From Calculus to Analysis

Description

This innovative textbook bridges the gap between undergraduate analysis and graduate measure theory by guiding students from the classical foundations of analysis to more modern topics like metric spaces and Lebesgue integration. Designed for a two-semester introduction to real analysis, the text gives special attention to metric spaces and topology to familiarize students with the level of abstraction and mathematical rigor needed for graduate study in real analysis. Fitting in between analysis textbooks that are too formal or too casual, From Classical to Modern Analysis is a comprehensive, yet straightforward, resource for studying real analysis. To build the foundational elements of real analysis, the first seven chapters cover number systems, convergence of sequences and series, as well as more advanced topics like superior and inferior limits, convergence of functions, and metric spaces. Chapters 8 through 12 explore topology in and continuity on metric spaces and introduce the Lebesgue integrals. The last chapters are largely independent and discuss various applications of the Lebesgue integral. Instructors who want to demonstrate the uses of measure theory and explore its advanced applications with their undergraduate students will find this textbook an invaluable resource. Advanced single-variable calculus and a familiarity with reading and writing mathematical proofs are all readers will need to follow the text. Graduate students can also use this self-contained and comprehensive introduction to real analysis for self-study and review. Cover......Page 1 From Calculus to Analysis......Page 4 Copyright......Page 5 Preface......Page 8 Contents......Page 10 1.1 The Algebra of the Reals......Page 12 Exercises......Page 15 1.2 Natural Numbers and Integers......Page 17 Exercises......Page 22 1.3 Rational Numbers and Real Numbers......Page 24 Exercises......Page 32 1.4 Power Functions......Page 34 Exercises......Page 42 2.1 Sequences......Page 44 Exercises......Page 49 2.2 Monotone Sequences, Bolzano-Weierstrass Theorem, and Operations on Limits......Page 51 Exercises......Page 55 2.3 Series......Page 56 Exercises......Page 64 2.4 Absolute Convergence......Page 66 Exercises......Page 73 3.1 Power Series......Page 76 3.2 Trigonometric Functions......Page 78 Exercises......Page 88 3.3 Inverse Trigonometric Functions......Page 90 Exercises......Page 92 3.4 Exponential and Logarithmic Functions......Page 93 Exercises......Page 103 4.1 Power Series Expansions......Page 106 Exercises......Page 118 4.2 Wallis' Integrals, Euler's Formula, and Stirling's Formula......Page 119 Exercises......Page 129 4.3 Convergence of Infinite Products......Page 131 Exercises......Page 140 4.4 The Number π Is Irrational......Page 142 Exercises......Page 145 5.1 Continuity......Page 148 Exercises......Page 157 5.2 Limits of Functions and Derivatives......Page 158 Exercises......Page 166 5.3 Algebra of Derivatives and Mean Value Theorems......Page 168 Exercises......Page 176 5.4 Intervals, Continuity, and Inverse Functions......Page 178 Exercises......Page 185 6.1 Construction of the Integral......Page 188 Exercises......Page 199 6.2 Properties of the Integral......Page 200 Exercises......Page 210 6.3 Uniform Continuity......Page 212 Exercises......Page 216 7: Convergence of Functions......Page 218 Power Series......Page 224 Exercises......Page 228 Existence......Page 232 Uniqueness......Page 234 Exercises......Page 243 9: Countable and Uncountable Sets......Page 246 Exercises......Page 254 Further Reading......Page 256 List of Mathematicians Cited in the Text......Page 258 Index......Page 260