內容簡介
《實分析(英文版·第4版)》是實分析課程的優秀教材，被國外眾多著名大學（如史丹福大學、哈佛大學等）採用。全書分為三部分：第一部分為實變函式論，介紹一元實變函式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、代數和動態結構的一般理論。書中不僅包含數學定理和定義，而且還提出了富有啟發性的問題，以便讀者更深入地理解書中內容。
作品目錄
Lebesgue Integration for Functions of a Single Real Variable
Preliminaries on Sets, Mappings, and Relations
Unions and Intersections of Sets
Equivalence Relations, the Axiom of Choice, and Zorn's Lemma
1 The Real Numbers: Sets. Sequences, and Functions
The Field, Positivity, and Completeness Axioms
The Natural and Rational Numbers
Countable and Uncountable Sets
Open Sets, Closed Sets, and Borel Sets of Real Numbers
Sequences of Real Numbers
Continuous Real-Valued Functions of a Real Variable
2 Lebesgne Measure
Introduction
Lebesgue Outer Measure
The o'-Algebra of Lebesgue Measurable Sets
Outer and Inner Approximation of Lebesgue Measurable Sets
Countable Additivity, Continuity, and the Borel-Cantelli Lemma
Noumeasurable Sets
The Cantor Set and the Cantor Lebesgue Function
3 LebesgRe Measurable Functions
Sums, Products, and Compositions
Sequential Pointwise Limits and Simple Approximation
Littlewood's Three Principles, Egoroff's Theorem, and Lusin's Theorem
4 Lebesgue Integration
The Riemann Integral
The Lebesgue Integral of a Bounded Measurable Function over a Set of
Finite Measure
The Lebesgue Integral of a Measurable Nonnegative Function
The General Lebesgue Integral
Countable Additivity and Continuity of Integration
Uniform Integrability: The Vifali Convergence Theorem
viii Contents
5 Lebusgue Integration: Fm'ther Topics
Uniform Integrability and Tightness: A General Vitali Convergence Theorem
Convergence in Measure
Characterizations of Riemaun and Lebesgue Integrability
6 Differentiation and Integration
Continuity of Monotone Functions
Differentiability of Monotone Functions: Lebesgue's Theorem
Functions of Bounded Variation: Jordan's Theorem
Absolutely Continuous Functions
Integrating Derivatives: Differentiating Indefinite Integrals
Convex Function
7 The Lp Spaces: Completeness and Appro~umation
Nor/ned Linear Spaces
The Inequalities of Young, HOlder, and Minkowski
Lv Is Complete: The Riesz-Fiseher Theorem
Approximation and Separability
8 The LP Spacesc Deailty and Weak Convergence
The Riesz Representation for the Dual of
Weak Sequential Convergence in Lv
Weak Sequential Compactness
The Minimization of Convex Functionals
II Abstract Spaces: Metric, Topological, Banach, and Hiibert Spaces
9. Metric Spaces: General Properties
Examples of Metric Spaces
Open Sets, Closed Sets, and Convergent Sequences
Continuous Mappings Between Metric Spaces
Complete Metric Spaces
Compact Metric Spaces
Separable Metric Spaces
10 Metric Spaces: Three Fundamental Thanreess
The Arzelb.-Ascoli Theorem
The Baire Category Theorem
The Banaeh Contraction Principle
H Topological Spaces: General Properties
Open Sets, Closed Sets, Bases, and Subbases
The Separation Properties
Countability and Separability
Continuous Mappings Between Topological Spaces
Compact Topological Spaces
Connected Topological Spaces
12 Topological Spaces: Three Fundamental Theorems
Urysohn's Lemma and the Tietze Extension Theorem
The Tychonoff Product Theorem
The Stone-Weierstrass Theorem
13 Continuous Linear Operators Between Bausch Spaces
Normed Linear Spaces
Linear Operators
Compactness Lost: Infinite Dimensional Normod Linear Spaces
The Open Mapping and Closed Graph Theorems
The Uniform Boundedness Principle
14 Duality for Normed Iinear Spaces
Linear Ftmctionals, Bounded Linear Functionals, and Weak Topologies
The Hahn-Banach Theorem
Reflexive Banach Spaces and Weak Sequential Convergence
Locally Convex Topological Vector Spaces
The Separation of Convex Sets and Mazur's Theorem
The Krein-Miiman Theorem
15 Compactness Regained: The Weak Topology
Alaoglu's Extension of Helley's Theorem
Reflexivity and Weak Compactness: Kakutani's Theorem
Compactness and Weak Sequential Compactness: The Eberlein-mulian
Theorem
Memzability of Weak Topologies
16 Continuous Linear Operators on Hilbert Spaces
The Inner Product and Orthogonality
The Dual Space and Weak Sequential Convergence
Bessers Inequality and Orthonormal Bases
bAdjoints and Symmetry for Linear Operators
Compact Operators
The Hilbert-Schmidt Theorem
The Riesz-Schauder Theorem: Characterization of Fredholm Operators
Measure and Integration: General Theory
17 General Measure Spaces: Their Propertles and Construction
Measures and Measurable Sets
Signed Measures: The Hahn and Jordan Decompositions
The Caratheodory Measure Induced by an Outer Measure
18 Integration Oeneral Measure Spaces
19 Gengral L Spaces:Completeness,Duality and Weak Convergence
20 The Construciton of Particular Measures
21 Measure and Topbogy
22 Invariant Measures
Bibiiography
index
