Real Analysis, 5th edition

Published by Pearson (July 7, 2023) © 2023

  • Halsey Royden
  • Patrick Fitzpatrick

eTextbook on Pearson+

ISBN-13: 9780136853473 (2023 update)

eTextbook rental includes

  • Instant access to eTextbook
  • Search, highlight, and notes
  • Create flashcards
Products list

Details

  • A print text you can rent
  • Fulfilled by eCampus.com
  • Option to keep after rental expires

Halsey L. Royden's Real Analysis has contributed to educating generations of mathematical analysis students. The 5th Edition of this classic text presents some important updates while presenting the measure theory, integration theory and elements of metric, topological, Hilbert and Banach spaces that a modern analyst should know. Part I continues to consider Lebesgue measure and integration for functions of a real variable. In this revision, the treatment of general measure and integration is moved to Part II rather than Part III; material formerly in Part II is placed in Part III and a brief Part IV. This brings measure and integration on Euclidean space closer to their origin, the case of real variables; it also presents the opportunity to foreshadow more strongly, in the context of general measure and integration, concepts which later appear in general spaces. The text assumes an undergraduate course on the fundamental concepts of analysis.

I: LEBESGUE INTEGRATION FOR FUNCTIONS OF A SINGLE REAL VARIABLE

Preliminaries on Sets, Mappings, and Relations

  • Unions and Intersections of Sets
  • Mappings Between 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
  1. Lebesgue Measure
    • 2.1 Introduction
    • 2.2 Outer Measure
    • 2.3 The σ-algebra of Lebesgue Measurable Sets
    • 2.4 Finer Properties of Measurable Sets
    • 2.5 Countable Additivity and Continuity of Measure, and the Borel-Cantelli Lemma
    • 2.6 Vitali's Example of a Nonmeasurable Set
    • 2.7 The Cantor Set and the Cantor-Lebesgue Function
  1. Lebesgue Measurable Functions
    • 3.1 Sums, Products, and Compositions
    • 3.2 Sequential Pointwise Limits and Simple Approximation
    • 3.3 Littlewood's Three Principles, Egoroff's Theorem and Lusin's Theorem
  1. Lebesgue Integration
    • 4.1 Comments on the Riemann Integral
    • 4.2 The Integral of a Bounded, Finitely Supported, Measurable Function
    • 4.3 The Integral of a Non-Negative Measurable Function
    • 4.4 The General Lebesgue Integral
    • 4.5 Countable Additivity and Continuity of Integration
  1. Lebesgue Integration: Further Topics
    • 5.1 Uniform Integrability and Tightness: The Vitali Convergence Theorems
    • 5.2 Convergence in the Mean and in Measure: A Theorem of Riesz
    • 5.3 Characterizations of Riemann and Lebesgue Integrability
  1. 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 Measurability: Images of Sets, Compositions of Functions
    • 6.7 Convex Functions
  1. The LΡ Spaces: Completeness and Approximation
    • 7.1 Normed Linear Spaces
    • 7.2 The Inequalities of Young, Hölder and Minkowski
    • 7.3  LΡ is Complete: Rapidly Cauchy Sequences and The Riesz-Fischer Theorem
    • 7.4 Approximation and Separability
  1. The  LΡ Spaces: Duality, Weak Convergence and Minimization
    • 8.1 Bounded Linear Functionals on a Normed Linear Space
    • 8.2 The Riesz Representation of the Dual of Lp, 1 ≤ p < ∞
    • 8.3 Weak Sequential Convergence in Lp
    • 8.4 The Minimization of Convex Functionals

