Mathematical Analysis: A Concise Introduction

Preface xi

Part I: Analysis of Functions of a Single Real Variable

1 The Real Numbers 1

1.1 Field Axioms 1

1.2 Order Axioms 4

1.3 Lowest Upper and Greatest Lower Bounds 8

1.4 Natural Numbers, Integers, and Rational Numbers 11

1.5 Recursion, Induction, Summations, and Products 17

2 Sequences of Real Number V 25

2.1 Limits 25

2.2 Limit Laws 30

2.3 Cauchy Sequences 36

2.4 Bounded Sequences 40

2.5 Infinite Limits 44

3 Continuous Functions 49

3.1 Limits of Functions 49

3.2. Limit Laws 52

3.3 One-Sided Limits and Infinite Limits 56

3.4 Continuity 59

3.5 Properties of Continuous Functions 66

3.6 Limits at Infinity 69

4 Differentiable Functions 71

4.1 Differentiability 71

4.2 Differentiation Rules 74

4.3 Rolle's Theorem and the Mean Value Theorem 80

5 The Riemann Integral I 85

5.1 Riemann Sums and the Integral 85

5.2 Uniform Continuity and Integrability of Continuous Functions 91

5.3 The Fundamental Theorem of Calculus 95

5.4 The Darboux Integral 97

6 Series of Real Numbers I 101

6.1 Series as a Vehicle to Define Infinite Sums 101

6.2 Absolute Convergence and Unconditional Convergence 108

7 Some Set Theory 117

7.1 The Algebra of Sets 117

7.2 Countable Sets 122

7.3 Uncountable Sets 124

8 The Riemann Integral II 127

8.1 Outer Lebesgue Measure 127

8.2 Lebesgue's Criterion for Riemann Integrability 131

8.3 More Integral Theorems 136

8.4 Improper Riemann Integrals 140

9 The Lebesgue Integral 145

9.1 Lebesgue Measurable Sets 147

9.2 Lebesgue Measurable Functions 153

9.3 Lebesgue Integration 158

9.4 Lebesgue Integrals versus Riemann Integrals 165

10 Series of Real Numbers II 169

10.1 Limits Superior and Inferior 169

10.2 The Root Test and the Ratio Test 172

10.3 Power Series 175

11 Sequences of Functions 179

11.1 Notions of Convergence 179

11.2 Uniform Convergence 182

12 Transcendental Functions 189

12.1 The Exponential Function 189

12.2 Sine and Cosine 193

12.3 L.' Hôpital's Rule 199

13 Numerical Methods 203

13.1 Approximation with Taylor Polynomials 204

13.2 Newton's Method 208

13.3 Numerical Integration 214

Part II: Analysis in Abstract Spaces

14 Integration on Measure Spaces 225

14.1 Measure Spaces 225

14.2 Outer Measures 230

14.3 Measurable Functions 234

14.4 Integration of Measurable Functions 235

14.5 Monotone and Dominated Convergence 238

14.6 Convergence in Mean, in Measure, and Almost Everywhere 242

14.7 Product Ϭ-Algebras 245

14.8 Product Measures and Fubini's Theorem 251

15 The Abstract Venues for Analysis 255

15.1 Abstraction I: Vector Spaces 255

15.2 Representation of Elements; Bases and Dimension 259

15.3 Identification of Spaces: Isomorphism 262

15.4 Abstraction II: Inner Product Spaces 264

15.5 Nicer Representations: Orthonormal Sets 267

15.6 Abstraction III: Norrned Spaces 269

15.7 Abstraction IV: Metric Spaces 275

15.8 L^P Spaces 278

15.9 Another Number Field: Complex Numbers 281

16 The Topology of Metric Spaces 287

16.1 Convergence of Sequences 287

16.2 Completeness 291

16.3 Continuous Functions 296

16.4 Open and Closed Sets 301

16.5 Compactness 309

16.6 The Normed Topology of R^d 316

16.7 Dense Subspaces 322

16.8 Connectedness 330

16.9 Locally Compact Spaces 333

17 Differentiation in Normed Spaces 341

17.1 Continuous Linear Functions 342

17.2 Matrix Representation of Linear Functions 348

17.3 Differentiability 353

17.4 The Mean Value Theorem 360

17.5 How Partial Derivatives Fit In 362

17.6 Multilinear Functions (Tensors) 369

17.7 Higher Derivatives 373

17.8 The. Implicit Function Theorem 380

18 Measure, Topology, and Differentiation 385

18.1 Lebesgue Measurable Sets in R^d 385

18.2 C^ꝏ and Approximation of Integrable Functions 391

18.3 Tensor Algebra and Determinants 397

18.4 Multidimensional Substitution 407

19 Introduction to Differential Geometry 421

19.1 Manifolds 421

19.2 Tangent Spaces and Differentiable Functions 427

19.3 Differential Forms, Integrals Over the Unit Cube 434

19.4 k-Forms and Integrals Over k-Chains 443

19.5 Integration on Manifolds 452

19.6 Stokes' Theorem 458

20 Hilbert Spaces 463

20.1 Orthonormal Bases 463

20.2 Fourier Series 467

20.3 The Riesz Representation Theorem 475

Part III: Applied Analysis

21 Physics Background 483

21.1 Harmonic Oscillators 484

21.2 Heat and Diffusion 486

21.3 Separation of Variables, Fourier Series, and Ordinary Differential Equa-tions 490

21.4 Maxwell's Equations 493

21.5 The Navier Stokes Equation for the Conservation of Mass 496

22 Ordinary Differential Equations 505

22.1 Burwell Space Valued Differential Equations 505

22.2 An Existence and Uniqueness Theorem 508

22.3 Linear Differential Equations 510

23 The Finite Element Method 513

23.1 Ritz-Galerkin Approximation 513

23.2 Wealth Differentiable Functions 518

23,3 Sobolev Spaces 524

23.4 Elliptic Differential Operators 532

23.5 Finite Elements 536

Conclusion and Outlook 544

Appendices

A Logic 545

A.1 Statements 545

A.2 Negations 546

B Set Theory 547

B. 1 The Zermelo-Fraenkel Axioms 547

B.2 Relations and Functions 548

C Natural Numbers, Integers, and Rational Numbers 549

C.1 The Natural Numbers 549

C.2 The Integers 550

C.3 The Rational Numbers 550

Bibliography 551

Index 553