The Way of Analysis Robert S. Strichartz Mathematics Department Cornell University Ithaca, New York Jones and Bartlett Publishers Boston London
Contents Preface xiii 1 Preliminaries 1 1.1 The Logic of Quantifiers 1 1.1.1 Rules of Quantifiers 1 1.1.2 Examples 4 1.1.3 Exercises 7 1.2 Infinite Sets 8 1.2.1 Countable Sets 8 1.2.2 Uncountable Sets 10 1.2.3 Exercises 13 1.3 Proofs 13 1.3.1 How to Discover Proofs 13 1.3.2 How to Understand Proofs 17 1.4 The Rational Number System 18 1.5 The Axiom of Choice* 21 2 Construction of the Real Number System 25 2.1 Cauchy Sequences 25 2.1.1 Motivation 25 2.1.2 The Definition 30 2.1.3 Exercises 37 2.2 The Reals as an Ordered Field 38 2.2.1 Defining Arithmetic 38 2.2.2 The Field Axioms 41 2.2.3 Order 45 2.2.4 Exercises 48
vi Contents 2.3 Limits and Completeness 50 2.3.1 Proof of Completeness 50 2.3.2 Square Roots 52 2.3.3 Exercises 54 2.4 Other Versions and Visions 56 2.4.1 Infinite Decimal Expansions 56 2.4.2 Dedekind Cuts* 59 2.4.3 Non-Standard Analysis* 63 2.4.4 Constructive Analysis* 66 2.4.5 Exercises 68 2.5 Summary 69 3 Topology of the Real Line 73 3.1 The Theory of Limits 73 3.1.1 Limits, Sups, and Infs 73 3.1.2 Limit Points 78 3.1.3 Exercises 84 3.2 Open Sets and Closed Sets 86 3.2.1 Open Sets 86 3.2.2 Closed Sets 91 3.2.3 Exercises 98 3.3 Compact Sets 99 3.3.1 Exercises 106 3.4 Summary 107 4 Continuous Functions 111 4.1 Concepts of Continuity Ill 4.1.1 Definitions Ill 4.1.2 Limits of Functions and Limits of Sequences... 119 4.1.3 Inverse Images of Open Sets 121 4.1.4 Related Definitions 123 4.1.5 Exercises 125 4.2 Properties of Continuous Functions 127 4.2.1 Basic Properties 127 4.2.2 Continuous Functions on Compact Domains... 131 4.2.3 Monotone Functions 134 4.2.4 Exercises 138 4.3 Summary 140
Contents vii 5 Differential Calculus 143 5.1 Concepts of the Derivative 143 5.1.1 Equivalent Definitions 143 5.1.2 Continuity and Continuous Differentiability... 148 5.1.3 Exercises 152 5.2 Properties of the Derivative 153 5.2.1 Local Properties 153 5.2.2 Intermediate Value and Mean Value Theorems. 157 5.2.3 Global Properties, 162 5.2.4 Exercises 163 5.3 The Calculus of Derivatives. 165 5.3.1 Product and Quotient Rules 165 5.3.2 The Chain Rule 168 5.3.3 Inverse Function Theorem 171 5.3.4 Exercises 176 5.4 Higher Derivatives and Taylor's Theorem 177 5.4.1 Interpretations of the Second Derivative 177 5.4.2 Taylor's Theorem 181 5.4.3 L'Hopital's Rule* 185 5.4.4 Lagrange Remainder Formula* 188 5.4.5 Orders of Zeros* 190 5.4.6 Exercises 192 5.5 Summary 195 6 Integral Calculus 201 6.1 Integrals of Continuous Functions 201 6.1.1 Existence of the Integral 201 6.1.2 Fundamental Theorems of Calculus 207 6.1.3 Useful Integration Formulas 212 6.1.4 Numerical Integration 214 6.1.5 Exercises 217 6.2 The Riemann Integral 219 6.2.1 Definition of the Integral 219 6.2.2 Elementary Properties of the Integral 224 6.2.3 Functions with a Countable Number of Discontinuities* 227 6.2.4 Exercises 231 6.3 Improper Integrals* 232
