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实分析和泛函分析（第3版）（英文版） (书店编码:2366585) |
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主要内容:
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.
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本书目录:
PART ONE General Topology
CHAPTERⅠ Sets
1. Some Basic Terminology
2. Denumerahle Sets
3. Zorn''s Lemma
CHAPTERⅡ Topological Spaces
1. Open and Closed Sets
2. Connected Sets
3. Compact Spaces
4. Separation by Continuous Functions
5. Exercises
CHAPTERⅢ Continuous Functions on Compact Sets
1. The Stone-Weierstrass Theorem
2. Ideals of Continuous Functions
3. Ascoli''s Theorem
4. Exercises
PART TWO Banach and Hilbert Spaces
CHAPTERⅣ 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
1. The Krein-Milman Theorem
2. Mazur''s Theorem
6. Exercises
CHAPTERⅣ HIIbert Space
1. Hermitian Forms
2. Functionals and Operators
3. Exercises
PART THREE Integration
CHAPTERⅥ The General Integral
1. Measured Spaces, Measurable Maps, and Positive Measures
2. The Integral of Step Maps
3. The L1-Compledon
4. Properties of the Integral: First Part
5. Properties of the Integral: Second Part
6. Approximations
7. Extension of Positive Measures from Algebras to q-Algebras
8. Product Measures and Integration on a Product Space
9. The Lebesgue Integral in Rp
10. Exercises
CHAPTERⅦ Duality and Representation Theorems
1. The Hilbert Space L2 u
2. Duality Between L1 u and L #
3. Complex and Vectorial Measures
4. Complex or Vectorial Measures and Duality
5. The LB Spaces, 1 < p <
6. The Law of Large Numbers
7. Exercises
CHAPTERⅧ Duality and Representation Theorems
1. The Hilbert Space L2 u
2. Duality Between L1 u and L #
3. Complex and Vectorial Measures
4. Complex or Vectorial Measures and Duality
5. The LB Spaces, 1 < p <
6. The Law of Large Numbers
7. Exercises
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PART FOUR Calculus
PART FIVE Functional Analysis
PART SIX Global Analysis
Bibliography
Table of Notation
Index
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