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

Published online by Cambridge University Press:  18 May 2009

G. Emmanuele
G. Emmanuele
Affiliation:
Department of Mathematics, University of Cataina, Viale A. Doria 6, Catania 95125, Italy
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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.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 31 , Issue 2 , May 1989 , pp. 137 - 140
Copyright
Copyright © Glasgow Mathematical Journal Trust 1989

References

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