《实分析》
《实分析》 内容介绍:
《实分析》(英文版第3版)是一本优秀的教材,主要分三部分:第一部分为实变函数论,第二部分为抽象空间,第三部分为一般测度与积分论。书中不仅包含数学定理和定义,而且还提出了挑战性的问题,以便读者更深入地理解书中的内容。《实分析》(英文版第3版)的题材是数学教学的共同基础,包含许多数学家的研究成果。
《实分析》 目录大纲:
Prologue to the Student 1
I Set Theory 6
1 Introduction 6
2 Functions 9
3 Unions, intersections, and complements 12
4 Algebras of sets 17
5 The axiom of choice and infinite direct products 19
6 Countable sets 20
7 Relations and equivalences 23
8 Partial orderings and the maximal principle 24
9 Well ordering and the countable ordinals 26
Part One
THEORY OF FUNCTIONS OF A
REAL VARIABLE
2 The Real Number System 31
1 Axioms for the real numbers 31
2 The natural and rational numbers as subsets of R 34
3 The extended real numbers 36
4 Sequences of real numbers 37
5 Open and closed sets of real numbers 40
6 Continuous functions 47
7 Borel sets 52
3 Lebesgue Measure 54
I Introduction 54
2 Outer measure 56
3 Measurable sets and Lebesgue measure 58
*4 A nonmeasurable set 64
5 Measurable functions 66
6 Littlewood's three principles 72
4 The Lebesgue Integral 75
1 The Riemann integral 75
2 The Lebesgue integral of a bounded function over a set of finite
measure 77
3 The integral of a nonnegative function 85
4 The general Lebesgue integral 89
*5 Convergence in measure 95
S Differentiation and Integration 97
1 Differentiation of monotone functions 97
2 Functions of bounded variation 102
3 Differentiation of an integral 104
4 Absolute continuity 108
5 Convex functions 113
6 The Classical Banach Spaces 118
1 The Lp spaces 118
2 The Minkowski and Holder inequalities 119
3 Convergence and completeness 123
4 Approximation in Lp 127
5 Bounded linear functionals on the Lp spaces 130
Part Two
ABSTRACT SPACES
7 Metric Spaces 139
1 Introduction 139
2 Open and closed sets 141
3 Continuous functions and homeomorphisms 144
4 Convergence and completeness 146
5 Uniform continuity and uniformity 148
6 Subspaces 151
7 Compact metric spaces 152
8 Baire category 158
9 Absolute Gs 164
10 The Ascoli-Arzela Theorem 167
8 Topological Spaces ltl
I Fundamental notions 171
2 Bases and countability 175
3 The separation axioms and continuous real-valued
functions 178
4 Connectedness 182
5 Products and direct unions of topological spaces 184
*6 Topological and uniform properties 187
*7 Nets 188
9 Compact and Locally Compact Spaces 190
I Compact spaces 190
2 Countable compactness and the Bolzano-Weierstrass
property 193
3 Products of compact spaces 196
4 Locally compact spaces 199
5 a-compact spaces 203
*6 Paracompact spaces 204
7 Manifolds 206
*8 The Stone-Cech compactification 209
9 The Stone-Weierstrass Theorem 210
10 Banach Spaces 217
I Introduction 217
2 Linear operators 220
3 Linear functionals and the Hahn-Banach Theorem 222
4 The Closed Graph Theorem 224
5 Topological vector spaces 233
6 Weak topologies 236
7 Convexity 239
8 Hilbert space 245
Part Three
GENERAL MEASURE AND INTEGRATION
THEORY
11 Measure and Integration 253
1 Measure spaces 253
2 Measurable functions 259
3 Integration 263
4 General Convergence Theorems 268
5 Signed measures 270
6 The Radon-Nikodym Theorem 276
7 The Lp-spaces 282
12 Measure and Outer Measure 288
1 Outer measure and measurability 288
2 The Extension Theorem 291
3 The Lebesgue-Stieltjes integral 299
4 Product measures 303
5 Integral operators 313
*6 Inner measure 317
*7 Extension by sets of measure zero 325
8 Caratheodory outer measure 326
9 Hausdorff measure 329
13 Measure and Topology 331
1 Baire sets and Borel sets 331
2 The regularity of Baire and Borel measures 337
3 The construction of Borel measures 345
4 Positive linear functionals and Borel measures 352
5 Bounded linear functionals on C(X) 355
14 Invariant Measures 361
1 Homogeneous spaces 361
2 Topological equicontinuity 362
3 The existence ofinvariant measures 365
4 Topological groups 370
5 Group actions and quotient spaces 376
6 Unicity ofinvariant measures 378
7 Groups ofdiffeomorphisms 388
15 Mappings of Measure Spaces 392
1 Point mappings and set mappings 392
2 Boolean algebras 394
3 Measure algebras 398
4 Borel equivalences 401
5 Borel measures on complete separable metric spaces 406
6 Set mappings and point mappings on complete separable
metric spaces 412
7 The isometries of Lp 415
16 The Daniell Integral 419
1 Introduction 419
2 The Extension Theorem 422
3 Uniqueness 427
4 Measurability and measure 429
Bibliography 435
Index of Symbols 437
Subject Index 439
微信扫一扫关注公众号
0 个评论
你也许想看:
《无穷分析引论(上、下)》
[瑞士] 欧拉.山西教育出版社.1997-1“”
《Characteristic Classes.》
John Milnor,James D. Stasheff.Princeton University Press.1974-8-1“”
《数学那玩意》
韩旭.浙江大学.2010-11“《数学那玩意:自主招生秘籍》按如下模式编写:基础知识+例题+练习。在基础知识中,我先介绍一些基本的公式、定理、方法,并适...”
