Exercise 24, § 2

Let $\mu$ be a positive measure on T, u a linear mapping from $L^1(\mu)$ into a Banach space F, transforming the unit ball of $L^1(\mu)$ into a relatively compact subset of F for $\sigma(F, F')$. Now prove that there exists a $\mu$-measurable mapping f from T into F, such that $|f(t)| \leq \|u\|$ for all $t \in T$ and that one has

for every function $g \in \overline{L^1(\mu)}$ (Dunford-Pettis-Phillips theorem). (First reduce to the case where T is compact by using Proposition 4 of Chapter V, § 1, No. 4. The unit ball B of $L^\infty(\mu) \subset L^1(\mu)$ is then relatively compact for $\sigma(L^1, L^\infty)$ (Chapter V, § 5, Exercise 17); using Exercise 23, show that the closed vector subspace of F generated by $u(L^1)$ is of countable type. One can therefore reduce to the case where F is of countable type and where $\overline{u(L^1)} = F$. The closure A in F of the image by u of the unit ball of $L^1$ is then compact for $\sigma(F, F')$ (Chapter IV, § 4, Exercise 19); show that it is metrizable for this topology (Topological Vector Spaces, Chapter IV, § 2, No. 2, corollary of Proposition 3). Observe that A then identifies with the unit ball of the dual of the normed space G obtained by endowing $F'$ with the norm equal to the gauge of $A^0$. Show that G is of countable type, and reduce in this way to applying the Dunford-Pettis theorem (corollary 2 of Theorem 1)).