Integration and Nonlinear Transformations In Hilbert Space

Leonard Gross
1960 Transactions of the American Mathematical Society  
The purpose of this paper is to generalize to Hilbert space a well known theorem of Jacobi on the transformation of integrals under a change of coordinates in En. In the process we develop some of the relevant integration theory over Hilbert space. Among the various theories [14; 3; 4; 5; 9; 10 ], of integration over a real Hilbert space 77 we shall utilize the one [14; 3; 4; 5], which seems most relevant to the quantum theory of fields and which is intimately connected with the Wiener
more » ... The formulation of the theory which we adopt is given in Segal [14, and we refer the reader to this paper for details. In outline this formulation consists in associating with 77 in an invariant way a probability space (S, m) and a map/-->/ which assigns a measurable function / on (S, m) to each tame function / on 77, a tame function being one which, roughly speaking, depends only on a finite number of coordinates in 77. The theory of integration over 77 is then the theory of integration over (5, m) with particular emphasis on those questions which arise from the relation of 77 to (S, m). In case H is real P2(0, 1) the probability space may be taken to be Wiener space. We first describe (Theorem 1) a class of continuous functions on 77 which are determined by their continuity properties alone and which correspond in a natural way to measurable functions on (S, m), i.e., functions/ other than tame functions for which/makes sense. The type of continuity involved plays a central role in the remainder of the paper and generalizes ordinary continuity in P" in the relevant way for the purposes of integration theory. Theorem 2 relates the convergence of sequences of these functions to convergence in probability of the associated measurable functions. Theorem 3 is a density theorem for the group of automorphisms of the algebra La(S, m) and provides a background for the approximation method used in the following theorem. Theorem 4 is a generalization of Jacobi's theorem for a transformation of the form T = I+K where 7 is the identity operator and K is nonlinear, small and smooth. In Theorem 5 we avoid the smallness requirement by considering a one-parameter group of nonlinear transformations on 77 on whose generator sufficient conditions are imposed which ensure that the elements of the group satisfy a Jacobi-like theorem. The case of linear transformations has already been investigated by Segal [16].
doi:10.2307/1993432 fatcat:vaktenv2qzbjzjlnmsqfqhpquy