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A Divergence Theorem for Hilbert Space

1972; American Mathematical Society; Volume: 164; Linguagem: Inglês

10.2307/1995985

1088-6850

Victor Goodman,

Point processes and geometric inequalities

Let B be a real separable Banach space.A suitable linear imbedding of a real separable Hubert 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 Hubert 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.1. Introduction.Let B be a real separable Banach space with norm || • ||.Consider a linear imbedding i of a real separable Hubert space into B with dense range.Gross [5] has shown that if the function \¿-|| is a measurable norm on the Hubert space, then the space B carries a Borel probability measure px uniquely characterized by the identity Px({xeB : iy,xy < r}) = (2Tr\¿*y\2)-1'2 f exp [-(2\¿*y\2Y1s2]ds.J -00Here, y is any element of the topological dual space of B, ¿* is the adjoint of the imbedding map, and | • | is the norm on the dual Hubert space.The space B, together with the Hubert space and the imbedding, is said to be an abstract Wiener space, andpx is said to be abstract Wiener measure on B. The above measure is easily identified if the associated Hubert space is finite dimensional.Since any dense linear imbedding of a finite dimensional space is invertible, a range space B of such an imbedding is isomorphic to the Hubert space.Under the isomorphism, the measure px is given by Px = (27r)-dlmfl'2 exp [-|x|2/2] dx where dx is Lebesgue measure on the Hubert space and | ■ | is the Hubert norm.