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-space
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 principle of uniform boundedness and the Banach-Steinhaus theorem
3. The Open Map 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. Continuous functions that are zero at infinity
5. The Stone-Weierstrass Theorem in Product Spaces
6. Continuous functions on compact metric spaces
7. Bases in Hilbert spaces

Chapter 5 Ascoli's 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: Standardized spaces
Appendix B: Hilbert Spaces
Appendix C: Ordered sets and characterizations of the field of real numbers
Appendix D: Differentiation of Vector Functions and the Mean Value Inequality

Bibliography

List of Special Topics Covered in Exercises

Index of Notes

Alphabetical Index