Problems and Solutions for Undergraduate Real Analysis

✍ Scribed by Kit-Wing Yu

• Year: 2018
• Tongue: English
• Leaves: 412
• Category: Library

✦ Synopsis

The aim of Problems and Solutions for Undergraduate Real Analysis I, as the name reveals, is to assist undergraduate students or first-year students who study mathematics in learning their first rigorous real analysis course. The wide variety of problems, which are of varying difficulty, include the following topics:

• Elementary Set Algebra
• The Real Number System
• Countable and Uncountable Sets
• Elementary Topology on Metric Spaces
• Sequences in Metric Spaces
• Series of Numbers
• Limits and Continuity of Functions
• Differentiation
• the Riemann-Stieltjes Integral

Furthermore, the main features of this book are listed as follows:

1. The book contains 230 problems, which cover the topics mentioned above, with detailed and complete solutions. As a matter of fact, my solutions show every detail, every step and every theorem that I applied.
2. Each chapter starts with a brief and concise note of introducing the notations, terminologies, basic mathematical concepts or important/famous/frequently used theorems (without proofs) relevant to the topic.
3. Three levels of difficulty have been assigned to problems so that you can sharpen your mathematics step-by-step.
4. Different colors are used frequently in order to highlight or explain problems, examples, remarks, main points/formulas involved, or show the steps of manipulation in some complicated proofs. (ebook only)
5. An appendix about mathematical logic is included. It tells students what concepts of logic (e.g. techniques of proofs) are necessary in advanced mathematics.

✦ Table of Contents

Preface
List of Figures
List of Tables
Contents
Chapter 1. Elementary Set Algebra
1.1 Fundamental Concepts
1.2 Sets, Functions and Relations
Chapter 2. The Real Number System
2.1 Fundamental Concepts
2.2 Rational and Irrational Numbers
2.3 Absolute Values
2.4 The Completeness Axiom
Chapter 3. Countable and Uncountable Sets
3.1 Fundamental Concepts
3.2 Problems on Countable and Uncountable Sets
Chapter 4. Elementary Topology on Metric Spaces
4.1 Fundamental Concepts
4.2 Open Sets and Closed Sets
4.3 Compact Sets
4.4 The Heine-Borel Theorem
4.5 Connected Sets
Chapter 5. Sequences in Metric Spaces
5.1 Fundamental Concepts
5.2 Convergence of Sequences
5.3 Upper and Lower Limits
5.4 Cauchy Sequences and Complete Metric Spaces
5.5 Recurrence Relations
Chapter 6. Series of Numbers
6.1 Fundamental Concepts
6.2 Convergence of Series of Nonnegative Terms
6.3 Alternating Series and Absolute Convergence
6.4 The Series Σ_{n=1}^∞ a_n b_n and Multiplication of Series
6.5 Power Series
Chapter 7. Limits and Continuity of Functions
7.1 Fundamental Concepts
7.2 Limits of Functions
7.3 Continuity and Uniform Continuity of Functions
7.4 The Extreme Value Theorem and the Intermediate Value Theorem
7.5 Discontinuity of Functions
7.6 Monotonic Functions
Chapter 8. Differentiation
8.1 Fundamental Concepts
8.2 Properties of Derivatives
8.3 The Mean Value Theorem for Derivatives
8.4 L’Hôspital’s Rule
8.5 Higher Order Derivatives and Taylor’s Theorem
Chapter 9. The Riemann-Stieltjes Integral
9.1 Fundamental Concepts
9.2 Integrability of Real Functions
9.3 Applications of Integration Theorems
9.4 The Mean Value Theorems for Integrals
Chapter 10. Sequences and Series of Functions
10.1 Fundamental Concepts
10.2 Uniform Convergence for Sequences of Functions
10.3 Uniform Convergence for Series of Functions
10.4 Equicontinuous Families of Functions
10.5 Approximation by Polynomials
Chapter 11. Improper Integrals
11.1 Fundamental Concepts
11.2 Evaluations of Improper Integrals
11.3 Convergence of Improper Integrals
11.4 Miscellaneous Problems on Improper Integrals
Chapter 12. Lebesgue Measure
12.1 Fundamental Concepts
12.2 Lebesgue Outer Measure
12.3 Lebesgue Measurable Sets
12.4 Necessary and Sufficient Conditions for Measurable Sets
Chapter 13. Lebesgue Measurable Functions
13.1 Fundamental Concepts
13.2 Lebesgue Measurable Functions
13.3 Applications of Littlewood’s Three Principles
Chapter 14. Lebesgue Integration
14.1 Fundamental Concepts
14.2 Properties of Integrable Functions
14.3 Applications of Fatou’s Lemma
Chapter 15. Differential Calculus of Functions of Several Variables
15.1 Fundamental Concepts
15.2 Differentiation of Functions of Several Variables
15.3 The Mean Value Theorem for Differentiable Functions
15.4 The Inverse Function Theorem and the Implicit Function Theorem
15.5 Higher Order Derivatives
Chapter 16. Integral Calculus of Functions of Several Variables
16.1 Fundamental Concepts
16.2 Jordan Measurable Sets
16.3 Integration on ℝ^n
16.4 Applications of the Mean Value Theorem
16.5 Applications of the Change of Variables Theorem
Appendix
A. Language of Mathematics
A.1 Fundamental Concepts
A.2 Statements and Logical Connectives
A.3 Quantifiers and their Basic Properties
A.4 Necessity and Sufficiency
A.5 Techniques of Proofs
Index
Bibliography
[1]-[16]
[17]-[34]