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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A divergence theorem for Hilbert space


Author: Victor Goodman
Journal: Trans. Amer. Math. Soc. 164 (1972), 411-426
MSC: Primary 46G05; Secondary 26A96, 28A40
MathSciNet review: 0298417
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Abstract: Let B be a real separable Banach space. A suitable linear imbedding of a real separable Hilbert space into B with dense range determines a probability measure on B which is known as abstract Wiener measure. In this paper it is shown that certain submanifolds of B carry a surface measure uniquely defined in terms of abstract Wiener measure. In addition, an identity is obtained which relates surface integrals to abstract Wiener integrals of functions associated with vector fields on regions in B. The identity is equivalent to the classical divergence theorem if the Hilbert space is finite dimensional. This identity is used to estimate the total measure of certain surfaces, and it is established that in any space B there exist regions whose boundaries have finite surface measure.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0298417-9
Keywords: Integral calculus on Banach spaces, abstract Wiener spaces, Hilbert space, divergence theorem, Gauss' theorem
Article copyright: © Copyright 1972 American Mathematical Society