This is the second 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 taught in two quarters of twenty-five to thirty lectures each.
Author(s): Terence Tao
Series: Texts and Readings in Mathematics, 38
Edition: 4
Publisher: Springer
Year: 2023
Language: English
Pages: 220
City: Singapore
Tags: Metric Spaces; Continuous Functions; Uniform Convergence; Power Series; Fourier Series; Lebesgue Measure; Lebesgue Integration
Preface to the First Edition
Preface to Subsequent Editions
Contents
About the Author
1 Metric Spaces
1.1 Definitions and Examples
1.2 Some Point-Set Topology of Metric Spaces
1.3 Relative Topology
1.4 Cauchy Sequences and Complete Metric Spaces
1.5 Compact Metric Spaces
2 Continuous Functions on Metric Spaces
2.1 Continuous Functions
2.2 Continuity and Product Spaces
2.3 Continuity and Compactness
2.4 Continuity and Connectedness
2.5 Topological Spaces (Optional)
3 Uniform Convergence
3.1 Limiting Values of Functions
3.2 Pointwise and Uniform Convergence
3.3 Uniform Convergence and Continuity
3.4 The Metric of Uniform Convergence
3.5 Series of Functions; the Weierstrass M-Test
3.6 Uniform Convergence and Integration
3.7 Uniform Convergence and Derivatives
3.8 Uniform Approximation by Polynomials
4 Power Series
4.1 Formal Power Series
4.2 Real Analytic Functions
4.3 Abel's Theorem
4.4 Multiplication of Power Series
4.5 The Exponential and Logarithm Functions
4.6 A Digression on Complex Numbers
4.7 Trigonometric Functions
5 Fourier Series
5.1 Periodic Functions
5.2 Inner Products on Periodic Functions
5.3 Trigonometric Polynomials
5.4 Periodic Convolutions
5.5 The Fourier and Plancherel Theorems
6 Several Variable Differential Calculus
6.1 Linear Transformations
6.2 Derivatives in Several Variable Calculus
6.3 Partial and Directional Derivatives
6.4 The Several Variable Calculus Chain Rule
6.5 Double Derivatives and Clairaut's Theorem
6.6 The Contraction Mapping Theorem
6.7 The Inverse Function Theorem in Several Variable Calculus
6.8 The Implicit Function Theorem
7 Lebesgue Measure
7.1 The Goal: Lebesgue Measure
7.2 First Attempt: Outer Measure
7.3 Outer Measure Is not Additive
7.4 Measurable Sets
7.5 Measurable Functions
8 Lebesgue Integration
8.1 Simple Functions
8.2 Integration of Non-negative Measurable Functions
8.3 Integration of Absolutely Integrable Functions
8.4 Comparison with the Riemann Integral
8.5 Fubini's Theorem
Index
