Article
MSC:
26B30,
46B99,
46E15,
46E40,
46G10,
47B38 | MR 1239124 | Zbl 0799.46046 | DOI: 10.21136/MB.1993.125924
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Keywords:
Riesz type representation theorem; Fréchet type representation theorem; representation theorems; linear continuous operators on spaces of Banach valued regulated functions of several real variables; bilinear continuous operators on cartesian products; functions of bounded variation; interior integral; geometry of Banach spaces; spaces of regulated functions of a real variable taking values in Banach spaces; regulated functions
Riesz type representation theorem; Fréchet type representation theorem; representation theorems; linear continuous operators on spaces of Banach valued regulated functions of several real variables; bilinear continuous operators on cartesian products; functions of bounded variation; interior integral; geometry of Banach spaces; spaces of regulated functions of a real variable taking values in Banach spaces; regulated functions
Summary:
We present two types of representation theorems: one for linear continuous operators on space of Banach valued regulated functions of several real variables and the other for bilinear continuous operators on cartesian products of spaces of regulated functions of a real variable taking values on Banach spaces. We use generalizations of the notions of functions of bounded variation in the sense of Vitali and Fréchet and the Riemann-Stieltjes-Dushnik or interior integral. A few applications using geometry of Banach spaces are given.
References:
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[3] M. Morse, W. Transue: The Fréchet variation in the small, sector limits, and left decompositions. Canadian Journal of Math. 2 (1950), 344-374. DOI 10.4153/CJM-1950-033-1 | MR 0037340
[4] G. C. Rocha-Filho: Integral de Riemann Vetorial e Geometria dos Espaços de Banach. doctoral thesis, IME-USP, 1979.
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[6] M. Fréchet: Sur les fonctionnelles bilineaires. Trans. Amer. Math. Soc. (1915), 215-234. DOI 10.2307/1988990 | MR 1501010
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