Brouwer Degree

Abstract

The main technical tool of the second part of these notes, and one of the most useful topological tools in analysis, is the Leray-Schauder degree. The setting for the Leray-Schauder degree is, in general, infinite-dimensional: normed linear spaces. In the first part of the notes, before proving the Schauder fixed point theorem for maps of such spaces, we first studied the corresponding finite-dimensional setting, that is, euclidean spaces. We proved the finite-dimensional version of Schauder’s theorem, the Brouwer fixed point theorem, and then used the Schauder projection to extend to the infinite-dimensional version. The finite-dimensional version of Leray-Schauder degree is called Brouwer degree and, like Brouwer’s fixed point theorem, its context is euclidean space. In this chapter, I will present the Brouwer degree and demonstrate certain properties of it. These properties are the ones we will need in order to extend to Leray-Schauder degree theory in the following chapter, again moving to infinite-dimensional spaces with the aid of the Schauder projection lemma. Just as we did not explore the many topological implications of the Brouwer fixed point theorem in the first part, here we will not be concerned with studying the Brouwer degree for its own sake, but instead we will proceed as efficiently as possible to the more general theory.