實分析或實數分析是處理實數及實函式的數學分析。專門研究數列，數列極限，微分，積分及函式序列，以及實函式的連續性，光滑性以及其他相關性質。實分析常以基礎集合論，函式概念定義等等開始。
基本介紹
- 書名：實數分析
- 作者：Elias M.Stein、Rami Shakarchi
- ISBN：9787506282383
- 頁數：402頁
- 出版社：世界圖書出版公司北京公司
- 出版時間：2007-01
- 裝幀：平裝
- 開本：32開
- 語種：英語
內容簡介,圖書目錄,
內容簡介
本書由在國際上享有盛譽普林史達林頓大學教授Stein等撰寫而成，是一部為數學及相關專業大學二年級和三年級學生編寫的教材，理論與實踐並重。為了便於非數學專業的學生學習，全書內容簡明、易懂,讀者只需掌握微積分和線性代數知識。關於本書的詳細介紹，請見“影印版前言”。
本書已被哈佛大學和加利福尼亞理工學院選為教材。
與本書相配套的教材《傅立葉分析導論》和《複分析》也已影印出版。
圖書目錄
Foreword
Introduction
1 Fourier series: completion
實分析 2 Limits of continuous functions
3 Length of curves
4 Differentiation and integration
5 The problem of measure
Chapter 1. Measure Theory
1 Preliminaries
2 The exterior measure
3 Measurable sets and the Lebesgue measure
4 Measurable functions
4.1 Definition and basic properties
4.2 Approximation by simple functions or step functions
4.3 Littlewoods three principles
5* The Brunn-Minkowski inequality
6 Exercises
7 Problems
Chapter 2. Integration Theory
1 The Lebesgue integral: basic properties and convergence theorems
2 The space L1 of integrable functions
3 Fubinis theorem
3.1 Statement and proof of the theorem
3.2 Applications of Fubinis theorem
4* A Fourier inversion formula
5 Exercises
6 Problems
Chapter 3. Differentiation and Integration
1 Differentiation of the integral
1.1 The Hardy-Littlewood maximal function
1.2 The Lebesgue differentiation theorem
2 Good kernels and approximations to the identity
3 Differentiability of functions
3.1 Functions of bounded variation
3.2 Absolutely continuous functions
3.3 Differentiability of jump functions
4 Rectifiable curves and the isoperimetric inequality
4.1 Minkowski content of a curve
4.2* Isoperimetrie inequality
5 Exercises
6 Problems
Chapter 4. Hilbert Spaces: An Introduction
1 The Hilbert space L2
2 Hilbert spaces
2.1 Orthogonality
2.2 Unitary mappings
2.3 Pre-Hilbert spaces
3 Fourier series and Fatous theorem
3.1 Fatous theorem
4 Closed subspaees and orthogonal projections
5 Linear transformations
5.1 Linear flmetionals and the Riesz representation the-orem
5.2 Adjoints
5.3 Examples
6 Compact operators
7 Exercises
8 Problems
Chapter 5. Hilbert Spaces: Several Examples
1 The Fourier transform on L2
2 The Hardy space of the upper half-plane
3 Constant coefficient partial differential equations
3.1 Weak solutions
3.2 The main theorem and key estimate
4* The Dirichlet principle
4.1 Harmonic functions
4.2 The boundary value problem and Diriehlets principle
5 Exercises
6 Problems
Chapter 6.Abstract Measure and Integration Theory
Chapter 7.Hausdorff Measure and Fractals
Notes and References
Bibliography
Symbol Glossary
Index
