An Introduction to Analysis

James R. Kirkwood

• Publisher: Chapman and Hall/CRC
• Year: 2021
• ISBN:
  9781000403565
• Edition: 3
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

The third edition of this widely popular textbook is authored by a master teacher. This book provides a mathematically rigorous introduction to analysis of real­valued functions of one variable. This intuitive, student-friendly text is written in a manner that will help to ease the transition from primarily computational to primarily theoretical mathematics.

The material is presented clearly and as intuitive as possible while maintaining mathematical integrity. The author supplies the ideas of the proof and leaves the write-up as an exercise. The text also states why a step in a proof is the reasonable thing to do and which techniques are recurrent.

Examples, while no substitute for a proof, are a valuable tool in helping to develop intuition and are an important feature of this text. Examples can also provide a vivid reminder that what one hopes might be true is not always true.

Features of the Third Edition:

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• Begins with a discussion of the axioms of the real number system.

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• The limit is introduced via sequences.

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• Examples motivate what is to come, highlight the need for hypothesis in a theorem, and make abstract ideas more concrete.

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• A new section on the Cantor set and the Cantor function.

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• Additional material on connectedness.

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• Exercises range in difficulty from the routine "getting your feet wet" types of problems to the moderately challenging problems.

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• Topology of the real number system is developed to obtain the familiar properties of continuous functions.

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• Some exercises are devoted to the construction of counterexamples.

The author presents the material to make the subject understandable and perhaps exciting to those who are beginning their study of abstract mathematics.

Table of Contents

Preface

Introduction

• The Real Number System

• Sequences of Real Numbers

• Topology of the Real Numbers

• Continuous Functions

• Differentiation

• Integration

• Series of Real Numbers

• Sequences and Series of Functions

• Fourier Series

Bibliography

Hints and Answers to Selected Exercises

Index

Biography

James R. Kirkwood holds a Ph.D. from University of Virginia. He has authored fifteen, published mathematics textbooks on various topics including calculus, real analysis, mathematical biology and mathematical physics. His original research was in mathematical physics, and he co-authored the seminal paper in a topic now called Kirkwood-Thomas Theory in mathematical physics. During the summer, he teaches real analysis to entering graduate students at the University of Virginia. He has been awarded several National Science Foundation grants. His texts, Elementary Linear Algebra, Linear Algebra, and Markov Processes, are also published by CRC Press.

1

The Real Number System

The pattren of mathematics is to propose a set of axioms that a collection of objects and operations on the objects must satisfy and then to derive as many conclusions as possible about the system under consideration. The system is then the objects and operations together with the axioms. While courses in plane geometry emphasize this pattern, this is often not the case in algebra. Thus, one is often adept at algebraic manipulations without realizing why these operations are legitimate. Why, for example, is $3+2=2+3$ or $5\times 0=0$? Throughout the text we shall deal with the real number system. In this first chapter, we present the preliminary material that provides the foundation for our work, including the axioms of the real numbers, and some of the consequences of these axioms.

1.1 Sets and Functions

Sets and their Operations

Perhaps the two most fundamental concepts in mathematics are sets and functions. In this section we describe some of their properties. The termset is left undefined and is synonymous with collection or aggrega...