Problems and Solutions for Undergraduate Analysis

Preface vii 0 Sets and Mappings 1(8) 0.2 Mappings 1(2) 0.3 Natural Numbers and Induction 3(3) 0.4 Denumerable Sets 6(1) 0.5 Equivalence Relations 7(2) I Real Numbers 9(10) I.1 Algebraic Axioms 9(1) I.2 Ordering Axioms 10(3) I.3 Integers and Rational Numbers 13(2) I.4 The Completeness Axiom 15(4) II Limits and Continuous Functions 19(16) II.1 Sequences of Numbers 19(3) II.2 Functions and Limits 22(2) II.3 Limits with Infinity 24(5) II.4 Continuous Functions 29(6) III Differentiation 35(8) III.1 Properties of the Derivative 35(3) III.2 Mean Value Theorem 38(1) III.3 Inverse Functions 39(4) IV Elementary Functions 43(30) IV.1 Exponential 43(8) IV.2 Logarithm 51(14) IV.3 Sine and Cosine 65(6) IV.4 Complex Numbers 71(2) V The Elementary Real Integral 73(18) V.2 Properties of the Integral 73(7) V.3 Taylor's Formula 80(4) V.4 Asymptotic Estimates and Stirling's Formula 84(7) VI Normed Vector Spaces 91(20) VI.2 Normed Vector Spaces 91(5) VI.3 n-Space and Function Spaces 96(3) VI.4 Completeness 99(5) VI.5 Open and Closed Sets 104(7) VII Limits 111(14) VII.1 Basic Properties 111(2) VII.2 Continuous Maps 113(7) VII.3 Limits in Function Spaces 120(5) VIII Compactness 125(8) VIII.1 Basic Properties of Compact Sets 125(1) VIII.2 Continuous Maps on Compact Sets 126(3) VIII.4 Relation with Open Coverings 129(4) IX Series 133(32) IX.2 Series of Positive Numbers 133(13) IX.3 Non-Absolute Convergence 146(4) IX.5 Absolute and Uniform Convergence 150(6) IX.6 Power Series 156(4) IX.7 Differentiation and Integration of Series 160(5) X The Integral in One Variable 165(18) X.3 Approximation by Step Maps 165(5) X.4 Properties of the Integral 170(9) X.6 Relation Between the Integral and the Derivative 179(4) XI Approximation with Convolutions 183(6) XI.1 Dirac Sequences 183(2) XI.2 The Weierstrass Theorem 185(4) XII Fourier Series 189(28) XII.1 Hermitian Products and Orthogonality 189(10) XII.2 Trigonometric Polynomials as a Total Family 199(4) XII.3 Explicit Uniform Approximation 203(5) XII.4 Pointwise Convergence 208(9) XIII Improper Integrals 217(26) XIII.1 Definition 217(2) XIII.2 Criteria for Convergence 219(6) XIII.3 Interchanging Derivatives and Integrals 225(18) XIV The Fourier Integral 243(10) XIV.1 The Schwartz Space 243(4) XIV.2 The Fourier Inversion Formula 247(3) XIV.3 An Example of Fourier Transform Not in the Schwartz Space 250(3) XV Functions on n-Space 253(34) XV.1 Partial Derivatives 253(9) XV.2 Differentiability and the Chain Rule 262(4) XV.3 Potential Functions 266(1) XV.4 Curve Integrals 267(6) XV.5 Taylor's Formula 273(4) XV.6 Maxima and the Derivative 277(10) XVI The Winding Number and Global Potential Functions 287(6) XVI.2 The Winding Number and Homology 287(1) XVI.5 The Homotopy Form of the Integrability Theorem 288(2) XVI.6 More on Homotopies 290(3) XVII Derivatives in Vector Spaces 293(10) XVII.1 The Space of Continuous Linear Maps 293(2) XVII.2 The Derivative as a Linear Map 295(1) XVII.3 Properties of the Derivative 296(1) XVII.4 Mean Value Theorem 297(1) XVII.5 The Second Derivative 298(3) XVII.6 Higher Derivatives and Taylor's Formula 301(2) XVIII Inverse Mapping Theorem 303(24) XVIII.1 The Shrinking Lemma 303(7) XVIII.2 Inverse Mappings, Linear Case 310(8) XVIII.3 The Inverse Mapping Theorem 318(2) XVIII.5 Product Decompositions 320(7) XIX Ordinary Differential Equations 327(10) XIX.1 Local Existence and Uniqueness 327(4) XIX.3 Linear Differential Equations 331(6) XX Multiple Integrals 337(22) XX.1 Elementary Multiple Integration 337(6) XX.2 Criteria for Admissibility 343(2) XX.3 Repeated Integrals 345(1) XX.4 Change of Variables 346(12) XX.5 Vector Fields on Spheres 358(1) XXI Differential Forms 359 XXI.1 Definitions 359(3) XXI.2 Inverse Image of a Form 362(1) XXI.4 Stokes' Formula for Simplices 363