An elementary proof of an equivalence theorem relevant in the theory of optimization

The authors give an elementary proof of an equivalence theorem of analysis which is often used in optimization theory. The theorem asserts that certain conditions are equivalent to weak convergence inL ₁. One is the Dunford-Pettis condition concerning absolute integrability. Two others are expressed in terms of Nagumo functions, and can be thought of as growth properties. The original proofs of the various parts of the theorem are scattered in different and specialized mathematical publications. The authors feel it useful to present here a straightforward proof of the various parts in terms of standard Lebesgue integration theory.