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), 329345
MSC:
Primary 28.46; Secondary 46.00
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 672372, Notices Amer. Math. Soc. 17 (1970), 188] in this setting. A theory of extended integration on function spaces with Lebesgue and nonLebesgue 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 Lebesguetype 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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 N. Dunford and J. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
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 R. J. Easton and D. H. Tucker, A generalized Lebesguetype integral, Math. Ann. 181 (1969), 311324. MR 1513279
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 J. R. Edwards and S. G. Wayment, A unifying representation theorem, Math. Ann. 187 (1970), 317328. MR 0270181 (42:5074)
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 R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York, 1965. MR 36 #4308. MR 0221256 (36:4308)
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 I. G. Fihtengol'c and L. Kantorovič, Sur les opérations linéaires dans l'espace des fonctions bornées, Studia Math. 5 (1934), 6998.
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 T. H. Hildebrandt, On bounded linear functional operations, Trans. Amer. Math. Soc. 36 (1934), 868875. MR 1501772
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 M. A. Naĭmark, Normed rings, GITTL, Moscow, 1956; English transl., Noordhoff, Groningen, 1959. MR 19, 870; MR 22 #1824.
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 D. J. Uherka, Generalized Stieltjes integrals and a strong representation theorem for continuous linear maps on a function space, Math. Ann. 182 (1969), 6066. MR 40 #705. MR 0247439 (40:705)
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 S. G. Wayment, Absolute continuity and the Radon theorem, Ph.D. Thesis, University of Utah, Salt Lake City, Utah, 1968.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197102818673
PII:
S 00029947(1971)02818673
Keywords:
Integral representations,
linear operators,
locally convex topologies,
Lebesguetype topologies,
extended integrals,
vector valued functions,
vector valued set functions,
quasiGowurin,
bounded variation,
weak convergence
Article copyright:
© Copyright 1971 American Mathematical Society
