Dunford-Pettis Properties and Spaces of Operators

J. Elton used an application of Ramsey theory to show that if X is an infinite dimensional Banach space, then c(0) embeds in X, l(1) embeds in X, or there is a subspace of X that fails to have the Dunford-Pettis property. Bessaga and Pelczynski showed that if c(0) embeds in X*, then l(infinity) embeds in X*. Emmanuele and John showed that if co embeds in K(X, Y), then K(X, Y) is not complemented in L(X, Y). 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.