Principles of Mathematical Analysis

Walter Rudin 3종 세트: PMA, RCA, FA

목숨이 아깝다면 이 책을 멀리 하라!

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]

1 목차[편집]

이하 Appendix(부록)와 Exercises(예제)는 생략한다.

1.1 Chapter 1. The Real and Complex Number Systems[편집]

  • Introduction
  • Ordered Sets
  • Fields
  • The Real Field
  • The Extended Real Number System
  • The Complex Field
  • Euclidean Spaces

1.2 Chapter 2. Basic Topology[편집]

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

1.3 Chapter 3. Numerical Sequences and Series[편집]

  • 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[편집]

  • 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[편집]

  • 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[편집]

  • 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[편집]

  • 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[편집]

  • 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[편집]

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[편집]

  • 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[편집]

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

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