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實分析（英文版·原書第4版） 
定 價：139 元 叢書名：經(jīng)典原版書庫
9 7 6 8 4 7 6 1 6 1 5 1 5 
本書是實分析課程的教材，被國外眾多大學(xué)（如斯坦福大學(xué)、哈佛大學(xué)等）采用。全書分為三部分：第壹部分為實變函數(shù)論，介紹一元實變函數(shù)的勒貝格測度和勒貝格積分；第二部分為抽象空間，介紹拓撲空間、度量空間、巴拿赫空間和希爾伯特空間；第三部分為一般測度與積分理論，介紹一般度量空間上的積分，以及拓撲、代數(shù)和動態(tài)結(jié)構(gòu)的一般理論。書中不僅包含數(shù)學(xué)定理和定義，而且還提出了富有啟發(fā)性的問題，以便讀者更深入地理解書中內(nèi)容。
第一部分 一元實變量函數(shù)的Lebesgue積分
第0章 集合、映射與關(guān)系的預(yù)備知識3 0.1 集合的并與交3 0.2 集合間的映射4 0.3 等價關(guān)系、選擇公理以及Zorn引理5 第1章 實數(shù)集：集合、序列與函數(shù)7 1.1 域、正性以及完備性公理7 1.2 自然數(shù)與有理數(shù)11 1.3 可數(shù)集與不可數(shù)集13 1.4 實數(shù)的開集、閉集和Borel集16 1.5 實數(shù)序列20 1.6 實變量的連續(xù)實值函數(shù)25 第2章 Lebesgue測度29 2.1 引言29 2.2 Lebesgue外測度31 2.3 Lebesgue可測集的代數(shù)34 2.4 Lebesgue可測集的外逼近和內(nèi)逼近40 2.5 可數(shù)可加性、連續(xù)性以及Borel-Cantelli引理43 2.6 不可測集47 2.7 Cantor集和Cantor-Lebesgue函數(shù)49 第3章 Lebesgue可測函數(shù)54 3.1 和、積與復(fù)合54 3.2 序列的逐點極限與簡單逼近60 3.3 Littlewood的三個原理、Egoroff定理以及Lusin定理64 第4章 Lebesgue積分68 4.1 Riemann積分68 4.2 有限測度集上的有界可測函數(shù)的 Lebesgue積分71 4.3 非負可測函數(shù)的Lebesgue積分79 4.4 一般的Lebesgue積分85 4.5 積分的可數(shù)可加性與連續(xù)性90 4.6 一致可積性：Vitali收斂定理92 第5章 Lebesgue積分：深入課題97 5.1 一致可積性和緊性：一般的Vitali收斂定理97 5.2 依測度收斂99 5.3 Riemann可積與Lebesgue可積的刻畫102 第6章 微分與積分107 6.1 單調(diào)函數(shù)的連續(xù)性108 6.2 單調(diào)函數(shù)的可微性：Lebesgue定理109 6.3 有界變差函數(shù)：Jordan定理116 6.4 絕對連續(xù)函數(shù)119 6.5 導(dǎo)數(shù)的積分：微分不定積分124 6.6 凸函數(shù)130 第7章 Lp空間：完備性與逼近135 7.1 賦范線性空間135 7.2 Young、H鰈der與Minkowski不等式139 7.3 Lp是完備的：Riesz-Fischer定理144 7.4 逼近與可分性150 第8章 Lp空間：對偶與弱收斂155 8.1 關(guān)于Lp（1≤p＜∞）的對偶的Riesz表示定理155 8.2 Lp中的弱序列收斂162 8.3 弱序列緊性171 8.4 凸泛函的最小化174 第二部分 抽象空間：度量空間、 拓撲空間、Banach空間 和Hilbert空間 第9章 度量空間：一般性質(zhì)183 9.1 度量空間的例子183 9.2 開集、閉集以及收斂序列187 9.3 度量空間之間的連續(xù)映射190 9.4 完備度量空間193 9.5 緊度量空間197 9.6 可分度量空間204 第10章 度量空間：三個基本定理206 10.1 Arzelà-Ascoli定理206 10.2 Baire范疇定理211 10.3 Banach壓縮原理215 第11章 拓撲空間：一般性質(zhì)222 11.1 開集、閉集、基和子基222 11.2 分離性質(zhì)227 11.3 可數(shù)性與可分性228 11.4 拓撲空間之間的連續(xù)映射230 11.5 緊拓撲空間233 11.6 連通的拓撲空間237 第12章 拓撲空間：三個基本定理239 12.1 Urysohn引理和Tietze延拓定理239 12.2 Tychonoff乘積定理244 12.3 Stone-Weierstrass定理247 第13章 Banach空間之間的連續(xù)線性算子253 13.1 賦范線性空間253 13.2 線性算子256 13.3 緊性喪失：無窮維賦范線性空間259 13.4 開映射與閉圖像定理263 13.5 一致有界原理268 第14章 賦范線性空間的對偶271 14.1 線性泛函、有界線性泛函以及弱拓撲271 14.2 Hahn-Banach定理277 14.3 自反Banach空間與弱序列 收斂性282 14.4 局部凸拓撲向量空間286 14.5 凸集的分離與Mazur定理290 14.6 Krein-Milman定理295 第15章 重新得到緊性：弱拓撲298 15.1 Helly定理的Alaoglu推廣298 15.2 自反性與弱緊性：Kakutani定理300 15.3 緊性與弱序列緊性：Eberlein-mulian定理302 15.4 弱拓撲的度量化305 第16章 Hilbert空間上的連續(xù)線性算子308 16.1 內(nèi)積和正交性309 16.2 對偶空間和弱序列收斂313 16.3 Bessel不等式與規(guī)范正交基316 16.4 線性算子的伴隨與對稱性319 16.5 緊算子324 16.6 Hilbert-Schmidt定理326 16.7 Riesz-Schauder定理：Fredholm算子的刻畫329 第三部分 測度與積分：一般理論 第17章 一般測度空間：性質(zhì)與構(gòu)造337 17.1 測度與可測集337 17.2 帶號測度：Hahn與Jordan分解342 17.3 外測度誘導(dǎo)的Carathéodory測度346 17.4 外測度的構(gòu)造349 17.5 將預(yù)測度延拓為測度：Carathéodory-Hahn定理352 第18章 一般測度空間上的積分359 18.1 可測函數(shù)359 18.2 非負可測函數(shù)的積分365 18.3 一般可測函數(shù)的積分372 18.4 Radon-Nikodym定理381 18.5 Nikodym度量空間：Vitali-Hahn-Saks定理388 第19章 一般的Lp空間：完備性、對偶性和弱收斂性394 19.1 Lp(X, )（1≤p≤∞）的完備性394 19.2 關(guān)于Lp(X, )（1≤p<∞）的對偶的Riesz表示定理399 