Book:H.L. Royden/Real Analysis/Fourth Edition

Subject Matter

 * Real Analysis

Contents

 * Preface


 * 1 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
 * 1.1 The Field, Positivity, and Completeness Axioms
 * 1.2 The Natural and Rational Numbers
 * 1.3 Countable and Uncountable Sets
 * 1.4 Open Sets, Closed Sets, and Borel Sets of Real Numbers
 * 1.5 Sequences of Real Numbers
 * 1.6 Continuous Real-Valued Functions of a Real Variable


 * 2 Lebesgue Measure
 * 2.1 Introduction
 * 2.2 Lebesgue Outer Measure
 * 2.3 The $\sigma$-Algebra of Lebesgue Measurable Sets
 * 2.4 Outer and Inner Approximation of Lebesgue Measurable Sets
 * 2.5 Countable Additivity, Continuity, and the Borel-Cantelli Lemma
 * 2.6 Nonmeasurable Sets
 * 2.7 The Cantor Set and the Cantor-Lebesgue Function


 * 3 Lebesgue Measurable Function
 * 3.1 Sums, Products, and Compositions
 * 3.2 Sequential Pointwise Limits and Simple Approximations
 * 3.3 Littlewood's Three Principles, Egoroff's Theorem, and Lusin's Theorem


 * 4 Lebesgue Integration
 * 4.1 The Riemann Integral
 * 4.2 The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure
 * 4.3 The Lebesgue Integral of a Measurable Nonnegative Function
 * 4.4 The General Lebesgue Integral
 * 4.5 Countable Additivity and Continuity of Integration
 * 4.6 Uniform Integrability: The Vitali Convergence Theorem


 * 5 Lebesgue Integration: Further Topics
 * 5.1 Uniform Integrability and Tightness
 * 5.2 Convergence in Measure
 * 5.3 Characterizations of Riemann and Lebesgue Integrability


 * 6 Differentiation and Integration
 * 6.1 Continuity of Monotone Functions
 * 6.2 Differentiability of Monotone Functions: Lebesgue's Theorem
 * 6.3 Functions of Bounded Variation: Jordan's Theorem
 * 6.4 Absolutely Continuous Functions
 * 6.5 Integrating Derivatives: Differentiating Indefinite Integrals
 * 6.6 Convex Functions


 * 7 The $L^p$ Spaces: Completeness and Approximation
 * 7.1 Normed Linear Spaces
 * 7.2 The Inequalities of Young, Hölder, and Minkowski
 * 7.3 $L^p$ Is Complete: The Riesz-Fischer Theorem
 * 7.4 Approximation and Separability


 * 8 The $L^p$ Spaces: Duality and Weak Convergence
 * 8.1 The Riesz Representation for the Dual of $L^p$, $1 \le p < \infty$
 * 8.2 Weak Sequential Convergence in $L^p$
 * 8.3 Weak Sequential Compactness
 * 8.4 The Minimization of Convex Functionals


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


 * 9 Metric Spaces: General Properties
 * 9.1 Examples of Metric Spaces
 * 9.2 Open Sets, Closed Sets, and Convergent Sequences
 * 9.3 Continuous Mappings Between Metric Spaces
 * 9.4 Complete Metric Spaces
 * 9.5 Compact Metric Spaces
 * 9.6 Separable Metric Spaces


 * 10 Metric Spaces: Three Fundamental Theorems
 * 10.1 The Arzelà-Ascoli Theorem
 * 10.2 The Baire Category Theorem
 * 10.3 The Banach Contraction Principle


 * 11 Topological Spaces: General Properties
 * 11.1 Open Sets, Closed Sets, Bases, and Subbases
 * 11.2 The Separation Properties
 * 11.3 Countability and Separability
 * 11.4 Continuous Mappings Between Topological Spaces
 * 11.5 Compact Topological Spaces
 * 11.6 Connected Topological Spaces


 * 12 Topological Spaces: Three Fundamental Theorems
 * 12.1 Urysohn's Lemma and the Tietze Extension Theorem
 * 12.2 The Tychonoff Product Theorem
 * 12.3 The Stone-Weierstrass Theorem


