Vol. 31, No. 2, 1969

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Abstract Wiener spaces and applications to analysis

James Donald Kuelbs

Vol. 31 (1969), No. 2, 433–450

Let C denote the space of real-valued continuous functions on [0,1] which vanish at zero, and let C be the subspace of C consisting of functions whose derivative is square integrable. Then C is a Hilbert space under the inner product (x,y) = 01x(t)y(t)dt and, as is well known, C has Wiener measure zero. Nevertheless, in many instances theorems involving the Wiener integral depend to a large extent on C. An example of this occurs in the behavior of Wiener measure under translation. However, in other situations it is the relationship between C and C which is important. An important factor in this relationship was pinpointed by L. Gross in the definition of an “abstract Wiener space.” This paper develops further the relationship of C and C which is embodied in this concept.

A representation theorem for additive Borel measurable functionals on a separable real Banach space B is established as well as a result related to the uniform boundedness principle. In another theorem a “stochastic expansion” of an arbitrary element of the Banach space B is given and it is shown that if B has a Schauder basis then it can be arranged so that the stochastic expansion and the basis expansion agree.

In the last section of the paper the ratios of certain integrals over an abstract Wiener space are examined, some of which, in the case of classical Wiener space, were studied by Cameron, Martin, and Shapiro in order to solve nonlinear integral equations. A theorem indicating how to weakly invert certain one-to-one operators from B into B is proved and finally an application to nonlinear integral equations involving functions of infinitely many variables is made.

Mathematical Subject Classification
Primary: 28.46
Received: 11 December 1968
Published: 1 November 1969
James Donald Kuelbs