實分析

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．a(chǎn)nd 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(chǎn) 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．

內(nèi)容概要

　 《實分析（英文版·第4版）》是實分析課程的優(yōu)秀教材，被國外眾多著名大學(xué)（如斯坦福大學(xué)、哈佛大學(xué)等）采用。全書分為三部分：第一部分為實變函數(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。

作者簡介

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

書籍目錄

Lebesgue Integration for Functions of a Single Real VariablePreliminaries on Sets, Mappings, and RelationsUnions and Intersections of SetsEquivalence Relations, the Axiom of Choice, and Zorn's Lemma 1 The Real Numbers: Sets. Sequences, and FunctionsThe Field, Positivity, and Completeness AxiomsThe Natural and Rational NumbersCountable and Uncountable SetsOpen Sets, Closed Sets, and Borel Sets of Real Numbers Sequences of Real NumbersContinuous Real-Valued Functions of a Real Variable2 Lebesgne MeasureIntroductionLebesgue Outer MeasureThe o'-Algebra of Lebesgue Measurable SetsOuter and Inner Approximation of Lebesgue Measurable Sets  Countable Additivity, Continuity, and the Borel-Cantelli LemmaNoumeasurable SetsThe Cantor Set and the Cantor Lebesgue Function3 LebesgRe Measurable FunctionsSums, Products, and CompositionsSequential Pointwise Limits and Simple ApproximationLittlewood's Three Principles, Egoroff's Theorem, and Lusin's Theorem4 Lebesgue IntegrationThe Riemann IntegralThe Lebesgue Integral of a Bounded Measurable Function over a Set ofFinite MeasureThe Lebesgue Integral of a Measurable Nonnegative FunctionThe General Lebesgue IntegralCountable Additivity and Continuity of IntegrationUniform Integrability: The Vifali Convergence Theoremviii Contents5 Lebusgue Integration: Fm'ther TopicsUniform Integrability and Tightness: A General Vitali Convergence TheoremConvergence in MeasureCharacterizations of Riemaun and Lebesgue Integrability6 Differentiation and IntegrationContinuity of Monotone FunctionsDifferentiability of Monotone Functions: Lebesgue's TheoremFunctions of Bounded Variation: Jordan's TheoremAbsolutely Continuous FunctionsIntegrating Derivatives: Differentiating Indefinite IntegralsConvex Function7 The Lp Spaces: Completeness and Appro~umationNor/ned Linear SpacesThe Inequalities of Young, HOlder, and Minkowski Lv Is Complete: The Riesz-Fiseher TheoremApproximation and Separability8 The LP Spacesc Deailty and Weak ConvergenceThe Riesz Representation for the Dual ofWeak Sequential Convergence in Lv  Weak Sequential CompactnessThe Minimization of Convex FunctionalsII  Abstract Spaces: Metric, Topological, Banach, and Hiibert Spaces9. Metric Spaces: General PropertiesExamples of Metric SpacesOpen Sets, Closed Sets, and Convergent SequencesContinuous Mappings Between Metric Spaces Complete Metric SpacesCompact Metric SpacesSeparable Metric Spaces10 Metric Spaces: Three Fundamental ThanreessThe Arzelb.-Ascoli TheoremThe Baire Category TheoremThe Banaeh Contraction PrincipleH Topological Spaces: General PropertiesOpen Sets, Closed Sets, Bases, and SubbasesThe Separation PropertiesCountability and SeparabilityContinuous Mappings Between Topological SpacesCompact Topological SpacesConnected Topological Spaces12 Topological Spaces: Three Fundamental TheoremsUrysohn's Lemma and the Tietze Extension Theorem The Tychonoff Product TheoremThe Stone-Weierstrass Theorem13 Continuous Linear Operators Between Bausch SpacesNormed Linear SpacesLinear OperatorsCompactness Lost: Infinite Dimensional Normod Linear SpacesThe Open Mapping and Closed Graph TheoremsThe Uniform Boundedness Principle14 Duality for Normed Iinear SpacesLinear Ftmctionals, Bounded Linear Functionals, and Weak TopologiesThe Hahn-Banach TheoremReflexive Banach Spaces and Weak Sequential ConvergenceLocally Convex Topological Vector SpacesThe Separation of Convex Sets and Mazur's TheoremThe Krein-Miiman Theorem15 Compactness Regained: The Weak TopologyAlaoglu's Extension of Helley's TheoremReflexivity and Weak Compactness: Kakutani's TheoremCompactness and Weak Sequential Compactness: The Eberlein-mulianTheoremMemzability of Weak Topologies 16 Continuous Linear Operators on Hilbert SpacesThe Inner Product and OrthogonalityThe Dual Space and Weak Sequential ConvergenceBessers Inequality and Orthonormal BasesbAdjoints and Symmetry for Linear OperatorsCompact OperatorsThe Hilbert-Schmidt TheoremThe Riesz-Schauder Theorem: Characterization of Fredholm Operators Measure and Integration: General Theory17 General Measure Spaces: Their Propertles and ConstructionMeasures and Measurable SetsSigned Measures: The Hahn and Jordan Decompositions  The Caratheodory Measure Induced by an Outer Measure18 Integration Oeneral Measure Spaces19 Gengral L Spaces:Completeness,Duality and Weak Convergence20 The Construciton of Particular Measures21 Measure and Topbogy22 Invariant MeasuresBibiiographyindex

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《實分析(英文版·第4版)》 :新增了50％的習(xí)題。擴充了基本結(jié)果。包括給出葉果洛夫定理和烏霄松引理的證明。介紹了博雷爾一坎特利引理、切比霄夫不等式、快速柯西序列及測度和積分所共有的連續(xù)性質(zhì).以及若干其他概念。

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