Topological aspects of the characterization of Hida distributions – a remark

In the recent years, the dual pair of smooth and generalized random variables on the White Noise space, (S) and (S)*, has found many applications. For example, stochastic (partial) differential equations [L0U 90, L0U 91, Po 92, Po 93], quantum field theory [PS 93] and Feynman integrals [FPS 91, KS 92, LLS 93]. The main advantage of (S) and (S)* is the S-Transform, which in a nice way characterizes the pair. This transform maps generalized Hida distributions into a space of complex valued functions on S(IR). This space of functions is called the space of U-functionals. Moreover, the S-Transform turns out to be a bijection onto this space [PS 91]. In most applications, one is really working on the space of U-functionals. For this reason, it is natural to topologize the U-functional space. The aim of this paper is to construct the U-functional space using inductive and projective limits of Banach spaces. This construction is in light of the construction of (S) and (S)* quite natural. With the given topologies, we show our main result: The S-Transform is a homeomorphism