Linear Functionals on Certain Spaces of Abstractly-Valued Functions

Abstract

This paper is concerned with the study of certain classes of abstractly-valued functions which arise naturally in connection with the theory of integration of functions with values in a Banach space. These classes correspond to the usual classes C, LP, M, etc. which are well known in analysis. The subject matter of the paper is divided into seven sections as follows: ?2 contains most of the technical apparatus, which consists of the Lebesgue integral of abstract functions as defined by Bochner,2 and several types of integrals based on the notion of the Riemann-Stieltjes integral. Since, in the case of abstractly-valued functions, there may be a distinction between an indefinite integral and an absolutely continuous function,3 we have found it important to introduce the class of functions of finite p-variation, corresponding to the classical criterion for a function's being the indefinite integral of a function of class LV. For certain Banach spaces it is known that the distinction mentioned above does not arise.4 Such a space is said to satisfy condition (D) [for the precise statement see ?21. We have obtained some information about the implications of this condition and its relation to other properties of the space [see especially ??4, 5 and 7]. ?3 contains the determination of the most general linear functionals on the various function classes under discussion. ?4 is devoted to a discussion of weak convergence and its relation to condition (D). In ?5 the interrelations of condition (D), weak completeness, and reflexiveness of Banach spaces is taken up. In ?6 we have results on the Parseval relation and properties of the Fourier coefficients of abstract functions. Counter examples to show the divergence from classical theory are given. In ?7 we return to the question of reflexivity and condition (D), obtaining a criterion for spaces which satisfy this condition. Finally, in ?8 it is shown how by the methods of Fourier series it is possible to characterize