J Austral Math Soc Ser A 47 pp22--31, 1989.

Archimedes Integrals and Conuclear Spaces

B. Jeffries and S. Okada

Abstract

The lack of completeness with respect to the semivariation norm, of the space of Banach space valued functions, Pettis integrable with respect to a measure m, often impedes the direct extensions of results involving integral representations, true in the finite-dimensional setting, to the general vector space setting. It is shown here that the space of functions with values in a space Y, m-Archimedes integrable in a Banach space X embedded into Y, is complete with respect to convergence in semivariation, provided the embedding from X into Y is completely summing. The result is applied to the case when Y is a conuclear space, in particular, when X is a function space continuously included in a space of distributions.

1980 AMS Subject Classification (1985 Revision): primary 38B05, 46G10; secondary 47B10, 46A12

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Authors

B. Jefferies

Department of Mathematics, The University of Wollongong, Wollongong, N.S.W. 2500, Australia.

S. Okada

School of Mathematics and Physical Sciences, Murdoch University, W.A. 6150, Australia.