Integral representations for continuous linear operators in the setting of convex topological vector spaces

Authors:
J. R. Edwards and S. G. Wayment

Journal:
Trans. Amer. Math. Soc. **157** (1971), 329-345

MSC:
Primary 28.46; Secondary 46.00

DOI:
https://doi.org/10.1090/S0002-9947-1971-0281867-3

MathSciNet review:
0281867

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Abstract: Suppose and are locally convex Hausdorff spaces, is arbitrary and is a ring of subsets of . The authors prove the analog of the theorem stated in [Abstract 672-372, Notices Amer. Math. Soc. 17 (1970), 188] in this setting. A theory of extended integration on function spaces with Lebesgue and non-Lebesgue type convex topologies is then developed. As applications, integral representations for continuous transformations into for the following function spaces (which have domain and range ) are obtained: (1) and are arbitrary, is a convex topology on the simple functions over is a set function on with values in , and is the Lebesgue-type space generated by ; (2) is a normal space and is the space of continuous functions each of whose range is totally bounded, with the topology of uniform convergence; (3) is a locally compact Hausdorff space, is the space of continuous functions of compact support with the topology of uniform convergence; (4) is a locally compact Hausdorff space and is the space of continuous functions with the topology of uniform convergence on compact subsets. In the above and may be replaced by topological Hausdorff spaces under certain additional compensating requirements.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0281867-3

Keywords:
Integral representations,
linear operators,
locally convex topologies,
Lebesgue-type topologies,
extended integrals,
vector valued functions,
vector valued set functions,
quasi-Gowurin,
bounded variation,
weak convergence

Article copyright:
© Copyright 1971
American Mathematical Society