Introduction to Analysis, Global Edition

Part I. ONE-DIMENSIONALTHEORY

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

 

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 

3. Functions on R

3.1 Two-sided limits

3.2 One-sided limits and limits atinfinity

3.3 Continuity

3.4 Uniform continuity

 

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'Hopital'sRule

4.5 Inverse function theorems 

5 Integrability on R

5.1 The Riemann integral

5.2 Riemann sums

5.3 The Fundamental Theorem ofCalculus

5.4 Improper Riemann integration

*5.5 Functions of boundedvariation

*5.6 Convex functions 

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 

7. Infinite Series of Functions

7.1 Uniform convergence ofsequences

7.2 Uniform convergence of series

7.3 Power series

7.4 Analytic functions

*7.5 Applications 

Part II. MULTIDIMENSIONAL THEORY 

8. Euclidean Spaces

8.1 Algebraic structure

8.2 Planes and lineartransformations

8.3 Topology of Rⁿ

8.4 Interior, closure, and boundary 

9. Convergence in Rⁿ

9.1 Limits of sequences

9.2 Heine-Borel Theorem

9.3 Limits of functions

9.4 Continuous functions

*9.5 Compact sets

*9.6 Applications 

10. Metric Spaces

10.1 Introduction

10.2 Limits of functions

10.3 Interior, closure, boundary

10.4 Compact sets

10.5 Connected sets

10.6 Continuous functions

10.7 Stone-Weierstrass Theorem 

11. Differentiability on Rⁿ

11.1 Partial derivatives andpartial integrals

11.2 The definition ofdifferentiability

11.3 Derivatives, differentials, andtangent planes

11.4 The Chain Rule

11.5 The Mean Value Theorem andTaylor's Formula

11.6 The Inverse Function Theorem

*11.7 Optimization 

12. Integration on Rⁿ

12.1 Jordan regions

12.2 Riemann integration on Jordanregions

12.3 Iterated integrals

12.4 Change of variables

*12.5 Partitions of unity

*12.6 The gamma function andvolume 

13. Fundamental Theorems of VectorCalculus

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 

*14. Fourier Series

*14.1 Introduction

*14.2 Summability of Fourierseries

*14.3 Growth of Fouriercoefficients

*14.4 Convergence of Fourierseries

*14.5 Uniqueness 

Appendices

A. Algebraic laws

B. Trigonometry

C. Matrices and determinants

D. Quadric surfaces

E. Vector calculus and physics

F. Equivalence relations 

References

Answers and Hints to Selected Exercises

Subject Index

Notation Index 

*Enrichment section