Universal properties of spaces of generalized functions

Abstract

Through the presentation of several examples, we motivate that universal properties are the simplest way to solve a given mathematical problem. To illustrate this point, we present the co-universal property of Schwartz distributions, as the simplest way to have derivatives of continuous functions. We also discuss Colombeau algebra as the simplest quotient algebra where representatives of zero are infinitesimal. Furthermore, we explore generalized smooth functions as the universal way to associate set-theoretical maps defined by nets of smooth functions (e.g. regularizations of distributions) and having arbitrary derivatives. Each of these properties results in a characterization up to isomorphisms of the corresponding space. The present work requires only the notions of category, functor, natural transformation and Schwartz distributions, and introduces the notion of universal solution using a simple and non-abstract language.