A Riesz representation theorem in the setting of locally convex spaces

Abstract

Introduction. Let H be a compact Hausdorff space, let E and F be locally convex topological vector spaces over the real or complex field where Fis Hausdorff. Let C(H, E) be the space of continuous functions from H into E with the topology of uniform convergence. The general problem is to find an integral representation theorem for a continuous linear transformation T from C(H, E) into F. The well-known Riesz representation theorem [3] gives a Stieltjes integral representation for T when H is a closed interval and E and F are the real numbers. Tucker [6] gives a Stieltjes' integral type representation for T in the case of H, a closed interval, and E and F, linear normed spaces. This paper was generalized by Uherka [7] to the case of H, a compact space, and E and F, linear normed spaces. Also see Swong's paper [5] in which H is a compact Hausdorff space and E and F are locally convex topological vector spaces. This last paper takes a fundamentally different approach from that of Tucker and Uherka, and Swong writes y'T as an integral where y' is in the topological dual F' of F. However, Swong is able to write T as an integral under various additional assumptions on E, F or T. The approach taken in this paper more closely follows that of Tucker and Uherka where a much smaller class of sets is employed in the definition of integral. The result is that the integral converges in the s00 topology, as compared to the weak topology in Swong's paper. And T is written as an integral.