An Introduction to FUNCTIONAL SPACES

Abstract

In mathematics there are many important spaces of distributions and functions. In this thesis we discuss a framework to work with such spaces: the theory of functional spaces. We start by giving a formal, axiomatic definition of a functional space, which uses the language of distributions and locally convex vector spaces, and then we give an extensive treatment of various constructions that can be used to create one functional space from another. This is done both in the Euclidean setting and in the setting of vector bundles over manifolds. Of particular importance is the connection between these two settings: sufficiently well-behaved functional spaces on Euclidean space can be used as a model for functional spaces on vector bundles. Finally, we show that under certain conditions we can extend our modeling procedure to fiber bundles and that the resulting spaces have the structure of infinite dimensional (Fréchet) manifolds.