# Principles of Mathematical Analysis

Walter Rudin 3종 세트: PMA, RCA, FA

Principles of Mathematical Analysis(PMA)는 해석학 교재로 가장 많이 쓰이는 책 중 하나이다. 해석학 교재 하면 대부분의 사람이 PMA를 추천한다Chapter 9~11는 보지 않는 것이 정신건강에 좋다. 물론 Rudin의 책답게 설명이 많지 않은 깔끔한(?) 서술이 되어 있다. 출판사는 McGraw Hill, Inc.이다. 참고로 아마존 평점은 4.2 out of 5 stars로, 높은 편이다.

## 1 목차

이하 Appendix(부록)와 Exercises(예제)는 생략한다.

### 1.1 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

### 1.2 Chapter 2. Basic Topology

• Finite, Countable, and Uncountable Sets
• Metric Spaces
• Compact Sets
• Perfect Sets
• Connected Sets

### 1.3 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

### 1.4 Chapter 4. Continuity

• Limit of Functions
• Continuous Functions
• Continuity and Compactness
• Continuity and Connectedness
• Discontinuities
• Monotonic Functions
• Infinite Limits and Limits at Infinity

### 1.5 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

### 1.6 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

### 1.7 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

### 1.8 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

### 1.9 Chapter 9. Functions of Several Variables

이하 보지 마세요, 차라리 Wade 해석학을 보시길

• 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

### 1.10 Chapter 10. Integration of Differential Forms

• Integration
• Primitive Mappings
• Partitions of Unity
• Change of Variables
• Simplexes and Chains
• Stokes' Theorem
• Closed Forms and Exact Forms
• Vector Analysis

### 1.11 Chapter 11. The Lebesgue Theory

• Set Functions
• Construction of the Lebesgue Measure
• Measure Spaces
• Measurable Functions
• Simple Functions
• Integration
• Comparison with the Riemann Integral
• Functions of Class $\displaystyle{ \mathscr L^2 }$