实分析

作者简介

《实分析(英文版·第4版)》是实分析课程的优秀教材，被国外众多著名大学（如斯坦福大学、哈佛大学等）采用。全书分为三部分：第一部分为实变函数论.介绍一元实变函数的勒贝格测度和勒贝格积分:第二部分为抽象空间。介绍拓扑空间、度量空间、巴拿赫空间和希尔伯特空间；第三部分为一般测度与积分理论。介绍一般度量空间上的积分.以及拓扑、代数和动态结构的一般理论。书中不仅包含数学定理和定义，而且还提出了富有启发性的问题，以便读者更深入地理解书中内容。

书籍目录

Contents

Preface iii

Lebesgue Integration for Functions of Single Real Variable

Preliminaries on Sets, Mappings, and Relations

UnionsandIntersectionsofSets

Equivalence Relations, the Axiom of Choice, and Zorn’s Lemma .

The Real Numbers: Sets, Sequences, and Functions

1.1 The Field, Positivity, and Completeness Axioms 7

1.2 TheNaturalandRationalNumbers 11

1.3 CountableandUncountableSets . 13

1.4 Open Sets, Closed Sets, and Borel Sets of Real Numbers 16

1.5 SequencesofRealNumbers . 20

1.6 Continuous Real-Valued Functions of a Real Variable . 25

Lebesgue Measure 29

2.1 Introduction . 29

2.2 LebesgueOuterMeasure 31

2.3 The σ-AlgebraofLebesgueMeasurableSets . 34

2.4 Outer and Inner Approximation of Lebesgue Measurable Sets 40

2.5 Countable Additivity, Continuity, and the Borel-Cantelli Lemma . 43

2.6 NonmeasurableSets 47

.2.7 The Cantor Set and the Cantor-Lebesgue Function 49

Lebesgue Measurable Functions 54

3.1 Sums,Products,andCompositions 54

3.2 Sequential Pointwise Limits and Simple Approximation 60

3.3 Littlewood’s Three Principles, Egoroff’s Theorem, and Lusin’s Theorem 64

Lebesgue Integration 68

4.1 TheRiemannIntegral 68

4.2 The Lebesgue Integral of a Bounded Measurable Function over a Set of

FiniteMeasure 71

4.3 The Lebesgue Integral of a Measurable Nonnegative Function 79

4.4 TheGeneralLebesgueIntegral 85

4.5 Countable Additivity and Continuity of Integration 90

4.6 Uniform Integrability: The Vitali Convergence Theorem 92

Lebesgue Integration: Further Topics 97

5.1 Uniform Integrability and Tightness: A General Vitali Convergence Theorem 97

5.2 ConvergenceinMeasure 99

5.3 Characterizations of Riemann and Lebesgue Integrability 102

Differentiation and Integration 107

6.1 ContinuityofMonotoneFunctions 108

6.2 Differentiability of Monotone Functions: Lebesgue’s Theorem 109

6.3 Functions of Bounded Variation: Jordan’s Theorem 116

6.4 AbsolutelyContinuousFunctions . 119

6.5 Integrating Derivatives: Differentiating Indefinite Integrals . 124

6.6 ConvexFunctions . 130

7The Lp Spaces: Completeness and Approximation 135

7.1 NormedLinearSpaces . 135

7.2 The Inequalities of Young, H older, and Minkowski 139¨

7.3 Lp IsComplete:TheRiesz-FischerTheorem 144

7.4 ApproximationandSeparability 150

8The Lp Spaces: Duality and Weak Convergence 155

8.1 The Riesz Representation for the Dual of Lp, 1 155

8.2 Weak Sequential Convergence in Lp 162

8.3 WeakSequentialCompactness 171

8.4 TheMinimizationofConvexFunctionals174

II Abstract Spaces: Metric, Topological, Banach, and Hilbert Spaces 181

Metric Spaces: General Properties 183

9.1 ExamplesofMetricSpaces 183

9.2 Open Sets, Closed Sets, and Convergent Sequences 187

9.3 ContinuousMappingsBetweenMetricSpaces 190

9.4 CompleteMetricSpaces 193

9.5 CompactMetricSpaces . 197

9.6 SeparableMetricSpaces 204

10 Metric Spaces: Three Fundamental Theorems 206

10.1TheArzela-AscoliTheorem `. 206

10.2TheBaireCategoryTheorem 211

10.3TheBanachContractionPrinciple. 215

11 Topological Spaces: General Properties 222

11.1 OpenSets,ClosedSets,Bases,andSubbases. 222

11.2TheSeparationProperties 227

11.3CountabilityandSeparability 228

11.4 Continuous Mappings Between Topological Spaces 230

11.5CompactTopologicalSpaces. 233

11.6ConnectedTopologicalSpaces 237

12 Topological Spaces: Three Fundamental Theorems 239

12.1 Urysohn’s Lemma and the Tietze Extension Theorem . 239

12.2TheTychonoffProductTheorem . 244

12.3TheStone-WeierstrassTheorem 247

13 Continuous Linear Operators Between Banach Spaces 253

13.1NormedLinearSpaces . 253

13.2LinearOperators . 256

13.3 Compactness Lost: Infinite Dimensional Normed Linear Spaces 259

13.4 TheOpenMappingandClosedGraphTheorems . 263

13.5TheUniformBoundednessPrinciple 268

14 Duality for Normed Linear Spaces 271

14.1 Linear Functionals, Bounded Linear Functionals, and Weak Topologies 271

14.2TheHahn-BanachTheorem . 277

14.3 Reflexive Banach Spaces and Weak Sequential Convergence 282

14.4 LocallyConvexTopologicalVectorSpaces 286

14.5 The Separation of Convex Sets and Mazur’s Theorem . 290

14.6TheKrein-MilmanTheorem. 295

15 Compactness Regained: The Weak Topology 298

