A divergence theorem for Hilbert space

Goodman, Victor

doi:10.1090/S0002-9947-1972-0298417-9

A divergence theorem for Hilbert space
HTML articles powered by AMS MathViewer

by Victor Goodman PDF

Trans. Amer. Math. Soc. 164 (1972), 411-426 Request permission

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.

References

Robert Bonic and John Frampton, Differentiable functions on certain Banach spaces, Bull. Amer. Math. Soc. 71 (1965), 393–395. MR 172084, DOI 10.1090/S0002-9904-1965-11310-6
J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press, New York-London, 1960. MR 0120319
I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR 0435831
Leonard Gross, Measurable functions on Hilbert space, Trans. Amer. Math. Soc. 105 (1962), 372–390. MR 147606, DOI 10.1090/S0002-9947-1962-0147606-6
Leonard Gross, Abstract Wiener spaces, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 31–42. MR 0212152
Leonard Gross, Potential theory on Hilbert space, J. Functional Analysis 1 (1967), 123–181. MR 0227747, DOI 10.1016/0022-1236(67)90030-4
J. Kurzweil, On approximation in real Banach spaces, Studia Math. 14 (1954), 214–231 (1955). MR 68732, DOI 10.4064/sm-14-2-214-231
Serge Lang, Introduction to differentiable manifolds, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155257
Gilbert Stengle, A divergence theorem for Gaussian stochastic process expectations, J. Math. Anal. Appl. 21 (1968), 537–546. MR 225376, DOI 10.1016/0022-247X(68)90262-X
J. H. M. Whitfield, Differentiable functions with bounded nonempty support on Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 145–146. MR 184073, DOI 10.1090/S0002-9904-1966-11458-1