**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: $95.00

AMS Member Price: $76.00

MAA Member Price: $85.50

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

List Price: $93.00

AMS Member Price: $74.40

MAA Member Price: $83.70

# 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