Dunford–Pettis property

Summary

A Banach space has the Dunford–Pettis property when the following holds. Pairing a weakly null sequence of vectors with a weakly null sequence of functionals always gives a null sequence of numbers. The spaces $L^1(\mu)$ and $C(K)$ have it, by results of Dunford and Pettis and of Grothendieck. No infinite-dimensional reflexive space has it. The property is a tool to show that some operators on these classical spaces are compact.