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实分析-(影印版) 版权信息
- ISBN：9787040226652
- 条形码：9787040226652 ; 978-7-04-022665-2
- 装帧：暂无
- 册数：暂无
- 重量：暂无
- 所属分类：>>
实分析-(影印版) 内容简介
本书是一本内容十分翔实的实分析教材。它包含集论，点集拓扑。测度与积分，Lebesgue函数空间，Banach空间与Hilbert空间，连续函数空间，广义函数与弱导数，Sobolev空间与Sobolev嵌入定理等；同时还包含 Lebesgue微分定理，Stone-Weierstrass逼近定理，Ascoli—Arzela定理， Calderon—Zygmund分解定理，Fefferman—Stein定理。Marcinkiewlcz插定理等实分析中有用的内容。
本书内容由浅入深。读者具有扎实的数学分析知识基础便可学习本书，学完本书的读者将具备学习分析所需要的实变与泛函(不包括算子理论)的准备知识和训练。
实分析-(影印版) 目录
Preface
Acknowledgments
Preliminaries
1 Countable sets
2 The Cantor set
3 Cardinality
3.1 Some examples
4 Cardinality of some infinite Cartesian products
5 Orderings, the maximal principle, and the axiom of choice
6 Well-ordering
6.1 The first uncountable
Problems and Complements
Ⅰ Topologies and Metric Spaces
1 Topological spaces
1.1 Hausdorff and normal spaces
2 Urysohn's lemma
3 The Tietze extension theorem
4 Bases, axioms of countability, and product topologies
4.1 Product topologies
5 Compact topological spaces
5.1 Sequentially compact topological spaces
6 Compact subsets of RN
7 Continuous functions on countably compact spaces
8 Products of compact spaces
9 Vector spaces
9.1 Convex sets
9.2 Linear maps and isomorphisms
10 Topological vector spaces
10.1 Boundedness and continuity
11 Linear functionals
12 Finite-dimensional topological vector spaces
12.1 Locally compact spaces
13 Metric spaces
13.1 Separation and axioms of countability
13.2 Equivalent metrics
13.3 Pseudometrics
14 Metric vector spaces
14.1 Maps between metric spaces
15 Spaces of continuous functions
15.1 Spaces of continuously differentiable functions
16 On the structure of a complete metric space
17 Compact and totally bounded metric spaces
17.1 Precompact subsets of X
Problems and Complements
Ⅱ Measuring Sets
1 Partitioning open subsets of RN
2 Limits of sets, characteristic functions, and or-algebras
3 Measures
3.1 Finite,a-finite, and complete measures
3.2 Some examples
4 Outer measures and sequential coverings
4.1 The Lebesgue outer measure in RN
4.2 The Lebesgue-Stieltjes outer measure
5 The Hausdorff outer measure in RN
6 Constructing measures from outer measures
7 The Lebesgue--Stieltjes measure on R
7.1 Borel measures
8 The Hausdorff measure on RN
9 Extending measures from semialgebras to a-algebras
9.1 On the Lebesgue-Stieltjes and Hausdorff measures
10 Necessary and sufficient conditions for measurability
11 More on extensions from semialgebras to a-algebras
12 The Lebesgue measure of sets in RN
12.1 A necessary and sufficient condition of naeasurability
13 A nonmeasurable set
……
Ⅲ The Lebesgue Integral
Ⅳ Topics on Measurable Functions of Real Variables
Ⅴ The Lp(E)Spaces
Ⅵ Banach Spaces
Ⅶ Spaces of Continuous Functions,Distributions,and Weak
Ⅷ Topics on Integrable Functions of Real Variables
Ⅸ Embeddings of W1,p(E)into Lq(E)
References
Index
Acknowledgments
Preliminaries
1 Countable sets
2 The Cantor set
3 Cardinality
3.1 Some examples
4 Cardinality of some infinite Cartesian products
5 Orderings, the maximal principle, and the axiom of choice
6 Well-ordering
6.1 The first uncountable
Problems and Complements
Ⅰ Topologies and Metric Spaces
1 Topological spaces
1.1 Hausdorff and normal spaces
2 Urysohn's lemma
3 The Tietze extension theorem
4 Bases, axioms of countability, and product topologies
4.1 Product topologies
5 Compact topological spaces
5.1 Sequentially compact topological spaces
6 Compact subsets of RN
7 Continuous functions on countably compact spaces
8 Products of compact spaces
9 Vector spaces
9.1 Convex sets
9.2 Linear maps and isomorphisms
10 Topological vector spaces
10.1 Boundedness and continuity
11 Linear functionals
12 Finite-dimensional topological vector spaces
12.1 Locally compact spaces
13 Metric spaces
13.1 Separation and axioms of countability
13.2 Equivalent metrics
13.3 Pseudometrics
14 Metric vector spaces
14.1 Maps between metric spaces
15 Spaces of continuous functions
15.1 Spaces of continuously differentiable functions
16 On the structure of a complete metric space
17 Compact and totally bounded metric spaces
17.1 Precompact subsets of X
Problems and Complements
Ⅱ Measuring Sets
1 Partitioning open subsets of RN
2 Limits of sets, characteristic functions, and or-algebras
3 Measures
3.1 Finite,a-finite, and complete measures
3.2 Some examples
4 Outer measures and sequential coverings
4.1 The Lebesgue outer measure in RN
4.2 The Lebesgue-Stieltjes outer measure
5 The Hausdorff outer measure in RN
6 Constructing measures from outer measures
7 The Lebesgue--Stieltjes measure on R
7.1 Borel measures
8 The Hausdorff measure on RN
9 Extending measures from semialgebras to a-algebras
9.1 On the Lebesgue-Stieltjes and Hausdorff measures
10 Necessary and sufficient conditions for measurability
11 More on extensions from semialgebras to a-algebras
12 The Lebesgue measure of sets in RN
12.1 A necessary and sufficient condition of naeasurability
13 A nonmeasurable set
……
Ⅲ The Lebesgue Integral
Ⅳ Topics on Measurable Functions of Real Variables
Ⅴ The Lp(E)Spaces
Ⅵ Banach Spaces
Ⅶ Spaces of Continuous Functions,Distributions,and Weak
Ⅷ Topics on Integrable Functions of Real Variables
Ⅸ Embeddings of W1,p(E)into Lq(E)
References
Index
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