 * 13 Continuous Linear Operators Between Banach Spaces
 * 13.1 Normed Linear Spaces
 * 13.2 Linear Operators
 * 13.3 Compactness Lost: Infinite Dimensional Normed Linear Spaces
 * 13.4 The Open Mapping and Closed Graph Theorems
 * 13.5 The Uniform Boundedness Principle


 * 14 Duality for Normed Linear Spaces
 * 14.1 Linear Functionals, Bounded Linear Functionals, and Weak Topologies
 * 14.2 The Hahn-Banach Theorem
 * 14.3 Reflexive Banach Spaces and Weak Sequential Convergence
 * 14.4 Locally Convex Topological Vector Spaces
 * 14.5 The Separation of Convex Sets and Mazur's Theorem
 * 14.6 The Krein-Milman Theorem


 * 15 Compactness Regained: The Weak Topology
 * 15.1 Alaoglu's Extension of Helley's Theorem
 * 15.2 Reflexivity and Weak Compactness: Kakutani's Theorem
 * 15.3 Compactness and Weak Sequential Compactness: The Eberlein-Šmulian Theorem
 * 15.4 Metrizability of Weak Topologies


 * 16 Continuous Linear Operators on Hilbert Spaces
 * 16.1 The Inner Product and Orthogonality
 * 16.2 The Dual Space and Weak Sequential Convergence
 * 16.3 Bessel's Inequality and Orthonormal Bases
 * 16.4 Adjoints and Symmetry for Linear Operators
 * 16.5 Compact Operators
 * 16.6 The Hilbert-Schmidt Theorem
 * 16.7 The Riesz-Schauder Theorem: Characterization of Fredholm Operators


 * III Measure and Integration: General Theory


 * 17 General Measure Spaces: Their Properties and Construction
 * 17.1 Measures and Measurable Sets
 * 17.2 Signed Measures: The Hahn and Jordan Decompositions
 * 17.3 The Carathéodory Measure Induced by an Outer Measure
 * 17.4 The Construction of Outer Measures
 * 17.5 The Carathéodory-Hahn Theorem: The Extension of a Premeasure to a Measure


 * 18 Integration Over General Measure Spaces
 * 18.1 Measurable Functions
 * 18.2 Integration of Nonnegative Measurable Functions
 * 18.3 Integration of General Measurable Functions
 * 18.4 The Radon-Nikodym Theorem
 * 18.5 The Nikodym Metric Space: The Vitali-Hahn-Saks Theorem


 * 19 General $L^p$ Spaces: Completeness, Duality, and Weak Convergence
 * 19.1 The Completeness of $L^p(X,\mu)$, $1 \le p \le \infty$
 * 19.2 The Riesz Representation Theorem for the Dual of $L^p(X,\mu)$, $1 \le p \le \infty$
 * 19.3 The Kantorovich Representation Theorem for the Dual of $L^\infty(X,\mu)$
 * 19.4 Weak Sequential Compactness in $L^p(X,\mu)$, $1 < p < 1$
 * 19.5 Weak Sequential Compactness in $L^1(X,\mu)$: The Dunford-Pettis Theorem


 * 20 The Construction of Particular Measures
 * 20.1 Product Measures: The Theorems of Fubini and Tonelli
 * 20.2 Lebesgue Measure on Euclidean Space $R^n$
 * 20.3 Cumulative Distribution Functions and Borel Measures on $R$
 * 20.4 Carathéodory Outer Measures and Hausdorff Measures on a Metric Space


 * 21 Measure and Topology
 * 21.1 Locally Compact Topological Spaces
 * 21.2 Separating Sets and Extending Functions
 * 21.3 The Construction of Radon Measures
 * 21.4 The Representation of Positive Linear Functionals on $C_c(X)$: The Riesz-Markov Theorem
 * 21.5 The Riesz Representation Theorem for the Dual of $C(X)$
 * 21.6 Regularity Properties of Baire Measures


 * 22 Invariant Measures
 * 22.1 Topological Groups: The General Linear Group
 * 22.2 Kakutani's Fixed Point Theorem
 * 22.3 Invariant Borel Measures on Compact Groups: von Neumann's Theorem
 * 22.4 Measure Preserving Transformations and Ergodicity: The Bogoliubov-Krilov Theorem


 * Bibliography
 * Index