Corollary 2 (Dunford–Pettis theorem)

— Let $F$ be a Hausdorff locally convex space containing a countable dense subset. For every function $f \in \mathscr{L}_{F'}^\infty$ and every function $g \in \mathscr{L}^1$, let $w_f(\tilde{g}) = \int gf\, d\mu \in F'$ (Lemma 4). The mapping $f \mapsto w_f$ defines, by passage to the quotient, a linear bijection of $L_{F_s}^\infty$ onto the space $\mathscr{R}(L^1, F')$ of linear mappings $u$ of $L^1$ into $F'$ such that the image under $u$ of the unit ball of $L^1$ is an equicontinuous subset of $F'$. If $F$ is a normed space (in which case $\mathscr{R}(L^1, F')$ is the space of continuous linear mappings of $L^1$ into the strong dual of $F$), the bijection $f \mapsto w_f$ is an isometry.

Taking into account the fact that $L^\infty$ is the dual of $L^1$, this follows from the preceding corollary and from the Appendix, No. 1, Prop. 1 and its corollary.