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I know the rather standard fact in Banach space theory that every separable Banach space is a quotient of $\ell_1$. Is it true that every (possibly non-separable) Banach space is a quotient of some $L_1$ space?

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Let $D $ be a dense subset of $S_X $. Define $T\colon \ell^1(D)\to X$ by $$ T((a_d)_{d\in D}) := \sum_d a_d d $$ Then $T $ maps the open unit ball of $\ell^1 (D) $ onto that of $X $, hence is a quotient map.

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