Let K be a compact Hausdorff space and E, F Banach spaces. We denote by C(K, E) the Banach space of all continuous. E-valued functions defined on K, with the supremum norm. It is well known ([6], [7]) that every operator (= bounded linear operator) T from C(K, E) to F has a finitely additive representing measure m of bounded semi-variation, defined on the Borel σ-field Σ of K and with values in L(E, F″) (the space of all operators from E into the second dual of F), in such a way thatwhere the integral is considered in Dinculeanu's sense.
A Banach space E is said to have the Dunford-Pettis property if for every pair of weakly null sequences (x,) c E and (x') c E' one has lim(x,, x') 0. Following Diestel [1] we shall say that a Banach space E is hereditarily
Let K tie a compact Hausdorff space and l e t E be a Banach So he answered negatively a question which was posed some years ago.We prove in this paper that for a large class of compacts K (the scattered compacts), C(K, E) has either the Dunford-Pettis property, or the reciprocal Dunford-Pettis property, or the Dieudonne property, or property V if and only if E has the same property.Also some properties of the operators defined on C(K, E) are studied.
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