Quotients of $L_1$

Question

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?

Answer 1

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.