Authors’ summary: J. Elton [“Weakly Null Normalized Sequences in Banach Spaces” (Ph. D. dissertation, Yale University) (1979; per bibl.)] used an application of Ramsey theory to show that if is an infinite-dimensional Banach space, then embeds in , embeds in , or there is a subspace of that fails to have the Dunford–Pettis property. C. Bessaga and A. Pełczyński [Stud. Math. 17, 151–164 (1958; Zbl 0084.09805)] showed that if embeds in , then embeds in . G. Emmanuele and K. John [Czech. Math. J. 47, No. 1, 19–32 (1997; Zbl 0903.46006)] showed that if embeds in , then is not complemented in .
In the paper under review, classical results from Schauder basis theory are used in a study of Dunford–Pettis sets and strong Dunford–Pettis sets to extend each of the preceding theorems. The space of - continuous operators is also studied.