《学数学,就这么简单!》
漱山士郎.科学出版社.2011-8“《学数学,就这么简单!》内容简介:我们生活的世界有形形色色的事物和现象,其中都必定包含着“科学”的成分。在这些成分中,有...”
《数学之旅》
[英]Tom Jackson.人民邮电出版社.2014-5-1“《数学之旅》主要讲述了数学发展史上的100个重大发现,通过这些重大发现展现出数学的发展和进步历程。从史前到中世纪,文艺复...”
《Things to Make and Do in the Fourth Dimension》
Matt Parker.Farrar, Straus and Giroux.2014-12-2“A revolutionary book from the stand-up mathematician that ma...”
《同调代数导论》
[美国] 韦伯尔著.机械工业出版社.2004-11“同调代数领域在20世纪后半叶己演进成为数学研究人员的一种基本工具。本书论述了关于当今同调代数的基本概念,并阐述了同调代数...”
《流形的拓扑学》
苏竞存.武汉大学出版社.2005-5“拓扑学的方法与结果在各个数学分支中有着广泛的应用,因此适当选择其中的内容供各个分支的研究者与教师之用是一个很重要的工作。...”
《无穷的玩艺》
路沙·彼得.大连理工大学.2008-4“《无穷的玩艺:数学的探索与旅行》是著名数学家写的数学普及读物。一部引人入胜的名著,不用任何公式。着重讨论数学的思想方法。...”
《数学趣闻集锦(下)》
(美)T·帕帕斯张远南,张昶.上海教育出版社.1998-12-1“本书内容包括:海洋波浪的数学、四维立方体的展开、七巧板、毕达哥拉斯定理的一种优雅证明、令人困惑的无穷大等。”
《Differential Forms in Algebraic Topology》
R. Bott,L.W. Tu.Springer-Verlag Berlin and Heidelberg GmbH & Co. K.1982-12-31“”
《李群》
Daniel Bump.世界图书出版公司.2009-8“本书作者采取了与许多教材以紧李群的表示论作为理论基础不同的安排,并精心挑选一系列材料,以给予读者更广阔的视野。为介绍紧李...”
《表示论基本教程》
William Fulton,Joe Harris.世界图书出版公司.2005-6“本书以英文的形式介绍了表示论基本教程。”
《中学数学用表》
人民教育出版社数学室 编.人民教育出版社.1983-10“《中学数学用表》主要是供中学生在没有计算器的计算时迅速查找运算结果的近似值。现在随着计算器的普及,这本《数学用表》逐渐成...”
《女士品茶》
萨尔斯伯格.中国统计出版社.2004-11-01“《20世纪统计怎样变革了科学:女士品茶》以某位喝茶的英国女士的假设学说为起点,引出了近代数理统计的开创者——费歇尔,以及...”
《数学基础》
汪芳庭.科学出版社.2001-09-01“本书概述了数学基础的历史,介绍了现代数学主体的基础——ZFC集论,重点讲述四种数(自然数、实数、序数和基数)的理论.书中...”
《An Elementary Introduction to the Wolfram Language》
Stephen Wolfram.Wolfram Media, Inc..2015-12-11“”
《Grothendieck-Serre Correspondence》
Jean-Pierre Serre,Catriona Maclean Pierre Colmez.American Mathematical Society.2003-12-16“This extraordinary volume contains a large part of the mathe...”
《The Apprenticeship of a Mathematician》
Andre Weil.Birkhauser.2002-02-20“"Extremely readable recollections of the author... A rare te...”
《奇妙的数王国》
李毓佩.中国少年儿童出版社.2002-01“《奇妙的数王国》一场莫名其妙的战争:“打仗啦!打仗啦!”弟弟小华一溜烟似地跑进了屋。哥哥小强正在专心做题,小华这一喊,把...”