**Mathematical Surveys and Monographs**

Volume: 23;
1986;
242 pp;
Softcover

MSC: Primary 47;
Secondary 54; 55; 58

Print ISBN: 978-0-8218-2770-3

Product Code: SURV/23.S

List Price: $93.00

Individual Member Price: $74.40

**Electronic ISBN: 978-1-4704-1250-0
Product Code: SURV/23.S.E**

List Price: $93.00

Individual Member Price: $74.40

# Introduction to Various Aspects of Degree Theory in Banach Spaces

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*E. H. Rothe*

Since its development by Leray and Schauder in the 1930's, degree theory in Banach spaces has proved to be an important tool in tackling many analytic problems, including boundary value problems in ordinary and partial differential equations, integral equations, and eigenvalue and bifurcation problems. With this volume E. H. Rothe provides a largely self-contained introduction to topological degree theory, with an emphasis on its function-analytical aspects. He develops the definition and properties of the degree as much as possible directly in Banach space, without recourse to finite-dimensional theory. A basic tool used is a homotopy theorem for certain linear maps in Banach spaces which allows one to generalize the distinction between maps with positive determinant and those with negative determinant in finite-dimensional spaces.

Rothe's book is addressed to graduate students who may have only a rudimentary knowledge of Banach space theory. The first chapter on function-analytic preliminaries provides most of the necessary background. For the benefit of less experienced mathematicians, Rothe introduces the topological tools (subdivision and simplicial approximation, for example) only to the degree of abstraction necessary for the purpose at hand. Readers will gain insight into the various aspects of degree theory, experience in function-analytic thinking, and a theoretic base for applying degree theory to analysis.

Rothe describes the various approaches that have historically been taken towards degree theory, making the relationships between these approaches clear. He treats the differential method, the simplicial approach introduced by Brouwer in 1911, the Leray-Schauder method (which assumes Brouwer's degree theory for the finite-dimensional space and then uses a limit process in the dimension), and attempts to establish degree theory in Banach spaces intrinsically, by an application of the differential method in the Banach space case.

#### Table of Contents

# Table of Contents

## Introduction to Various Aspects of Degree Theory in Banach Spaces

- Contents iii4 free
- Preface v6 free
- Introduction 18 free
- Chapter 1. Function-Analytic Preliminaries 1320 free
- Chapter 2. The Leray-Schauder Degree for Differentiate Maps 2633
- Chapter 3. The Leray-Schauder Degree for Not Necessarily Differentiable Maps 5966
- Chapter 4. The Poincare-Bohl Theorem and Some of Its Applications 7683
- Chapter 5. The Product Theorem and Some of Its Consequences 8895
- Chapter 6. The Finite-Dimensional Case 112119
- Chapter 7. On Spheres 138145
- Chapter 8. Some Extension and Homotopy Theorems 172179
- Chapter 9. The Borsuk Theorem and Some of Its Consequences 192199
- Appendix A. The Linear Homotopy Theorem 199206
- 1. Motivation for the theorem and the method of proof 199206
- 2. Background material from spectral theory in a complex Banach space Z 200207
- 3. The complexification Z of a real Banach space E 204211
- 4. On the index j of linear nonsingular L.-S. maps on complex and real Banach spaces 208215
- 5. Proof of the linear homotopy theorem 214221
- 6. The multiplication theorem for the indices 228235

- Appendix B. Proof of the Sard-Smale Theorem 4.4 of Chapter 2 231238
- References 238245
- Index 241248