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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．

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
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