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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 $\sigma(L^1,L^\infty)$.

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    $\begingroup$ The book of Albiac and Kalton. $\endgroup$ – Bill Johnson Aug 29 '14 at 17:06
  • $\begingroup$ You can also find it in the article by J. Diestel, Uniform integrability: an introduction. I believe you can freely access this on the web. $\endgroup$ – user19038 Aug 29 '14 at 22:40
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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.

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