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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

An isomorphism and isometry theorem for a class of linear functionals


Author: William D. L. Appling
Journal: Trans. Amer. Math. Soc. 199 (1974), 131-140
MSC: Primary 28A25
MathSciNet review: 0352385
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0352385-1
PII: S 0002-9947(1974)0352385-1
Keywords: $ \upsilon $-integral, isomorphism, isometry, transformation representation
Article copyright: © Copyright 1974 American Mathematical Society