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

DOI:
https://doi.org/10.1090/S0002-9947-1974-0352385-1

MathSciNet review:
0352385

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Abstract: Suppose is a set, is a field of subsets of and is the set of all real-valued, finitely additive functions defined on . Two principal notions are considered in this paper. The first of these is that of a subset of , defined by certain closure properties and called a -set. The second is that of a collection of linear transformations from into with special boundedness properties. Given a -set which is a linear space, an isometric isomorphism is established from the dual of onto the set of all elements of with range a subset of . 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 is demonstrated.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0352385-1

Keywords:
-integral,
isomorphism,
isometry,
transformation representation

Article copyright:
© Copyright 1974
American Mathematical Society