實分析和泛函分析(第3版)（簡體書）

《實分析和泛函分析(第3版)(英文版)》內容簡介：This book is meant as a text for a first year graduate course in analysis. Any standard course in undergraduate analysis will constitute sufficient preparation for its understanding, for instance, my Undergraduate Analysis. I assume that the reader is acquainted with notions of uniform convergence and the like.
In this third edition, I have reorganized the book by covering integration before functional analysis. Such a rearrangement fits the way courses are taught in all the places I know of. I have added a number of examples and exercises, as well as some material about integration on the real line (e.g. on Dirac sequence approximation and on Fourier analysis), and some material on functional analysis (e.g. the theory of the Gelfand transform in Chapter XVI). These upgrade previous exercises to sections in the text.

PART ONE
General Topology
CHAPTER Ⅰ
Sets
l. Some Basic Terminology
2. Denumerable Sets
3. Zorns 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. Ascolis 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. Mazurs 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 Taylors 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