An isomorphism and isometry theorem for a class of linear functionals

Abstract

Suppose U U is a set, F {\mathbf {F}} is a field of subsets of U U and p A B {\mathfrak {p}_{AB}} is the set of all real-valued, finitely additive functions defined on F {\mathbf {F}} . Two principal notions are considered in this paper. The first of these is that of a subset of p A B {\mathfrak {p}_{AB}} , defined by certain closure properties and called a C C -set. The second is that of a collection C \mathcal {C} of linear transformations from p A B {\mathfrak {p}_{AB}} into p A B {\mathfrak {p}_{AB}} with special boundedness properties. Given a C C -set M M which is a linear space, an isometric isomorphism is established from the dual of M M onto the set of all elements of C \mathcal {C} with range a subset of M M . As a corollary it is demonstrated that the above-mentioned isomorphism and isometry theorem, together with a previous representation theorem of the author (J. London Math. Soc. 44 (1969), pp. 385-396), imply an analogue of a dual representation theorem of Edwards and Wayment (Trans. Amer. Math. Soc. 154 (1971), pp. 251-265). Finally, a “pseudo-representation theorem” for the dual of p A B {\mathfrak {p}_{AB}} is demonstrated.