viii Contents 6.3.1 Definitions and Examples 232 6.3.2 Exercises 235 6.4 Summary 236 7 Sequences and Series of Functions 241 7.1 Complex Numbers 241 7.1.1 Basic Properties of C 241 7.1.2 Complex-Valued Functions 247 7.1.3 Exercises 249 7.2 Numerical Series and Sequences 250 7.2.1 Convergence and Absolute Convergence 250 7.2.2 Rearrangements 256 7.2.3 Summation by Parts* 260 7.2.4 Exercises 262 7.3 Uniform Convergence 263 7.3.1 Uniform Limits and Continuity 263 7.3.2 Integration and Differentiation of Limits 268 7.3.3 Unrestricted Convergence* 272 7.3.4 Exercises 274 7.4 Power Series 276 7.4.1 The Radius of Convergence 276 7.4.2 Analytic Continuation 281 7.4.3 Analytic Functions on Complex Domains*... 286 7.4.4 Closure Properties of Analytic Functions*... 288 7.4.5 Exercises 294 7.5 Approximation by Polynomials 296 7.5.1 Lagrange Interpolation 296 7.5.2 Convolutions and Approximate Identities... 297 7.5.3 The Weierstrass Approximation Theorem... 301 7.5.4 Approximating Derivatives 305 7.5.5 Exercises 307 7.6 Equicontinuity 309 7.6.1 The Definition of Equicontinuity 309 7.6.2 The Arzela-Ascoli Theorem 312 7.6.3 Exercises 314 7.7 Summary 316
Contents ix 8 Transcendental Functions 323 8.1 The Exponential and Logarithm 323 8.1.1 Five Equivalent Definitions 323 8.1.2 Exponential Glue and Blip Functions 329 8.1.3 Functions with Prescribed Taylor Expansions*. 332 8.1.4 Exercises 335 8.2 Trigonometric Functions 337 8.2.1 Definition of Sine and Cosine 337 8.2.2 Relationship Between Sines, Cosines, and Complex Exponentials 344 8.2.3 Exercises 349 8.3 Summary 350 9 Euclidean Space and Metric Spaces 355 9.1 Structures on Euclidean Space 355 9.1.1 Vector Space and Metric Space 355 9.1.2 Norm and Inner Product 358 9.1.3 The Complex Case 364 9.1.4 Exercises 366 9.2 Topology of Metric Spaces 368 9.2.1 Open Sets 368 9.2.2 Limits and Closed Sets 373 9.2.3 Completeness 374 9.2.4 Compactness 377 9.2.5 Exercises 384 9.3 Continuous Functions on Metric Spaces 386 9.3.1 Three Equivalent Definitions 386 9.3.2 Continuous Functions on Compact Domains... 391 9.3.3 Connectedness 393 9.3.4 The Contractive Mapping Principle,397 9.3.5 The Stone-Weierstrass Theorem* 399 9.3.6 Nowhere Differentiate Functions, and Worse*. 403 9.3.7 Exercises 409 9.4 Summary 412 10 Differential Calculus in Euclidean Space 419 10.1 The Differential 419 10.1.1 Definition of Differentiability 419
