Supplements to the Exercises in Chapters 1–7 of Walter Rudin's Principles of Mathematical Analysis, Third Edition

✍ Scribed by George M. Bergman

• Year: 2022
• Tongue: English
• Leaves: 122
• Category: Library

✦ Synopsis

"This packet contains both additional exercises relating to the material in Chapters 1–7 of Rudin['s Principles of Mathematical Analysis, 3rd Edition], and information on Rudin’s exercises for those chapters."

✦ Table of Contents

Preface
Chapter 1. The Real and Complex Number Systems.
1.1. INTRODUCTION.
1.2. ORDERED SETS.
1.3. FIELDS.
1.4. THE REAL FIELD.
1.5. THE EXTENDED REAL NUMBER SYSTEM.
1.6. THE COMPLEX FIELD.
1.7. EUCLIDEAN SPACES.
1.8. APPENDIX to Chapter 1. (Constructing R by Dedekind cuts.)
Chapter 2. Basic Topology.
2.1. FINITE, COUNTABLE, AND UNCOUNTABLE SETS.
2.2. METRIC SPACES.
2.3. COMPACT SETS.
2.4. PERFECT SETS.
2.5. CONNECTED SETS.
2.6. Separable metric spaces (developed only in exercises).
Chapter 3. Numerical Sequences and Series.
3.1. CONVERGENT SEQUENCES.
3.2. SUBSEQUENCES.
3.3. CAUCHY SEQUENCES.
3.4. UPPER AND LOWER LIMITS.
3.5. SOME SPECIAL SEQUENCES.
3.6. SERIES.
3.7. SERIES OF NONNEGATIVE TERMS (Convergence by grouping).
3.8. THE NUMBER e.
3.9. THE ROOT AND RATIO TESTS.
3.10. POWER SERIES.
3.11. SUMMATION BY PARTS.
3.12. ABSOLUTE CONVERGENCE.
3.13. ADDITION AND MULTIPLICATION OF SERIES.
3.14. REARRANGEMENTS.
Chapter 4. Continuity.
4.1. LIMITS OF FUNCTIONS.
4.2. CONTINUOUS FUNCTIONS.
4.3. CONTINUITY AND COMPACTNESS (and uniform continuity).
4.4. CONTINUITY AND CONNECTEDNESS.
4.5. DISCONTINUITIES.
4.6. MONOTONIC FUNCTIONS.
4.7. INFINITE LIMITS AND LIMITS AT INFINITY.
Chapter 5. Differentiation.
5.1. THE DERIVATIVE OF A REAL FUNCTION.
5.2. MEAN VALUE THEOREMS.
5.3. Restrictions on discontinuities of derivatives (called by Rudin THE CONTINUITY OF DERIVATIVES).
5.4. L'HOSPITAL'S RULE.
5.5. DERIVATIVES OF HIGHER ORDER.
5.6. TAYLOR'S THEOREM.
5.7. DIFFERENTIATION OF VECTOR-VALUED FUNCTIONS.
Chapter 6. The Riemann-Stieltjes integral.
6.1. The Riemann integral (beginning of Rudin's section DEFINITION AND EXISTENCE OF THE INTEGRAL).
6.2. The Riemann-Stieltjes integral (middle of Rudin's section DEFINITION AND EXISTENCE OF THE INTEGRAL).
6.3. Conditions for integrability (end of Rudin's section DEFINITION AND EXISTENCE OF THE INTEGRAL).
6.4. Basic properties (beginning of Rudin's section PROPERTIES OF THE INTEGRAL).
6.5. Step functions, differentiable , and change of variables (end of Rudin's section PROPERTIES OF THE INTEGRAL).
6.6. INTEGRATION AND DIFFERENTIATION (the Fundamental Theorem of Calculus).
6.7. INTEGRATION OF VECTOR-VALUED FUNCTIONS.
6.8. RECTIFIABLE CURVES.
Chapter 7. Sequences and series of functions.
7.1. DISCUSSION OF THE MAIN PROBLEM.
7.2. UNIFORM CONVERGENCE.
7.3. UNIFORM CONVERGENCE AND CONTINUITY.
7.4. UNIFORM CONVERGENCE AND INTEGRATION.
7.5. UNIFORM CONVERGENCE AND DIFFERENTIATION.
7.6. EQUICONTINUOUS FAMILIES OF FUNCTIONS.
7.7. The Weierstrass Theorem, and a corollary (beginning of Rudin's section THE STONEWEIERSTRASS THEOREM).
7.8. Algebras of Functions, Uniform Closure, and Separation of Points (middle of Rudin's section THE STONE-WEIERSTRASS THEOREM).
7.9. The Stone-Weierstrass Theorem (end of Rudin's section THE STONE-WEIERSTRASS THEOREM).