Exercise 19, § 2

³ Let $\mu$ be a measure on $T$ such that the Banach space $L^1(\mu)$ is separable. Show that every continuous linear mapping of $L^1(\mu)$ into the strong dual $F'$ of an arbitrary Banach space $F$ may be obtained by passage to the quotient starting from a mapping of the form $g \mapsto \int gf d\mu$, where $f \in \mathscr{L}_{F'_s}^\infty$. (Reduce to the Dunford–Pettis theorem, using Exer. 19 of TVS, IV, §2.)