A companion to analysis: a second first and first second course in analysis

✍ Scribed by Körner, Thomas William

• Publisher: American Mathematical Society
• Year: 2004
• Tongue: English
• Leaves: 1014
• Series: Graduate studies in mathematics 62
• Edition: 1
• Category: Library

✦ Synopsis

A Companion to Analysis, with Partial Solutions for Questions in Appendix K

✦ Table of Contents

Title Page
Contents
Introduction
Chapter 1. The Real Line
§1.1. Why do we bother?
§1.2. Limits
§1.3. Continuity
§1.4. The fundamental axiom
§1.5. The axiom of Archimedes
§1.6. Lion hunting
§1.7. The mean value inequality
§1.8. Full circle
§1.9. Are the real numbers unique?
Chapter 2. A First Philosophical Interlude
§2.1. Is the intermediate value theorem obvious?
Chapter 3. Other Versions of the Fundamental Axiom
§3.1. The supremum
§3.2. The Bolzano—Weierstrass theorem
§3.3. Some general remarks
Chapter 4. Higher Dimensions
§4.1. Bolzano-Weierstrass in Higher Dimensions
§4.2. Open and closed sets
§4.3. A central theorem of analysis
§4.4. The mean value theorem
§4.5. Uniform continuity
§4.6. The general principle of convergence
Chapter 5. Sums and Suchlike
§5.1. Comparison tests
§5.2. Conditional convergence
§5.3. Interchanging limits
§5.4. The exponential function
§5.5. The trigonometric functions
§5.6. The logarithm
§5.7. Powers
§5.8. The fundamental theorem of algebra
Chapter 6. Differentiation
§6.1. Preliminaries
§6.2. The operator norm and the chain rule
§6.3. The mean value inequality in higher dimensions
Chapter 7. Local Taylor Theorems
§7.1. Some one-dimensional Taylor theorems
§7.2. Some many-dimensional local Taylor theorems
§7.3. Critical points
Chapter 8. The Riemann Integral
§8.1. Where is the problem ?
§8.2. Riemann integration
§8.3. Integrals of continuous functions
§8.4. First steps in the calculus of variations
§8.5. Vector-valued integrals
Chapter 9. Developments and Limitations of the Riemann Integral
§9.1. Why go further?
§9.2. Improper integrals
§9.3. Integrals over areas
§9.4. The Riemann- Stieltjes integral
§9.5. How long is a piece of string?
Chapter 10. Metric Spaces
§10.1. Sphere packing
§10.2. Shannon's theorem
§10.3. Metric spaces
§10.4. Norms and the interaction of algebra and analysis
§10.5. Geodesies
Chapter 11. Complete Metric Spaces
§11.1. Completeness
§11.2. The Bolzano-Weierstrass property
§11.3. The uniform norm
§11.4. Uniform convergence
§11.5. Power series
§11.6. Fourier series
Chapter 12. Contraction Mappings and Differential Equations
§12.1. Banach's contraction mapping theorem
§12.2. Existence of solutions of differential equations
§12.3. Local to global
§12.4. Green's function solutions
Chapter 13. Inverse and Implicit Functions
§13.1. The inverse function theorem
§13.2. The implicit function theorem
§13.3. Lagrange multipliers
Chapter 14. Completion
§14.1. What is the correct question?
§14.2. The solution
§14.3. Why do we construct the reals?
§14.4. How do we construct the reals?
§14.5. Paradise lost?
Appendix A. Ordered Fields
Appendix B. Countability
Appendix C. The Care and Treatment of Counterexamples
Appendix D. A More General View of Limits
Appendix E. Traditional Partial Derivatives
Appendix F. Another Approach to the Inverse Function Theorem
Appendix G. Completing Ordered Fields
Appendix H. Constructive Analysis
Appendix I. Miscellany
Appendix J. Executive Summary
Appendix K. Exercises
Bibliography
Index
Back Cover
Partial Solutions for Questions in Appendix K