TITLE: Proof of the Dunford-Pettis theorem
QUESTION [6 upvotes]: I would like to know where to find a complete proof of the Dunford-Pettis theorem:
A sequence $(f_n)_{n\geq 0} \subset L^1$ is uniformly integrable if and only if it is relatively compact for the weak topology $\sigma(L^1,L^\infty)$.

REPLY [5 votes]: This is surely in many places, but a reasonably complete proof is given in Section III.2, Theorem 15 of
J. Diestel, J. J. Uhl, Vector measures. American Mathematical Society 1977.