Abstract
This chapter is devoted to the properties of the Brouwer degree that we will need for the Leray-Schauder degree. In all that follows, we assume we have a map \( f = \bar U \to R^n \) such that F = f-1(0) is admissible in U, that is, compact and disjoint from ∂U, so the Brouwer degree d(f, U) is well-defined. The properties of the degree are given names for easy identification; the terminology I’m using for this purpose is pretty much standard. Some of the properties will carry over to the infinite-dimensional case and others are needed in order to make the transition to that more general setting. The following simple lemma will be helpful in verifying some of those properties.
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