Contents

Preface v

Introduction 1

CHAPTER 1. Function-Analytic Preliminaries 13

CHAPTER 2. The Leray-Schauder Degree for Differentiate Maps 26

1. The degree for linear maps 26

2. The degree if yo is a regular value for the map / 27

3. The one-dimensional case and the case of polynomial maps 31

4. The degree for a not necessarily regular value yo 39

5. Notes 51

CHAPTER 3. The Leray-Schauder Degree for Not Necessarily

Differentiable Maps 59

1. An extension lemma 59

2. An application of the extension lemma 62

3. The degree theory for finite layer maps 64

4. Another application of the extension lemma 65

5. Two additional properties of the Leray-Schauder degree 67

6. Generalized L.-S. maps 69

7. Notes 71

CHAPTER 4. The Poincare-Bohl Theorem and Some of Its Applications 76

1. The Poincare-Bohl theorem and the winding number 76

2. The interpretation of degree and winding number as intersection

numbers 83

3. Notes 84

CHAPTER 5. The Product Theorem and Some of Its Consequences 88

1. The product theorem 88

2. The invariance of the domain 97

3. The Jordan-Leray theorem 103

4. Notes 108

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