An isomorphism and isometry theorem for a class of linear functionals

Abstract:

Suppose $U$ is a set, ${\mathbf {F}}$ is a field of subsets of $U$ and ${\mathfrak {p}_{AB}}$ is the set of all real-valued, finitely additive functions defined on ${\mathbf {F}}$. Two principal notions are considered in this paper. The first of these is that of a subset of ${\mathfrak {p}_{AB}}$, defined by certain closure properties and called a $C$-set. The second is that of a collection $\mathcal {C}$ of linear transformations from ${\mathfrak {p}_{AB}}$ into ${\mathfrak {p}_{AB}}$ with special boundedness properties. Given a $C$-set $M$ which is a linear space, an isometric isomorphism is established from the dual of $M$ onto the set of all elements of $\mathcal {C}$ with range a subset of $M$. 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 ${\mathfrak {p}_{AB}}$ is demonstrated.