Applications of Topology to Analysis

Name: Applications of Topology to Analysis

Author(s): e Chaim Samuel Hönig

Pages: 221

Publication: IMPA, 1976

ISBN: 978-85-244-0160-2

Edition: 1

Introduction

Notations

Chapter 1 General Topology
1. Topological Spaces
2. Metric Spaces
3. Compactness
4. Compact metric spaces
5. Continuous functions
6. Product spaces
7. Examples of metric spaces
8. Examples of continuous functions
9. Other categories of topological spaces

Chapter 2 The Method of Successive Approximations
1. Banach’s fixed point theorem
2. Ordinary differential equations
3. Integral equations
4. Partial differential equations
5. The implicit function theorem
6. Linear equations in Banach spaces

Chapter 3 Baire’s Theorem

1. Baire’s theorem
2. The uniform limitation principle and the Banach-Steinhaus theorem
3. The open application theorem and the closed graph theorem

Chapter 4 The Stone-Weierstrass Theorem

1. The Stone-Weierstrass Theorem
2. The classical Weierstrass Theorem
3. Extension to locally compact spaces
4. Zero continuous functions at infinity
5. The Stone-Weierstrass Theorem on product spaces
6. Continuous functions on compact metric spaces
7. Bases in Hilbert spaces

Chapter 5 The Ascoli Theorem
1. Ascoli’s Theorem
2. Applications of Ascoli’s theorem

Chapter 6 Brouwer’s and Schauder ‘s Theorems
1. Brouwer’s Theorem
2. Application
3. Schauder’s Theorem
Appendix A: Normed Spaces
Appendix B: Hilbert Spaces
Appendix C: Ordered Sets and Characterizations of the Body of Real Numbers
Appendix D: Differentiation of Vector Functions and the Mean Inequality

Bibliography

List of Special Topics Covered in Exercises

Index of Notations

Alphabetical Index