Elementary Classical Analysis

Designed for courses in advanced calculus and introductory real analysis,Elementary Classical Analysisstrikes a careful balance between pure and applied mathematics with an emphasis on specific techniques important to classical analysis without vector calculus or complex analysis. Intended for students of engineering and physical science as well as of pure mathematics.

PREFACE

ix

(2)

PREFACE TO THE FIRST EDITION

xi

INTRODUCTION: SETS AND FUNCTIONS

1

(24)

Supplement on the Axioms of Set Theory

7

(11)

Worked Examples

18

(2)

Exercises

20

(5)

1 THE REAL LINE AND EUCLIDEAN SPACE

25

(78)

1.1 Ordered Fields and the Number System

25

(10)

1.2 Completeness and the Real Number System

35

(10)

1.3 Least Upper Bounds

45

(4)

1.4 Cauchy Sequences

49

(3)

1.5 Cluster Points; lim inf and lim sup

52

(5)

1.6 Euclidean Space

57

(7)

1.7 Norms, Inner Products, and Metrics

64

(6)

1.8 The Complex Numbers

70

(9)

Theorem Proofs

79

(16)

Worked Examples

95

(2)

Exercises

97

(6)

2 THE TOPOLOGY OF EUCLIDEAN SPACE

103

(48)

2.1 Open Sets

104

(4)

2.2 Interior of a Set

108

(2)

2.3 Closed Sets

110

(3)

2.4 Accumulation Points

113

(3)

2.5 Closure of a Set

116

(2)

2.6 Boundary of a Set

118

(2)

2.7 Sequences

120

(3)

2.8 Completeness

123

(2)

2.9 Series of Real Numbers and Vectors

125

(5)

Theorem Proofs

130

(10)

Worked Examples

140

(3)

Exercises

143

(8)

3 COMPACT AND CONNECTED SETS

151

(26)

3.1 Compactness

151

(4)

3.2 The Heine-Borel Theorem

155

(2)

3.3 Nested Set Property

157

(3)

3.4 Path-Connected Sets

160

(3)

3.5 Connected Sets

163

(2)

Theorem Proofs

165

(5)

Worked Examples

170

(2)

Exercises

172

(5)

4 CONTINUOUS MAPPINGS

177

(60)

4.1 Continuity

177

(5)

4.2 Images of Compact and Connected Sets

182

(2)

4.3 Operations on Continuous Mappings

184

(4)

4.4 The Boundedness of Continuous Functions on Compact Sets

188

(3)

4.5 The Intermediate Value Theorem

191

(3)

4.6 Uniform Continuity

194

(2)

4.7 Differentiation of Functions of One Variable

196

(8)

4.8 Integration of Functions of One Variable

204

(7)

Theorem Proofs

211

(16)

Worked Examples

227

(4)

Exercises

231

(6)

5 UNIFORM CONVERGENCE

237

(90)

5.1 Pointwise and Uniform Convergence

237

(7)

5.2 The Weierstrass M Test

244

(3)

5.3 Integration and Differentiation of Series

247

(7)

5.4 The Elementary Functions

254

(14)

5.5 The Space of Continuous Functions

268

(4)

5.6 The Arzela-Ascoli Theorem

272

(3)

5.7 The Contraction Mapping Principle and Its Applications

275

(8)

5.8 The Stone-Weierstrass Theorem

283

(4)

5.9 The Dirichlet and Abel Tests

287

(2)

5.10 Power Series and Cesaro and Abel Summability

289

(5)

Theorem Proofs

294

(19)

Worked Examples

313

(3)

Exercises

316

(11)

6 DIFFERENTIABLE MAPPINGS

327

(64)

6.1 Definition of the Derivative

327

(4)

6.2 Matrix Representation

331

(3)

6.3 Continuity of Differentiable Mappings; Differentiable Paths

334

(6)

6.4 Conditions for Differentiability

340

(5)

6.5 The Chain Rule

345

(4)

6.6 Product Rule and Gradients

349

(4)

6.7 The Mean Value Theorem

353

(2)

6.8 Taylor's Theorem and Higher Derivatives

355

(7)

6.9 Maxima and Minima

362

(5)

Theorem Proofs

367

(13)

Worked Examples

380

(3)

Exercises

383

(8)

7 THE INVERSE AND IMPLICIT FUNCTION THEOREMS AND RELATED TOPICS

391

(54)

7.1 Inverse Function Theorem

392

(5)

7.2 Implicit Function Theorem

397

(4)

7.3 The Domain-Straightening Theorem

401

(2)

7.4 Further Consequences of the Implicit Function Theorem

403

(4)

7.5 An Existence Theorem for Ordinary Differential Equations

407

(4)

7.6 The Morse Lemma

411

(3)

7.7 Constrained Extrema and Lagrange Multipliers

414

(6)

Theorem Proofs

420

(15)

Worked Examples

435

(3)

Exercises

438

(7)

8 INTEGRATION

445

(52)

8.1 Integrable Functions

445

(6)

8.2 Volume and Sets of Measure Zero

451

(3)

8.3 Lebesgue's Theorem

454

(3)

8.4 Properties of the Integral

457

(2)

8.5 Improper Integrals

459

(7)

8.6 Some Convergence Theorems

466

(3)

8.7 Introduction to Distributions

469

(3)

Theorem Proofs

472

(10)

Worked Examples

487

(3)

Exercises

490

(7)

9 FUBINI'S THEOREM AND THE CHANGE OF VARIABLES FORMULA

497

(46)

9.1 Introduction

497

(3)

9.2 Fubini's Theorem

500

(5)

9.3 Change of Variables Theorem

505

(3)

9.4 Polar Coordinates

508

(2)

9.5 Spherical and Cylindrical Coordinates

510

(3)

9.6 A Note on the Lebesgue Integral

513

(1)

9.7 Interchange of Limiting Operations

514

(7)

Theorem Proofs

521

(10)

Worked Examples

531

(4)

Exercises

535

(8)

10 FOURIER ANALYSIS

543

(120)

10.1 Inner Product Spaces

545

(6)

10.2 Orthogonal Families of Functions

551

(9)

10.3 Completeness and Convergence Theorems

560

(10)

10.4 Functions of Bounded Variation and Fejer Theory (Optional)

570

(3)

10.5 Computation of Fourier Series

573

(14)

10.6 Further Convergence Theorems

587

(6)

10.7 Applications

593

(12)

10.8 Fourier Integrals

605

(5)

10.9 Quantum Mechanical Formalism

610

(8)

Theorem Proofs

618

(26)

Worked Examples

644

(6)

Exercises

650

(13)

APPENDIX A: MISCELLANEOUS EXERCISES

663

(14)

APPENDIX B: REFERENCES AND SUGGESTIONS FOR FURTHER STUDY

677

(6)

APPENDIX C: ANSWERS AND SUGGESTIONS FOR SELECTED ODD-NUMBERED EXERCISES

683

(46)

INDEX

729

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Elementary Classical Analysis

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