_1(\aleph_1)$ bijectively to the points of this subset, and extend by linearity.