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

 

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
MathSciNet review: 0281867
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Abstract: Suppose $ X$ and $ Y$ are locally convex Hausdorff spaces, $ H$ is arbitrary and $ \Sigma $ is a ring of subsets of $ H$. 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 $ Y$ for the following function spaces $ F$ (which have domain $ H$ and range $ X$) are obtained: (1) $ H$ and $ \Sigma $ are arbitrary, $ \tau $ is a convex topology on the simple functions over $ \Sigma ,K$ is a set function on $ \Sigma $ with values in $ L[X,Y]$, and $ F$ is the Lebesgue-type space generated by $ K$; (2) $ H$ is a normal space and $ F$ is the space of continuous functions each of whose range is totally bounded, with the topology of uniform convergence; (3) $ H$ is a locally compact Hausdorff space, $ F$ is the space of continuous functions of compact support with the topology of uniform convergence; (4) $ H$ is a locally compact Hausdorff space and $ F$ is the space of continuous functions with the topology of uniform convergence on compact subsets. In the above $ X$ and $ Y$ may be replaced by topological Hausdorff spaces under certain additional compensating requirements.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0281867-3
PII: S 0002-9947(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 $ (p,q)$ variation, weak convergence
Article copyright: © Copyright 1971 American Mathematical Society