Table Of Content

Contents Page: vii Preface to the second and third editions Page: ix Preface to the first edition Page: xi About the Author Page: xvii 1 Metric spaces Page: 1 1.1 Definitions and examples Page: 1 1.2 Some point-set topology of metric spaces Page: 10 1.3 Relative topology Page: 15 1.4 Cauchy sequences and complete metric spaces Page: 17 1.5 Compact metric spaces Page: 21 2 Continuous functions on metric spaces Page: 28 2.1 Continuous functions Page: 28 2.2 Continuity and product spaces Page: 31 2.3 Continuity and compactness Page: 34 2.4 Continuity and connectedness Page: 36 2.5 Topological spaces (Optional) Page: 39 3 Uniform convergence Page: 45 3.1 Limiting values of functions Page: 46 3.2 Pointwise and uniform convergence Page: 48 3.3 Uniform convergence and continuity Page: 53 3.4 The metric of uniform convergence Page: 56 3.5 Series of functions; the Weierstrass M-test Page: 58 3.6 Uniform convergence and integration Page: 61 3.7 Uniform convergence and derivatives Page: 63 3.8 Uniform approximation by polynomials Page: 66 4 Power series Page: 75 4.1 Formal power series Page: 75 4.2 Real analytic functions Page: 78 4.3 Abel’s theorem Page: 83 4.4 Multiplication of power series Page: 87 4.5 The exponential and logarithm functions Page: 90 4.6 A digression on complex numbers Page: 93 4.7 Trigonometric functions Page: 101 5 Fourier series Page: 107 5.1 Periodic functions Page: 108 5.2 Inner products on periodic functions Page: 110 5.3 Trigonometric polynomials Page: 114 5.4 Periodic convolutions Page: 116 5.5 The Fourier and Plancherel theorems Page: 121 6 Several variable differential calculus Page: 127 6.1 Linear transformations Page: 127 6.2 Derivatives in several variable calculus Page: 134 6.3 Partial and directional derivatives Page: 137 6.4 The several variable calculus chain rule Page: 144 6.5 Double derivatives and Clairaut’s theorem Page: 147 6.6 The contraction mapping theorem Page: 149 6.7 The inverse function theorem in several variable calculus Page: 152 6.8 The implicit function theorem Page: 157 7 Lebesgue measure Page: 162 7.1 The goal: Lebesgue measure Page: 164 7.2 First attempt: Outer measure Page: 166 7.3 Outer measure is not additive Page: 174 7.4 Measurable sets Page: 176 7.5 Measurable functions Page: 183 8 Lebesgue integration Page: 187 8.1 Simple functions Page: 187 8.2 Integration of non-negative measurable functions Page: 193 8.3 Integration of absolutely integrable functions Page: 201 8.4 Comparison with the Riemann integral Page: 205 8.5 Fubini’s theorem Page: 207 Index Page: 213 Texts and Readings in Mathematics Page: 219

Description:

This is part two of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and then finally the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. The book also has appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of 25–30 lectures each. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.