A Riesz representation theorem

Abstract

Introduction. This paper deals with a Riesz representation theorem in the following setting: Let H be a compact Hausdorff space, let E and be locally convex Hausdorff topological vector spaces over the real or complex field. Let C(H, E) denote the continuous functions from H into E with the topology of uniform convergence. The purpose of this paper is to improve certain integral representation theorems for continuous linear transformations T from C(H, E) into The well-known Riesz representation theorem [10] gives a Stieltjes integral representation for T when H is a closed interval and when E and are the real numbers. There have been many generalizations of this theorem in the literature, and there have been two essentially different approaches giving rise to two different kinds of representation theorems. In one approach (see [1], [2], [5], [8], [9], [11]) y'T is written as an integral where y' is in the topological dual of The integral converges in the weak topology with a measure defined on the Borel sets with values in L(E, F), the space on continuous linear transformations of E into F. Under various additional assumptions on E, F, or T, T is written as an integral. (For example, this can be done if T is assumed to be compact or is assumed to be reflexive.) The most general of these theorems is due to Swong [11, Theorem 6.1, p. 283]. Another approach in the literature can be found in the papers [6], [12], [13], [14]. In these papers the method is of a constructive nature and T is written as an integral, and the convergence of the integral is in the e00 topology of F (norm topology in the case is a normed space), and T is thought of as a mapping of E into F by the canonical imbedding of into F. This is done by restricting the class of sets the measure is defined on. In [12] the class of sets is the intervals where H = [0, I], and E and are assumed to be normed spaces. In [6] and [13] the class of sets is the ring generated by compact Gj's. Recently in [14] Tucker and Wayment have been able to extend the measure to a larger proper subring of the Baire sets. In this paper it is shown that the restriction of the measure in [61, [12], [13], [14] was not necessary, and in fact T can be represented as an integral with convergence in the e topology with the measure defined on the Borel sets. This result thus implies and strengthens