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Extensions of the representation theorems of Riesz and Fréchet. (English) Zbl 0799.46046

Summary: We present two types of representation theorems: one for linear continuous operators on spaces 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 in 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.

MSC:

46E40 Spaces of vector- and operator-valued functions
46G10 Vector-valued measures and integration
46E15 Banach spaces of continuous, differentiable or analytic functions
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