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Two properties of Bochner integrals

Published online by Cambridge University Press:  17 April 2009

B. D. Craven
B. D. Craven
Affiliation:
University of Melbourne, Parkville, Victoria.
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Abstract

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Two theorems for Lebesgue integrals, namely the Gauss-Green Theorem relating surface and volume integrals, and the integration-by-parts formula, are shown to possess generalizations where the integrands take values in a Banach space, the integrals are Bochner integrals, and derivatives are Fréchet derivatives. For integration-by-parts, the integrand consists of a continuous linear map applied to a vector-valued function. These results were required for a generalization of the calculus of variations, given in another paper.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 3 , Issue 3 , December 1970 , pp. 363 - 368
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Craven, B.D., “On the Gauss-Green theorem”, J. Austral. Math. Soc. 8 (1968), 385–396.Google Scholar
[2]Hille, Einar and Phillips, Ralph S., Functional analysis and semi-groups (Colloquium Publ. 31, revised ed., Amer. Math. Soc., Providence, R.J., 1957).Google Scholar
[3]Yosida, Kôsaku, Functional analysis, 2nd ed. (Die Grundlehren der mathematischen Wissenschaften, Band 123, Springer-Verlag, Berlin, Heidelberg, New York, 1968).Google Scholar