Analysis I

Description

This is the first book of a two-volume textbook on real analysis. Both the volumes—Analysis I and Analysis II—are intended for honors undergraduates who have already been exposed to calculus. The emphasis is on rigor and foundations. The material starts at the very beginning—the construction of number systems and set theory (Analysis I, Chaps. 1–5), then on to the basics of analysis such as limits, series, continuity, differentiation, and Riemann integration (Analysis I, Chaps. 6–11 on Euclidean spaces, and Analysis II, Chaps. 1–3 on metric spaces), through power series, several variable calculus, and Fourier analysis (Analysis II, Chaps. 4–6), and finally to the Lebesgue integral (Analysis II, Chaps. 7–8). There are appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) is in two quarters of twenty-five to thirty lectures each. Preface to the First Edition Preface to Subsequent Editions Contents About the Author 1 Introduction 1.1 What Is Analysis? 1.2 Why Do Analysis? 2 Starting at the Beginning: The Natural Numbers 2.1 The Peano Axioms 2.2 Addition 2.3 Multiplication 3 Set Theory 3.1 Fundamentals 3.2 Russell's Paradox (Optional) 3.3 Functions 3.4 Images and Inverse Images 3.5 Cartesian Products 3.6 Cardinality of Sets 4 Integers and Rationals 4.1 The Integers 4.2 The Rationals 4.3 Absolute Value and Exponentiation 4.4 Gaps in the Rational Numbers 5 The Real Numbers 5.1 Cauchy Sequences 5.2 Equivalent Cauchy Sequences 5.3 The Construction of the Real Numbers 5.4 Ordering the Reals 5.5 The Least Upper Bound Property 5.6 Real Exponentiation, Part I 6 Limits of Sequences 6.1 Convergence and Limit Laws 6.2 The Extended Real Number System 6.3 Suprema and Infima of Sequences 6.4 Limsup, Liminf, and Limit Points 6.5 Some Standard Limits 6.6 Subsequences 6.7 Real Exponentiation, Part II 7 Series 7.1 Finite Series 7.2 Infinite Series 7.3 Sums of Non-negative Numbers 7.4 Rearrangement of Series 7.5 The Root and Ratio Tests 8 Infinite Sets 8.1 Countability 8.2 Summation on Infinite Sets 8.3 Uncountable Sets 8.4 The Axiom of Choice 8.5 Ordered Sets 9 Continuous Functions on R 9.1 Subsets of the Real Line 9.2 The Algebra of Real-Valued Functions 9.3 Limiting Values of Functions 9.4 Continuous Functions 9.5 Left and Right Limits 9.6 The Maximum Principle 9.7 The Intermediate Value Theorem 9.8 Monotonic Functions 9.9 Uniform Continuity 9.10 Limits at Infinity 10 Differentiation of Functions 10.1 Basic Definitions 10.2 Local Maxima, Local Minima, and Derivatives 10.3 Monotone Functions and Derivatives 10.4 Inverse Functions and Derivatives 10.5 L'Hôpital's Rule 11 The Riemann Integral 11.1 Partitions 11.2 Piecewise Constant Functions 11.3 Upper and Lower Riemann Integrals 11.4 Basic Properties of the Riemann Integral 11.5 Riemann Integrability of Continuous Functions 11.6 Riemann Integrability of Monotone Functions 11.7 A Non-riemann Integrable Function 11.8 The Riemann–Stieltjes Integral 11.9 The Two Fundamental Theorems of Calculus 11.10 Consequences of the Fundamental Theorems Appendix A The Basics of Mathematical Logic A.1 Mathematical Statements A.2 Implication A.3 The Structure of Proofs A.4 Variables and Quantifiers A.5 Nested Quantifiers A.6 Some Examples of Proofs and Quantifiers A.7 Equality Appendix B The Decimal System B.1 The Decimal Representation of Natural Numbers B.2 The Decimal Representation of Real Numbers Index