Exercise 19, § 2

Let $\mu$ be a measure on $T$ such that the Banach space $L^1(\mu)$ is of countable type. Now prove that every continuous linear mapping from $L^1(\mu)$ into the strong dual $F'$ of an arbitrary Banach space $F$ is obtained by passing to the quotient from a mapping of the form $g \to \int g f d\mu$, where $f \in \mathscr{L}_{F_s'}^\infty$. (Reduce to the Dunford-Pettis theorem by using Exercise 7 of Topological Vector Spaces, Chapter IV, § 5.)