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

Subject Matter

 * Real Analysis

Contents

 * Prologue to the Student


 * 1 Set Theory
 * 1 Introduction
 * 2 Functions
 * 3 Unions, intersections, and complements
 * 4 Algebras of sets
 * 5 The axiom of choice and infinite direct products
 * 6 Countable sets
 * 7 Relations and equivalences
 * 8 Partial orderings and the maximal principle
 * 9 Well ordering and the countable ordinals


 * Part One Theory of Functions of a Real Variable
 * 2 The Real Number System
 * 1 Axioms for the real numbers
 * 2 The natural and rational numbers as subsets of $\mathbf R$
 * 3 The extended real numbers
 * 4 Sequences of real numbers
 * 5 Open and closed sets of real numbers
 * 6 Continuous functions
 * 7 Borel sets


 * 3 Lebesgue Measure
 * 1 Introduction
 * 2 Outer measure
 * 3 Measurable sets and Lebesgue measure
 * * 4 A nonmeasurable set
 * 5 Measurable functions
 * 6 Littlewood's three principles


 * 4 The Lebesgue Integral
 * 1 The Riemann integral
 * 2 The Lebesgue integral of a bounded function over a set of finite measure
 * 3 The integral of a nonnegative function
 * 4 The general Lebesgue integral
 * * 5 Convergence in measure


 * 5 Differentiation and Integration
 * 1 Differentiation of monotone functions
 * 2 Functions of bounded variation
 * 3 Differentiation of an integral
 * 4 Absolute continuity
 * * 5 Convex functions


 * 6 The Classical Banach Spaces
 * 1 The $L^p$ spaces
 * 2 The Minkowski and Hölder inequalities
 * 3 Convergence and completeness
 * 4 Approximation in $L^p$
 * 5 Bounded linear functionals on the $L^p$ spaces


 * Part Two Abstract Spaces
 * 7 Metric Spaces
 * 1 Introduction
 * 2 Open and closed sets
 * 3 Continuous functions and homeomorphisms
 * 4 Convergence and completeness
 * 5 Uniform continuity and uniformity
 * 6 Subspaces
 * 8 Baire category


 * 8 Topological Spaces
 * 1 Fundamental notions
 * 2 Bases and countability
 * 3 The separation axioms and continuous real-valued functions
 * 4 Product spaces
 * 5 Connectedness
 * * 6 Absolute $\mathcal G_\delta$'s
 * * 7 Nets


 * 9 Compact Spaces
 * 1 Basic properties
 * 2 Countable compactness and the Bolzano-Weierstrass property
 * 3 Compact metric spaces
 * 4 Products of compact spaces
 * 4 Locally compact spaces
 * * 6 The Stone-Čech compactification
 * 7 The Stone-Weierstrass theorem
 * * The Ascoli theorem


 * 10 Banach Spaces
 * 1 Introduction
 * 2 Linear operators
 * 3 Linear functionals and the Hahn-Banach theorem
 * 4 The closed graph theorem
 * * 5 Topological vector spaces
 * * 6 Weak topologies
 * * 7 Convexity
 * 8 Hilbert space


 * Part Three General Measure and Integration Theory
 * 11 Measure and Integration
 * 1 Measure spaces
 * 2 Measurable functions
 * 3 Integration
 * * 4 General convergence theorems
 * 5 Signed measures
 * 6 The Radon-Nikodym Theorem
 * 7 The $L^p$ spaces


 * 12 Measure and Outer Measure
 * 1 Outer measure and measurability
 * 2 The extension theorem
 * * 3 The Lebesgue-Stieltjes integral
 * 4 Product measures
 * * 5 Inner measure
 * * 6 Extension by sets of measure zero
 * * 7 Carathéodory outer measure


 * 13 The Daniell Integral
 * 1 Introduction
 * 2 The extension theorem
 * 3 Uniqueness
 * 4 Measurability and measure


 * 14 Measure and Topology
 * 1 Baire sets and Borel sets
 * 2 Positive linear functionals and Borel measures
 * 3 Bounded linear functionals on $C \left({X}\right)$
 * * 4 The Borel extension of a measure


 * 15 Mappings of Measure Spaces
 * 1 Point mappings and set mappings
 * 2 Measure algebras
 * 3 Borel equivalences
 * 4 Set mappings and point mappings on complete separable metric spaces
 * 5 The isometries of $L^p$


 * Epilogue
 * Bibliography
 * Index of Symbols
 * Subject Index