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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.
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