The notion of multiple stochastic integration was first introduced
by Norbert Wiener in connection with his studies of the Brownian Motion
process. Given any 1-dimensional Brownian path X(t) and any function
f(t) which depends only on t, he wanted to be able to define the integral
f(t)dX(t) = 1(f); however, the integral makes no sense as a Stieltjes sum
since X(t) is not a function of bounded variation.
The problem becomes even more complex when one wants to consider
multiple stochastic integrals
••• f f(t.,...,t ) dXCO-.-dXCt ) = I (f)
J _i.n I 1 n 1 n n
J It J
where f is a real valued function defined on
K. Ito  was the
first to establish the existence of I (f) for symmetric functions f onIR+*
whose L -norms are finite. Using these multiple integrals, K. Ito was able
to establish an orthogonal decomposition of the space of L -Brownian
functionals. This decomposition agrees exactly with one obtained by R.
Cameron and W. Martin  using Hermite polynomials and clarifies much of
the work started by N. Wiener in "The Homogeneous Chaos" .
To outline these results let (ft,®,Pr) be a Lebesgue probability
space and (R,f,m) the real numbers with Lebesgue measure m. A Brownian
Received by the editors June 1980. This paper is a revised version
of the author's doctoral dissertation at Yale University, 1979, which
was supported by a Yale University graduate fellowship.