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Publisher:

Cambridge University Press

Publication Date:

2004

Number of Pages:

261

Format:

Paperback

Price:

50.00

ISBN:

0-521-60047-2

Category:

Textbook

[Reviewed by , on ]

Christopher Hammond

02/5/2005

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 Ã±8 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

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 Ã±8 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

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