15.1 Alaoglu’sExtensionofHelley’sTheorem . 298

15.2 Reflexivity and Weak Compactness: Kakutani’s Theorem 300

15.3 Compactness and Weak Sequential Compactness: The Eberlein-ˇ

Smulian Theorem 302

15.4MetrizabilityofWeakTopologies . 305

16 Continuous Linear Operators on Hilbert Spaces 308

16.1TheInnerProductandOrthogonality 309

16.2 The Dual Space and Weak Sequential Convergence 313

16.3 Bessel’sInequalityandOrthonormalBases . 316

16.4 AdjointsandSymmetryforLinearOperators 319

16.5CompactOperators 324

16.6TheHilbert-SchmidtTheorem 326

16.7 The Riesz-Schauder Theorem: Characterization of Fredholm Operators 329

III Measure and Integration: General Theory 335

17 General Measure Spaces: Their Properties and Construction 337

17.1MeasuresandMeasurableSets 337

17.2 Signed Measures: The Hahn and Jordan Decompositions 342

17.3 The Carath′346

eodory Measure Induced by an Outer Measure

17.4TheConstructionofOuterMeasures 349

17.5 The Carath′eodory-Hahn Theorem: The Extension of a Premeasure to a

Measure 352

18 Integration Over General Measure Spaces 359

18.1MeasurableFunctions 359

18.2 Integration of Nonnegative Measurable Functions 365

18.3 Integration of General Measurable Functions 372

18.4TheRadon-NikodymTheorem 381

18.5 The Nikodym Metric Space: The Vitali–Hahn–Saks Theorem 388

19 General Lp Spaces: Completeness, Duality, and Weak Convergence 394

19.1 The Completeness of LpX,μ1 ≤≤. 394

19.2 The Riesz Representation Theorem for the Dual of LpX,μ1 ≤≤ 399

19.3 The Kantorovitch Representation Theorem for the Dual of L∞X,μ. 404

19.4 Weak Sequential Compactness in LpX,μ1 [p[ 1. 407

19.5 Weak Sequential Compactness in L1X,μ: The Dunford-Pettis Theorem 409

20 The Construction of Particular Measures 414

20.1 Product Measures: The Theorems of Fubini and Tonelli 414

20.2 Lebesgue Measure on Euclidean Space Rn 424

20.3 Cumulative Distribution Functions and Borel Measures on 437

20.4 Caratheodory Outer Measures and Hausdorff Measures on a Metric Space ′. 441

21 Measure and Topology 446

21.1LocallyCompactTopologicalSpaces 447

21.2 SeparatingSetsandExtendingFunctions452

21.3TheConstructionofRadonMeasures 454

21.4 The Representation of Positive Linear Functionals on CcX:The Riesz-

MarkovTheorem . 457

21.5 The Riesz Representation Theorem for the Dual of CX 462

21.6 RegularityPropertiesofBaireMeasures 470

22 Invariant Measures 477

22.1 Topological Groups: The General Linear Group . 477

22.2Kakutani’sFixedPointTheorem . 480

22.3 Invariant Borel Measures on Compact Groups: von Neumann’s Theorem 485

22.4 Measure Preserving Transformations and Ergodicity: The Bogoliubov-Krilov

Theorem 488

Bibliography 495

Index 497

编辑推荐

《实分析(英文版·第4版)》 :新增了50％的习题。扩充了基本结果。包括给出叶果洛夫定理和乌霄松引理的证明。介绍了博雷尔一坎特利引理、切比霄夫不等式、快速柯西序列及测度和积分所共有的连续性质.以及若干其他概念。

前言

The first three editions of H．］．Royden’S Real Analysis have contributed to the education of generation so fm a them atical analysis students．This four the dition of Real Analysispreservesthe goal and general structure of its venerable predecessors——to present the measure theory．integration theory．and functional analysis that a modem analyst needs to know．The book is divided the three parts：Part I treats Lebesgue measure and Lebesgueintegration for functions of a single real variable；Part II treats abstract spaces topological spaces，metric spaces，Banach spaces，and Hilbert spaces；Part III treats integration over general measure spaces．together with the enrichments possessed by the general theory in the presence of topological，algebraic，or dynamical structure．The material in Parts II and III does not formally depend on Part I．However．a careful treatment of Part I provides the student with the opportunity to encounter new concepts in afamiliar setting，which provides a foundation and motivation for the more abstract conceptsdeveloped in the second and third parts．Moreover．the Banach spaces created in Part I．theLp spaces，are one of the most important dasses of Banach spaces．The principal reason forestablishing the completeness of the Lp spaces and the characterization of their dual spacesiS to be able to apply the standard tools of functional analysis in the study of functionals andoperators on these spaces．The creation of these tools is the goal of Part II．

内容概要

作者：（美国）罗伊登（Royden.H.L.） （美国）菲茨帕特里克（Fitzpatrick.P.M.）

章节摘录

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