x Contents 10.1.2 Partial Derivatives 423 10.1.3 The Chain Rule 428 10.1.4 Differentiation of Integrals 432 10.1.5 Exercises 435 10.2 Higher Derivatives 437 10.2.1 Equality of Mixed Partials 437 10.2.2 Local Extrema 441 10.2.3 Taylor Expansions 448 10.2.4 Exercises 452 10.3 Summary 454 11 Ordinary Differential Equations 459 11.1 Existence and Uniqueness 459 11.1.1 Motivation 459 11.1.2 Picard Iteration 467 11.1.3 Linear Equations 473 11.1.4 Local Existence and Uniqueness* 476 11.1.5 Higher Order Equations* 481 11.1.6 Exercises 483 11.2 Other Methods of Solution* 485 11.2.1 Difference Equation Approximation 485 11.2.2 Peano Existence Theorem 490 11.2.3 Power-Series Solutions 494 11.2.4 Exercises 500 11.3 Vector Fields and Flows* 501 11.3.1 Integral Curves 501 11.3.2 Hamiltonian Mechanics 505 11.3.3 First-Order Linear P.D.E.'s 506 11.3.4 Exercises 507 11.4 Summary 509 12 Fourier Series 515 12.1 Origins of Fourier Series 515 12.1.1 Fourier Series Solutions of P.D.E.'s 515 12.1.2 Spectral Theory 520 12.1.3 Harmonic Analysis 525 12.1.4 Exercises 528 12.2 Convergence of Fourier Series 531
Contents xi 12.2.1 Uniform Convergence for C 1 Functions... 531 12.2.2 Summability of Fourier Series 537 12.2.3 Convergence in the Mean 543 12.2.4 Divergence and Gibb's Phenomenon* 550 12.2.5 Solution of the Heat Equation* 555 12.2.6 Exercises 559 12.3 Summary 562 13 Implicit Functions, Curves, and Surfaces 567 13.1 The Implicit Function Theorem 567 13.1.1 Statement of the Theorem 567 13.1.2 The Proof* 573 13.1.3 Exercises 580 13.2 Curves and Surfaces 581 13.2.1 Motivation and Examples 581 13.2.2 Immersions and Embeddings 585 13.2.3 Parametric Description of Surfaces 591 13.2.4 Implicit Description of Surfaces 597 13.2.5 Exercises 600 13.3 Maxima and Minima on Surfaces 602 13.3.1 Lagrange Multipliers 602 13.3.2 A Second Derivative Test* 605 13.3.3 Exercises 609 13.4 Arc Length... 610 13.4.1 Rectifiable Curves 610 13.4.2 The Integral Formula for Arc Length 614 13.4.3 Arc Length Parameterization* 616 13.4.4 Exercises 617 13.5 Summary 618 14 The Lebesgue Integral 623 14.1 The Concept of Measure 623 14.1.1 Motivation 623 14.1.2 Properties of Length 627 14.1.3 Measurable Sets 631 14.1.4 Basic Properties of Measures 634 14.1.5 A Formula for Lebesgue Measure 636 14.1.6 Other Examples of Measures 639
xii Contents 14.1.7 Exercises 641 14.2 Proof of Existence of Measures* 643 14.2.1 Outer Measures 643 14.2.2 Metric Outer Measure 647 14.2.3 Hausdorff Measures* 650 14.2.4 Exercises 654 14.3 The Integral 655 14.3.1 Non-negative Measurable Functions 655 14.3.2 The Monotone Convergence Theorem 660 14.3.3 Integrable Functions 664 14.3.4 Almost Everywhere 667 14.3.5 Exercises 668 14.4 The Lebesgue Spaces L 1 and L 2 670 14.4.1 L 1 as a Banach Space 670 14.4.2 L 2 as a Hilbert Space 673 14.4.3 Fourier Series for L 2 Functions 676 14.4.4 Exercises 681 14.5 Summary 682 15 Multiple Integrals 691 15.1 Interchange of Integrals 691 15.1.1 Integrals of Continuous Functions 691 15.1.2 Fubini's Theorem 694 15.1.3 The Monotone Class Lemma* 700 15.1.4 Exercises 703 15.2 Change of Variable in Multiple Integrals 705 15.2.1 Determinants and Volume 705 15.2.2 The Jacobian Factor* 709 15.2.3 Polar Coordinates 714 15.2.4 Change of Variable for Lebesgue Integrals*... 717 15.2.5 Exercises 720 15.3 Summary 722 Index 727