Introduction to Real Analysis (Textbooks in Mathematics)

✍ Scribed by Manfred Stoll

• Publisher: Chapman and Hall/CRC
• Year: 2021
• Tongue: English
• Leaves: 583
• Edition: 3
• Category: Library

✦ Synopsis

This classic textbook has been used successfully by instructors and students for nearly three decades. This timely new edition offers minimal yet notable changes while retaining all the elements, presentation, and accessible exposition of previous editions. A list of updates is found in the Preface to this edition.

This text is based on the author’s experience in teaching graduate courses and the minimal requirements for successful graduate study. The text is understandable to the typical student enrolled in the course, taking into consideration the variations in abilities, background, and motivation. Chapters one through six have been written to be accessible to the average student,

w hile at the same time challenging the more talented student through the exercises.

Chapters seven through ten assume the students have achieved some level of expertise in the subject. In these chapters, the theorems, examples, and exercises require greater sophistication and mathematical maturity for full understanding.

In addition to the standard topics the text includes topics that are not always included in comparable texts.

• Chapter 6 contains a section on the Riemann-Stieltjes integral and a proof of Lebesgue’s t heorem providing necessary and sufficient conditions for Riemann integrability.

• Chapter 7 also includes a section on square summable sequences and a brief introduction to normed linear spaces.

• C hapter 8 contains a proof of the Weierstrass approximation theorem using the method of

aapproximate identities.

• The inclusion of Fourier series in the text allows the student to gain some exposure to this important subject.

• The final chapter includes a detailed treatment of Lebesgue measure and the Lebesgue integral, using inner and outer measure.

• The exercises at the end of each section reinforce the concepts.

• Notes provide historical comments or discuss additional topics.

✦ Table of Contents

Contents
Preface to the Third Edition
Preface to the First Edition
To the Student
Acknowledgments
1 The Real Numbers
1.1 Sets and Operations on Sets
1.2 Functions
1.3 Mathematical Induction
1.4 The Least Upper Bound Property
1.5 Consequences of the Least Upper Bound Property
1.6 Binary and Ternary Expansions
1.7 Countable and Uncountable Sets
Notes
Miscellaneous Exercises
Supplemental Reading
2 Topology of the Real Line
2.1 Metric Spaces
2.2 Open and Closed Sets
2.3 Compact Sets
2.4 Compact Subsets of R
2.5 The Cantor Set
Notes
Miscellaneous Exercises
Supplemental Reading
3 Sequences of Real Numbers
3.1 Convergent Sequences
3.2 Sequences of Real Numbers
3.3 Monotone Sequences
3.4 Subsequences and the Bolzano-Weierstrass Theorem
3.5 Limit Superior and Inferior of a Sequence
3.6 Cauchy Sequences
3.7 Series of Real Numbers
Notes
Miscellaneous Exercises
Supplemental Reading
4 Limits and Continuity
4.1 Limit of a Function
4.2 Continuous Functions
4.3 Uniform Continuity
4.4 Monotone Functions and Discontinuities
Notes
Miscellaneous Exercises
Supplemental Reading
5 Differentiation
5.1 The Derivative
5.2 The Mean Value Theorem
5.3 L’Hospital’s Rule
5.4 Newton’s Method
Notes
Miscellaneous Exercises
Supplemental Reading
6 Integration
6.1 The Riemann Integral
6.2 Properties of the Riemann Integral
6.3 Fundamental Theorem of Calculus
6.4 Improper Riemann Integrals
6.5 The Riemann-Stieltjes Integral
6.6 Numerical Methods
6.7 Proof of Lebesgue’s Theorem
Notes
Miscellaneous Exercises
Supplemental Reading
7 Series of Real Numbers
7.1 Convergence Tests
7.2 The Dirichlet Test
7.3 Absolute and Conditional Convergence
7.4 Square Summable Sequences
Notes
Miscellaneous Exercises
Supplemental Reading
8 Sequences and Series of Functions
8.1 Pointwise Convergence and Interchange of Limits
8.2 Uniform Convergence
8.3 Uniform Convergence and Continuity
8.4 Uniform Convergence and Integration
8.5 Uniform Convergence and Differentiation
8.6 The Weierstrass Approximation Theorem
8.7 Power Series Expansions
8.8 The Gamma Function
Notes
Miscellaneous Exercises
Supplemental Reading
9 Fourier Series
9.1 Orthogonal Functions
9.2 Completeness and Parseval’s Equality
9.3 Trigonometric and Fourier Series
9.4 Convergence in the Mean of Fourier Series
9.5 Pointwise Convergence of Fourier Series
Notes
Miscellaneous Exercises
Supplemental Reading
10 Lebesgue Measure and Integration
10.1 Introduction to Measure
10.2 Measure of Open Sets: Compact Sets
10.3 Inner and Outer Measure: Measurable Sets
10.4 Properties of Measurable Sets
10.5 Measurable Functions
10.6 Lebesgue Integral of a Bounded Function
10.7 The General Lebesgue Integral
10.8 Square Integrable Functions
Notes
Miscellaneous Exercises
Supplemental Reading
Bibliography
Hints and Solutions
Index