實分析和泛函分析

《實分析和泛函分析》是1997年世界圖書出版公司出版的圖書，作者是S.Lang 。本書是作為分析專業的一年級研究生教材使用，需要本科的相關知識作為學習前提。

基本介紹

• 書名：實分析和泛函分析
• ISBN：9787506233071
• 頁數：580頁
• 出版社：世界圖書出版公司
• 出版時間：1997年9月1日
• 裝幀：平裝
• 開本：32
• 版次：1

內容簡介

《實分析和泛函分析(第3版)(英文版)》內容簡介：Thisbookismeantasatextforafirstyeargraduatecourseinanalysis.Anystandardcourseinundergraduateanalysiswillconstitutesufficientpreparationforitsunderstanding,forinstance,myUndergraduateAnalysis.Iassumethatthereaderisacquaintedwithnotionsofuniformconvergenceandthelike.Inthisthirdedition,Ihavereorganizedthebookbycoveringintegrationbeforefunctionalanalysis.SucharearrangementfitsthewaycoursesaretaughtinalltheplacesIknowof.Ihaveaddedanumberofexamplesandexercises,aswellassomematerialaboutintegrationontherealline(e.g.onDiracsequenceapproximationandonFourieranalysis),andsomematerialonfunctionalanalysis(e.g.thetheoryoftheGelfandtransforminChapterXVI).Theseupgradepreviousexercisestosectionsinthetext. {zzjj}

目錄

PART ONE

General Topology

CHAPTER Ⅰ

Sets

l. Some Basic Terminology

2. Denumerable Sets

3. Zorn's Lemma

CHAPTER Ⅱ

Topological Spaces

1. Open and Closed Sets

2. Connected Sets

3. Compact Spaces

4. Separation by Continuous Functions

5. Exercises

CHAPTER Ⅲ

Continuous Functions on Compact Sets

l. The Stone-Weierstrass Theorem

2. Ideals of Continuous Functions

3. Ascoli's Theorem

4. Exercises

PART TWO

Banach and Hilbert Spaces

CHAPTER IV

Banach Spaces

1. Definitions, the Dual Space, and the Hahn-Banach Theorem

2. Banach Algebras

3. The Linear Extension Theorem

4. Completion of a Normed Vector Space

5. Spaces with Operators

Appendix: Convex Sets

I. The Krein-Milman Theorem

2. Mazur's Theorem

6. Exercises

CHAPTER V

Hilbert Space

1. Hermitian Forms

2. Functionals and Operators

3. Exercises

PART THREE

Integration

CHAPTER Ⅳ

The General Integral

1. Measured Spaces, Measurable Maps, arid Positive Measures

2. The Integral of Step Maps

3. The L1-Completion

4. Properties of the Integral: First Part

5. Properties of the Integral: Second Part

6. Approximations

7. Extension of Positive Measures from Algebras to a-Algebras

8. Product Measures and Integration on a Product Space

9. The Lebesgue Integral in Rp

10. Exercises

CHAPTER Ⅶ

Duality and Representation Theorems

1. The Hilbert Space L2(μ)

2. Duality Between L1(μ) and L∞(μ)

3. Complex and Vectorial Measures

4. Complex or Vectorial Measures and Duality

5. The Lp Spaces, 1 < p < ∞

6. The Law of Large Numbers

7. Exercises

CHAPTER Ⅷ

Some Applications of Integration

1. Convolution

2. Continuity and Differentiation Under the Integral Sign

3. Dirac Sequences

4. The Schwartz Space and Fourier Transform

5. The Fourier Inversion Formula

6. The Poisson Summation Formula

7. An Example of Fourier Transform Not in the Schwartz Space

8. Exercises

CHAPTER Ⅸ

Integration and Measures on Locally Compact Spaces

1. Positive and Bounded Functionals on Cc(X)

2. Positive Functionals as Integrals

3. Regular Positive Measures

4. Bounded Functionals as Integrals

5. Localization of a Measure and of the Integral

6. Product Measures on Locally Compact Spaces

7. Exercises

CHAPTER Ⅹ

Riemann-Stieltjes Integral and Measure

l. Functions of Bounded Variation and the Stiehjes Integral

2. Applications to Fourier Analysis

3. Exercises

CHAPTER Xl

Distributions

1. Definition and Examples

2. Support and Localization

3. Derivation of Distributions

4. Distributions with Discrete Support

CHAPTER Ⅻ

Integration on Locally Compact Groups

l. Topological Groups

2. The Haar Integral, Uniqueness

3. Existence of the Haar Integral

4. Measures on Factor Groups and Homogeneous Spaces

5. Exercises

PART FOUR

Calculus

CHAPTER ⅩⅢ

Differential Calculus

1. Integration in One Variable

2. The Derivative as a Linear Map

3. Properties of the Derivative

4. Mean Value Theorem

5. The Second Derivative

6. Higher Derivatives and Taylor's Formula

7. Partial Derivatives

8. Differentiating Under the Integral Sign

9. Differentiation of Sequences

10. Exercises

CHAPTER ⅩⅣ

Inverse Mappings and Differential Equations

1. The Inverse Mapping Theorem

2. The Implicit Mapping Theorem

3. Existence Theorem for Differential Equations

4. Local Dependence on Initial Conditions

5. Global Smoothness of the Flow

6. Exercises

PART FIVE

Functional Analysis

CHAPTER XV

The Open Mapping Theorem, Factor Spaces, and Duality

1. The Open Mapping Theorem

2. Orthogonality

3. Applications of the Open Mapping Theorem

CHAPTER XVl

The Spectrum

1. The Gelfand-Mazur Theorem

2. The Gelfand Transform

3. C*-Algebras

4. Exercises

CHAPTER XVll

Compact and Fredholm Operators

1. Compact Operators

2. Fredholm Operators and the Index

3. Spectral Theorem for Compact Operators

4. Application to Integral Equations

5. Exercises

……

PART SIX

Index