Gaussian Spaces

Abstract

As is well-known, one can not have a Lebesgue measure on infinite dimensional vector spaces. On the other hand, one can not set up a concrete functional analysis without reference to a measure. Gaussian measures offer a good choice as a reference measure because of several reasons. Firstly, they do exist on infinite dimensional spaces. Moreover, they have simple properties, and appear in many applications. Therefore, we shall use Gaussian measures as reference measures throughout this book. For our purposes, the measure of white noise (Example 1.2) is the most important example. Since the “sample paths” of white noise are not given by functions but rather by generalized functions, this measure is constructed on the Schwartz space l′(ℝ) of tempered distributions. l′(ℝ) is the dual of the Schwartz space of test functions l(ℝ), and the topological structure of l(ℝ) is the one of a nuclear countably Hilbert space (see, e.g., Gel’fand and Vilenkin (1968), and below). Thus we introduce in this chapter Gaussian spaces as spaces which are the topological dual of a countably Hilbert space, and which are equipped with a Gaussian measure. Such spaces are the basis of the whole book.