19.3 關(guān)于L∞(X, )的對偶的Kantorovitch表示定理404 19.4 Lp(X, )（1＜p＜∞）的弱序列緊性407 19.5 L1(X, )的弱序列緊性：Dunford-Pettis定理409 第20章 特定測度的構(gòu)造414 20.1 乘積測度：Fubini與Tonelli定理414 20.2 歐氏空間Rn上的Lebesgue測度424 20.3 累積分布函數(shù)與Borel測度437 20.4 度量空間上的Carathéodory外測度與Hausdorff測度441 第21章 測度與拓撲446 21.1 局部緊拓撲空間447 21.2 集合分離與函數(shù)延拓452 21.3 Radon測度的構(gòu)造454 21.4 Cc(X)上的正線性泛函的表示：Riesz-Markov定理457 21.5 C（X）的對偶的表示：Riesz-Kakutani表示定理462 21.6 Baire測度的正則性470 第22章 不變測度477 22.1 拓撲群：一般線性群477 22.2 Kakutani不動點定理480 22.3 緊群上的不變Borel測度：von Neumann定理485 22.4 測度保持變換與遍歷性：Bogoliubov-Krilov定理488 參考文獻495 Contents I Lebesgue Integration for Functions of a Single Real Variable 1 0 Preliminaries on Sets, Mappings, and Relations 3 UnionsandIntersectionsofSets ............................. 3 Mappings Between Sets............................. 4 Equivalence Relations, the Axiom of Choice, and Zorn’s Lemma . . . . . . . . . . 5 1 The Real Numbers: Sets, Sequences, and Functions 7 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 2 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 3 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 4 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 5 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 6 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 Inde.nite 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 ≤ p < ∞ ........... 155 8.2 Weak Sequential Convergence in Lp ....................... 162 8.3 WeakSequentialCompactness........................... 171 8.4 TheMinimizationofConvexFunctionals. . . . . . . . . . . . . . . . . . . . . 174 II Abstract Spaces: Metric, Topological, Banach, and Hilbert Spaces 181 9 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 Contents i. 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: In.nite 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 Re.exive 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 Re.exivity and Weak Compactness: Kakutani’s Theorem . . . . . . . . . . . 300 15.3 Compactness and Weak Sequential Compactness: The Eberlein-ˇSmulian 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 Lp.X, μ., 1 ≤p ≤∞ ................... 394 19.2 The Riesz Representation Theorem for the Dual of Lp.X, μ., 1 ≤p ≤∞ . . 399 19.3 The Kantorovitch Representation Theorem for the Dual of L∞.X, μ. .... 404 19.4 Weak Sequential Compactness in Lp.X, μ., 1 19.5 Weak Sequential Compactness in L1.X, 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 R ......... 437 20.4 Caratheodory Outer Measures and Hausdor ′ff Measures on a Metric Space 441 21 Measure and Topology 446 21.1LocallyCompactTopologicalSpaces ....................... 447 21.2 SeparatingSetsandExtendingFunctions. . . . . . . . . . . . . . . . . . . . . 452 21.3TheConstructionofRadonMeasures....................... 454 21.4 The Representation of Positive Linear Functionals on Cc.X.:The Riesz-MarkovTheorem .................... 457 21.5 The Riesz Representation Theorem for the Dual of C.X. ........... 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
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