The Way of Analysis. Robert S. Strichartz. Jones and Bartlett Publishers. Mathematics Department Cornell University Ithaca, New York

Transcription

1 The Way of Analysis Robert S. Strichartz Mathematics Department Cornell University Ithaca, New York Jones and Bartlett Publishers Boston London

2 Contents Preface xiii 1 Preliminaries The Logic of Quantifiers Rules of Quantifiers Examples Exercises Infinite Sets Countable Sets Uncountable Sets Exercises Proofs How to Discover Proofs How to Understand Proofs The Rational Number System The Axiom of Choice* 21 2 Construction of the Real Number System Cauchy Sequences Motivation The Definition Exercises The Reals as an Ordered Field Defining Arithmetic The Field Axioms Order Exercises 48

3 vi Contents 2.3 Limits and Completeness Proof of Completeness Square Roots Exercises Other Versions and Visions Infinite Decimal Expansions Dedekind Cuts* Non-Standard Analysis* Constructive Analysis* Exercises Summary 69 3 Topology of the Real Line The Theory of Limits Limits, Sups, and Infs Limit Points Exercises Open Sets and Closed Sets Open Sets Closed Sets Exercises Compact Sets Exercises Summary Continuous Functions Concepts of Continuity Ill Definitions Ill Limits of Functions and Limits of Sequences Inverse Images of Open Sets Related Definitions Exercises Properties of Continuous Functions Basic Properties Continuous Functions on Compact Domains Monotone Functions Exercises Summary 140

4 Contents vii 5 Differential Calculus Concepts of the Derivative Equivalent Definitions Continuity and Continuous Differentiability Exercises Properties of the Derivative Local Properties Intermediate Value and Mean Value Theorems Global Properties, Exercises The Calculus of Derivatives Product and Quotient Rules The Chain Rule Inverse Function Theorem Exercises Higher Derivatives and Taylor's Theorem Interpretations of the Second Derivative Taylor's Theorem L'Hopital's Rule* Lagrange Remainder Formula* Orders of Zeros* Exercises Summary Integral Calculus Integrals of Continuous Functions Existence of the Integral Fundamental Theorems of Calculus Useful Integration Formulas Numerical Integration Exercises The Riemann Integral Definition of the Integral Elementary Properties of the Integral Functions with a Countable Number of Discontinuities* Exercises Improper Integrals* 232

5 viii Contents Definitions and Examples Exercises Summary Sequences and Series of Functions Complex Numbers Basic Properties of C Complex-Valued Functions Exercises Numerical Series and Sequences Convergence and Absolute Convergence Rearrangements Summation by Parts* Exercises Uniform Convergence Uniform Limits and Continuity Integration and Differentiation of Limits Unrestricted Convergence* Exercises Power Series The Radius of Convergence Analytic Continuation Analytic Functions on Complex Domains* Closure Properties of Analytic Functions* Exercises Approximation by Polynomials Lagrange Interpolation Convolutions and Approximate Identities The Weierstrass Approximation Theorem Approximating Derivatives Exercises Equicontinuity The Definition of Equicontinuity The Arzela-Ascoli Theorem Exercises Summary 316

6 Contents ix 8 Transcendental Functions The Exponential and Logarithm Five Equivalent Definitions Exponential Glue and Blip Functions Functions with Prescribed Taylor Expansions* Exercises Trigonometric Functions Definition of Sine and Cosine Relationship Between Sines, Cosines, and Complex Exponentials Exercises Summary Euclidean Space and Metric Spaces Structures on Euclidean Space Vector Space and Metric Space Norm and Inner Product The Complex Case Exercises Topology of Metric Spaces Open Sets Limits and Closed Sets Completeness Compactness Exercises Continuous Functions on Metric Spaces Three Equivalent Definitions Continuous Functions on Compact Domains Connectedness The Contractive Mapping Principle, The Stone-Weierstrass Theorem* Nowhere Differentiate Functions, and Worse* Exercises Summary Differential Calculus in Euclidean Space The Differential Definition of Differentiability 419

7 x Contents Partial Derivatives The Chain Rule Differentiation of Integrals Exercises Higher Derivatives Equality of Mixed Partials Local Extrema Taylor Expansions Exercises Summary Ordinary Differential Equations Existence and Uniqueness Motivation Picard Iteration Linear Equations Local Existence and Uniqueness* Higher Order Equations* Exercises Other Methods of Solution* Difference Equation Approximation Peano Existence Theorem Power-Series Solutions Exercises Vector Fields and Flows* Integral Curves Hamiltonian Mechanics First-Order Linear P.D.E.'s Exercises Summary Fourier Series Origins of Fourier Series Fourier Series Solutions of P.D.E.'s Spectral Theory Harmonic Analysis Exercises Convergence of Fourier Series 531

8 Contents xi Uniform Convergence for C 1 Functions Summability of Fourier Series Convergence in the Mean Divergence and Gibb's Phenomenon* Solution of the Heat Equation* Exercises Summary Implicit Functions, Curves, and Surfaces The Implicit Function Theorem Statement of the Theorem The Proof* Exercises Curves and Surfaces Motivation and Examples Immersions and Embeddings Parametric Description of Surfaces Implicit Description of Surfaces Exercises Maxima and Minima on Surfaces Lagrange Multipliers A Second Derivative Test* Exercises Arc Length Rectifiable Curves The Integral Formula for Arc Length Arc Length Parameterization* Exercises Summary The Lebesgue Integral The Concept of Measure Motivation Properties of Length Measurable Sets Basic Properties of Measures A Formula for Lebesgue Measure Other Examples of Measures 639

9 xii Contents Exercises Proof of Existence of Measures* Outer Measures Metric Outer Measure Hausdorff Measures* Exercises The Integral Non-negative Measurable Functions The Monotone Convergence Theorem Integrable Functions Almost Everywhere Exercises The Lebesgue Spaces L 1 and L L 1 as a Banach Space L 2 as a Hilbert Space Fourier Series for L 2 Functions Exercises Summary Multiple Integrals Interchange of Integrals Integrals of Continuous Functions Fubini's Theorem The Monotone Class Lemma* Exercises Change of Variable in Multiple Integrals Determinants and Volume The Jacobian Factor* Polar Coordinates Change of Variable for Lebesgue Integrals* Exercises Summary 722 Index 727