Lecture Notes: Non-Standard Approach to J.F. Colombeauâ•Žs Theory of Generalized Functions

In these lecture notes we present an introduction to non-standard analysis especially written for the community of mathematicians, physi­ cists and engineers who do research on J. F. Colombeau’ theory of new generalized functions and its applications. The main purpose of our non-standard approach to Colombeau’ theory is the improvement of the properties of the scalars of the varieties of spaces of generalized functions: in our non-standard approach the sets of scalars of the func­ tional spaces always form algebraically closed non-archimedean Can­ tor complete fields. In contrast, the scalars of the functional spaces in Colombeau’s theory are rings with zero divisors. The improve­ ment of the scalars leads to other improvements and simplifications of Colombeau’s theory such as reducing the number of quantifiers and possibilities for an axiomatization of the theory. Some of the algebras we construct in these notes have already counterparts in Colombeau’s theory, other seems to be without counterpart. We present applica­ tions of the theory to PDE and mathematical physics. Although our approach is directed mostly to Colombeau’s community, the readers who are already familiar with non-standard methods might also find a short and comfortable way to learn about Colombeau’s theory: a new branch of functional analysis which naturally generalizes the Schwartz theory of distributions with numerous applications to partial differen­ tial equations, differential geometry, relativity theory and other areas of mathematics and physics. MSC: Functional Analysis (46F30); Generalized Solutions of PDE (35D05).