# Quotients of $L_1$

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?

## 1 Answer

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.