Introduction to Real Analysis

Manfred Stoll

• Publisher: Chapman and Hall/CRC
• Year: 2021
• Print ISBN:
  9780367683931
  9780367486884
• eBook ISBN:
  9781000345148
• Edition: 3
• Topic: Matematica
• Subtopic: Analisi funzionale

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.

1

The Real Numbers

The key to understanding many of the fundamental concepts of calculus, such as limits, continuity, and the integral, is the least upper bound property of the real number system ℝ. As we all know, the rational number system contains gaps. For example, there does not exist a rational number r such that $r^2=2$, i.e., $\sqrt{2}$ is irrational. The fact that the rational numbers do contain gaps makes them inadequate for any meaningful discussion of the above concepts.

The standard argument used in proving that the equation $r^2=2$ does not have a solution in the rational numbers goes as follows: Suppose that there exists a rational number r such that $r^2=2$. Write $r=\frac{m}{n}$ where m,n are integers which are not both even. Thus $m^2=2n^2$. Therefore m² is even, and hence m itself must be even. But then m², and hence also $2n^2$ are both divisible by 4. Therefore n² is even, and as a consequence n is also even. This however contradicts our assumption that not both m and n are even. The method of proof used in this example is proof by contradiction; namely, we assume the negation of t...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Contents
• Preface to the Third Edition
• Preface to the First Edition
• To the Student
• Acknowledgments
• 1 The Real Numbers
• 2 Topology of the Real Line
• 3 Sequences of Real Numbers
• 4 Limits and Continuity
• 5 Differentiation
• 6 Integration
• 7 Series of Real Numbers
• 8 Sequences and Series of Functions
• 9 Fourier Series
• 10 Lebesgue Measure and Integration
• Bibliography
• Hints and Solutions
• Index