Principles of Mathematical Analysis

Chapter 1: The Real and Complex Number Systems

Introduction

Ordered Sets

Fields

The Real Field

The Extended Real Number System

The Complex Field

Euclidean Spaces

Appendix

Exercises

Chapter 2: Basic Topology

Finite, Countable, and Uncountable Sets

Metric Spaces

Compact Sets

Perfect Sets

Connected Sets

Exercises

Chapter 3: Numerical Sequences and Series

Convergent Sequences

Subsequences

Cauchy Sequences

Upper and Lower Limits

Some Special Sequences

Series

Series of Nonnegative Terms

The Number e

The Root and Ratio Tests

Power Series

Summation by Parts

Absolute Convergence

Addition and Multiplication of Series

Rearrangements

Exercises

Chapter 4: Continuity

Limits of Functions

Continuous Functions

Continuity and Compactness

Continuity and Connectedness

Discontinuities

Monotonic Functions

Infinite Limits and Limits at Infinity

Exercises

Chapter 5: Differentiation

The Derivative of a Real Function

Mean Value Theorems

The Continuity of Derivatives

L'Hospital's Rule

Derivatives of Higher-Order

Taylor's Theorem

Differentiation of Vector-valued Functions

Exercises

Chapter 6: The Riemann-Stieltjes Integral

Definition and Existence of the Integral

Properties of the Integral

Integration and Differentiation

Integration of Vector-valued Functions

Rectifiable Curves

Exercises

Chapter 7: Sequences and Series of Functions

Discussion of Main Problem

Uniform Convergence

Uniform Convergence and Continuity

Uniform Convergence and Integration

Uniform Convergence and Differentiation

Equicontinuous Families of Functions

The Stone-Weierstrass Theorem

Exercises

Chapter 8: Some Special Functions

Power Series

The Exponential and Logarithmic Functions

The Trigonometric Functions

The Algebraic Completeness of the Complex Field

Fourier Series

The Gamma Function

Exercises

Chapter 9: Functions of Several Variables

Linear Transformations

Differentiation

The Contraction Principle

The Inverse Function Theorem

The Implicit Function Theorem

The Rank Theorem

Determinants

Derivatives of Higher Order

Differentiation of Integrals

Exercises

Chapter 10: Integration of Differential Forms

Integration

Primitive Mappings

Partitions of Unity

Change of Variables

Differential Forms

Simplexes and Chains

Stokes' Theorem

Closed Forms and Exact Forms

Vector Analysis

Exercises

Chapter 11: The Lebesgue Theory

Set Functions

Construction of the Lebesgue Measure

Measure Spaces

Measurable Functions

Simple Functions

Integration

Comparison with the Riemann Integral

Integration of Complex Functions

Functions of Class L²

Exercises

Bibliography

List of Special Symbols

Index

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