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Analysis: An Introduction

Richard Beals
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Christopher Hammond
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Richard Beals' Analysis: An Introduction is a serious textbook for serious students. Intended for advanced undergraduates, this book demands as much personal maturity from the reader as it does mathematical sophistication.

The distinguishing feature of this book is its breadth. It is typical for an introductory analysis text to treat certain fundamental topics with great care, making only passing references (if that) to more sophisticated applications. Beals takes a rather different approach. He clearly views this book as being an introduction to the entire area of analysis, rather than an exposition of a predetermined set of topics. Less than half of the book is dedicated to material which (in the reviewer's experience) would generally appear in a standard introductory course. The remainder deals with more advanced topics, as well as a variety of applications. The last third of the book, in fact, is devoted exclusively to Fourier series and differential equations.

While, technically speaking, this book could be used for a first course in analysis, the title is perhaps something of a misnomer. The important introductory concepts are all discussed, precisely and completely, but often as a stepping-stone to more sophisticated results. Take, for example, the chapter that deals with continuity. Beals spends less than six pages (including exercises) discussing the general properties of continuous functions; after that, he shifts his attention to the spaces C([a,b]) and the Weierstrass Approximation Theorem. While one could argue that six pages are sufficient to his purposes, this transition might seem a bit precipitate to someone encountering these concepts for the first time.

Beals' writing style is characterized by a certain austere elegance. The author has an admirable command of the English language, and he appears unaffected by the excessive informality that has afflicted so many undergraduate textbooks. Apart from a few casual remarks in the introduction, there is virtually no "padding" anywhere in the text. The lemmas, propositions, theorems, and corollaries come in rapid succession, with very little commentary in between. Beals clearly expects a level of discipline from his readers that is comparable to his own.

Analysis: An Introduction is most appropriate for an undergraduate who has already grappled with the main ideas from real analysis, and who is looking for a succinct, well-written treatise that connects these concepts to some of their most powerful applications. Beals' book has the potential to serve this audience very well indeed.

Christopher Hammond is Assistant Professor of Mathematics at Connecticut College.

Preface page ix
1 Introduction 1
1A. Notation and Motivation 1
1B*. The Algebra of Various Number Systems 5
1C*. The Line and Cuts 9
1D. Proofs, Generalizations, Abstractions, and Purposes 12
2 The Real and Complex Numbers 15
2A. The Real Numbers 15
2B*. Decimal and Other Expansions; Countability 21
2C*. Algebraic and Transcendental Numbers 24
2D. The Complex Numbers 26
3 Real and Complex Sequences 30
3A. Boundedness and Convergence 30
3B. Upper and Lower Limits 33
3C. The Cauchy Criterion 35
3D. Algebraic Properties of Limits 37
3E. Subsequences 39
3F. The Extended Reals and Convergence to ñ8 40
3G. Sizes of Things: The Logarithm 42
Additional Exercises for Chapter 3 43
4 Series 45
4A. Convergence and Absolute Convergence 45
4B. Tests for (Absolute) Convergence 48
4C*. Conditional Convergence 54
4D*. Euler's Constant and Summation 57
4E*. Conditional Convergence: Summation by Parts 58
Additional Exercises for Chapter 4 59
5 Power Series 61
5A. Power Series, Radius of Convergence 61
5B. Differentiation of Power Series 63
5C. Products and the Exponential Function 66
5D*. Abel's Theorem and Summation 70
6 Metric Spaces 73
6A. Metrics 73
6B. Interior Points, Limit Points, Open and Closed Sets 75
6C. Coverings and Compactness 79
6D. Sequences, Completeness, Sequential Compactness 81
6E*. The Cantor Set 84
7 Continuous Functions 86
7A. Definitions and General Properties 86
7B. Real- and Complex-Valued Functions 90
7C. The Space C(I) 91
7D*. Proof of the Weierstrass Polynomial Approximation Theorem 95
8 Calculus 99
8A. Differential Calculus 99
8B. Inverse Functions 105
8C. Integral Calculus 107
8D. Riemann Sums 112
8E*. Two Versions of Taylor's Theorem 113
Additional Exercises for Chapter 8 116
9 Some Special Functions 119
9A. The Complex Exponential Function and Related Functions 119
9B*. The Fundamental Theorem of Algebra 124
9C*. Infinite Products and Euler's Formula for Sine 125
10 Lebesgue Measure on the Line 131
10A. Introduction 131
10B. Outer Measure 133
10C. Measurable Sets 136
10D. Fundamental Properties of Measurable Sets 139
10E*. A Nonmeasurable Set 142
11 Lebesgue Integration on the Line 144
11A. Measurable Functions 144
11B*. Two Examples 148
11C. Integration: Simple Functions 149
11D. Integration: Measurable Functions 151
11E. Convergence Theorems 155
12 Function Spaces 158
12A. Null Sets and the Notion of Almost Everywhere" 158
12B*. Riemann Integration and Lebesgue Integration 159
12C. The Space L1 162
12D. The Space L2 166
12E*. Differentiating the Integral 168
Additional Exercises for Chapter 12 172
13 Fourier Series 173
13A. Periodic Functions and Fourier Expansions 173
13B. Fourier Coefficients of Integrable and Square-Integrable
Periodic Functions 176
13C. Dirichlet's Theorem 180
13D. Féjer's Theorem 184
13E. The Weierstrass Approximation Theorem 187
13F. L2-Periodic Functions: The Riesz-Fischer Theorem 189
13G. More Convergence 192
13H*. Convolution 195
14* Applications of Fourier Series 197
14A*. The Gibbs Phenomenon 197
14B*. AContinuous