Another proof of a result of N. J. Kalton, E. Saab and P. Saab on the Dieudonné property in C(K, E)

Abstract

Let K be a compact Hausdorff topological space and E be a Banach space not containing l1. Recently N. J. Kalton, E. Saab and P. Saab ([5]) obtained the results that under the above assumptions the usual space C(K, E) has the Dieudonné property; i.e. each weakly completely continuous operator on C(K, E) is weakly compact. They use topological results concerning multivalued mappings in their proof. In this short note we furnish a new and simpler proof of that result without using topological results but only well known theorems of Bourgain ([2]) and Talagrand ([8]) on weak compactness of sets of Bochner integrable functions; i.e. results in vector measure theory. At the end of the paper we present some applications of the result to Banach spaces of compact operators.

Continuous linear operators onC(K, X) and pointwise weakly precompact subsets ofC(K, X)

“…Assertions (c) and (d) of the foregoing theorem localize and, together with Lemma 2-4, unify the main results of the papers [5] and [13], respectively. See also the papers [3] and [10]. Assertion (6) localizes some parts of theorems 8 and 9 of [1].…”

Strongly Bounded Representing Measures and Convergence Theorems

“…For this reason, as well as for the convenience of the reader, we include a brief description of the bilinear integral we shall use and a proof of the convergence results we need. In the process, the technique and the results in [21] are extended.…”