Dunford–Pettis Properties and Spaces of Operators

Abstract

AbstractJ. Elton used an application of Ramsey theory to show that ifXis an infinite dimensional Banach space, thenc0embeds inX, ℓ1embeds inX, or there is a subspace ofXthat fails to have the Dunford–Pettis property. Bessaga and Pelczynski showed that ifc0embeds inX*, then ℓ∞embeds inX*. Emmanuele and John showed that ifc0embeds inK(X,Y), thenK(X,Y) is not complemented inL(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 spaceLw*(X*,Y) ofw*–wcontinuous operators is also studied.