*(English)*Zbl 1170.46009

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 $X$ is an infinite-dimensional Banach space, then ${c}_{0}$ embeds in $X$, ${\ell}_{1}$ embeds in $X$, or there is a subspace of $X$ 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 ${c}_{0}$ embeds in ${X}^{*}$, then ${\ell}_{\infty}$ embeds in ${X}^{*}$. G. Emmanuele and K. John [Czech. Math. J. 47, No. 1, 19–32 (1997; Zbl 0903.46006)] showed that if ${c}_{0}$ embeds in $K(X,Y)$, then $K(X,Y)$ is not complemented in $L(X,Y)$.*

*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 ${L}_{{w}^{*}}({X}^{*},Y)$ of ${w}^{*}$-$w$ continuous operators is also studied.*

##### MSC:

46B03 | Isomorphic theory (including renorming) of Banach spaces |

46B28 | Spaces of operators; tensor products; approximation properties |