An Introduction to Analysis (Global Edition)

Cover
Half Title
Title Page
Copyright
Dedication
Contents
Preface
Acknowledgments for the Global Edition
Chapter 1. The Real Number System
	1.1 Introduction
	1.2 Ordered Field Axioms
	1.3 Completeness Axiom
	1.4 Mathematical Induction
	1.5 Inverse Functions and Images
	1.6 Countable and Uncountable Sets
Chapter 2. Sequences in R
	2.1 Limits of Sequences
	2.2 Limit Theorems
	2.3 Bolzano–Weierstrass Theorem
	2.4 Cauchy Sequences
	∗2.5 Limits Supremum and Infimum
Chapter 3. Functions on R
	3.1 Two-Sided Limits
	3.2 One-Sided Limits and Limits at Infinity
	3.3 Continuity
	3.4 Uniform Continuity
Chapter 4. Differentiability on R
	4.1 The Derivative
	4.2 Differentiability Theorems
	4.3 The Mean Value Theorem
	4.4 Taylor’s Theorem and l’Hôpital’s Rule
	4.5 Inverse Function Theorems
Chapter 5. Integrability on R
	5.1 The Riemann Integral
	5.2 Riemann Sums
	5.3 The Fundamental Theorem of Calculus
	5.4 Improper Riemann Integration
	∗5.5 Functions of Bounded Variation
	∗5.6 Convex Functions
Chapter 6. Infinite Series of Real Numbers
	6.1 Introduction
	6.2 Series with Nonnegative Terms
	6.3 Absolute Convergence
	6.4 Alternating Series
	∗6.5 Estimation of Series
	∗6.6 Additional Tests
Chapter 7. Infinite Series of Functions
	7.1 Uniform Convergence of Sequences
	7.2 Uniform Convergence of Series
	7.3 Power Series
	7.4 Analytic Functions
	∗7.5 Applications
Chapter 8. Euclidean Spaces
	8.1 Algebraic Structure
	8.2 Planes and Linear Transformations
Chapter 9. Convergence in Rn
	9.1 Topology of Rn
	9.2 Interior, Closure, and Boundary
	∗9.3 Compact Sets
	9.4 Heine–Borel Theorem
	9.5 Limits of Sequences
	9.6 Limits of Functions
	9.7 Continuous Functions
	∗9.8 Applications
Chapter 10. Metric Spaces
	10.1 Introduction
	10.2 Interior, Closure, and Boundary
	10.3 Compact Sets
	10.4 Connected Sets
	10.5 Limits of Functions
	10.6 Continuous Functions
	∗10.7 Stone–Weierstrass Theorem
Chapter 11. Differentiability on Rn
	11.1 Partial Derivatives and Partial Integrals
	11.2 The Definition of Differentiability
	11.3 Derivatives, Differentials, and Tangent Planes
	11.4 The Chain Rule
	11.5 The Mean Value Theorem and Taylor’s Formula
	11.6 The Inverse Function Theorem
	∗11.7 Optimization
Chapter 12. Integration on Rn
	12.1 Jordan Regions
	12.2 Riemann Integration on Jordan Regions
	12.3 Iterated Integrals
	12.4 Change of Variables
	∗12.5 Partitions of Unity
	∗12.6 The Gamma Function and Volume
Chapter 13. Fundamental Theorems of Vector Calculus
	13.1 Curves
	13.2 Oriented Curves
	13.3 Surfaces
	13.4 Oriented Surfaces
	13.5 Theorems of Green and Gauss
	13.6 Stokes’s Theorem
Chapter 14. Fourier Series
	∗14.1 Introduction
	∗14.2 Summability of Fourier Series
	∗14.3 Growth of Fourier Coefficients
	∗14.4 Convergence of Fourier Series
	∗14.5 Uniqueness
Appendices
	Appendix A. Algebraic Laws
	Appendix B. Trigonometry
	Appendix C. Matrices and Determinants
	Appendix D. Quadric Surfaces
	Appendix E. Vector Calculus and Physics
	Appendix F. Equivalence Relations
References
Answers and Hints to Selected Exercises
Subject Index
Notation Index