II: MEASURE AND INTEGRATION: GENERAL THEORY

  1. General Measure Spaces: Their Properties and Construction
    • 9.1 Measurable Sets and Measure Spaces
    • 9.2 Measures Induced by an Outer-measure
    • 9.3 The Carathéodory-Hahn Theorem
  1. Particular Measures: Lebesgue Measure on Euclidean Space, Borel Measures, and Signed Measure
    • 10.1 Lebesgue Measure on Euclidean Space
    • 10.2 Lebesgue Measurability and Measure of Images of Mappings
    • 10.3 Regularity of Borel Measures on Rn, and Cumulative Distribution Functions
    • 10.4 Carathéodory Outer-measures and Hausdorff Measures
    • 10.5 Signed Measures: the Hahn and Jordan Decompositions
  1. Integration Over General Measure Spaces
    • 11.1 Measurable Functions: the Egoroff and Lusin Theorems
    • 11.2 Integration of Non-negative Measurable Functions: Fatou's Lemma, the Monotone Convergence Theorem and Beppo Levi's Theorem
    • 11.3 Integration of General Measurable Functions: the Dominated Convergence Theorem and the Vitali Convergence Theorem
    • 11.4 The Radon-Nikodym Theorem
    • 11.5 Product Measures: the Tonelli and Fubini Theorems
    • 11.6 Products of Lebesgue measure on Euclidean spaces: Cavalieri's Principle
  1. General Lp Spaces: Completeness, Convolution, and Duality
    • 12.1 The Spaces Lp(X; μ); 1 ≤ p ≤ ∞
    • 12.2 Convolution, Smooth Approximation and a Smooth Urysohn's Lemma
    • 12.3 The Riesz Representation Theorem for the Dual of Lp(X; μ); 1 ≤ p < ∞
    • 12.4 Weak Sequential Compactness in Lp(X; μ); 1 < p < ∞
    • 12.5 The Kantorovitch Representation Theorem for the Dual of L∞ (X; μ)

III: ABSTRACT SPACES: METRIC, TOPOLOGICAL, BANACH, AND HILBERT SPACES

  1. Metric Spaces: General Properties
    • 13.1 Examples of Metric Spaces
    • 13.2 Open Sets, Closed Sets, and Convergent Sequences
    • 13.3 Continuous Mappings Between Metric Spaces
    • 13.4 Complete Metric Spaces
    • 13.5 Compact Metric Spaces
    • 13.6 Separable Metric Spaces
  1. Metric Spaces: Three Fundamental Theorems and Applications
    • 14.1 The Arzelà-Ascoli Theorem
    • 14.2 The Banach Contraction Principle
    • 14.3 The Baire Category Theorem
    • 14.4 The Nikodym Metric Space: The Vitali-Hahn-Saks Theorem and the Dunford-Pettis Theorem
  1. Topological Spaces: General Properties
    • 15.1 Open Sets, Closed Sets, Bases, and Subbases
    • 15.2 The Separation Properties
    • 15.3 Countability and Separability
    • 15.4 Continuous Mappings Between Topological Spaces
    • 15.5 Compact Topological Spaces
    • 15.6 Connected Topological Spaces
  1. Topological Spaces: Three Fundamental Theorems
    • 16.1 Urysohn's Lemma and the Tietze Extension Theorem
    • 16.2 The Tychonoff Product Theorem
    • 16.3 The Stone-Weierstrass Theorem
  1. Continuous Linear Operators Between Banach Spaces
    • 17.1 Normed Linear Spaces
    • 17.2 Linear Operators
    • 17.3 Compactness Lost: Infinite Dimensional Normed Linear Spaces
    • 17.4 The Open Mapping and Closed Graph Theorems
    • 17.5 The Uniform Boundedness Principle
  1. Duality for Normed Linear Spaces
    • 18.1 Linear Functionals, Bounded Linear Functionals, and Weak Topologies
    • 18.2 The Hahn-Banach Theorem
    • 18.3 Reflexive Banach Spaces and Weak Sequential Convergence
    • 18.4 Locally Convex Topological Vector Spaces
    • 18.5 The Separation of Convex Sets and Mazur's Theorem
    • 18.6 The Krein-Milman Theorem
  1. Compactness Regained: The Weak Topology
    • 19.1 Alaoglu's Extension of Helly's Theorem
    • 19.2 Reflexivity and Weak Compactness: Kakutani's Theorem
    • 19.3 Compactness and Weak Sequential Compactness: The Eberlein-Å mulian Theorem
    • 19.4 Metrizability of Weak Topologies
  1. Continuous Linear Operators on Hilbert Spaces
    • 20.1 The Inner Product and Orthogonality
    • 20.2 Bessel's Inequality and Orthonormal Bases
    • 20.3 The Dual Space and Weak Sequential Convergence
    • 20.4 Symmetric Operators
    • 20.5 Compact Operators
    • 20.6 The Hilbert-Schmidt Theorem
    • 20.7 The Riesz-Schauder Theorem: Characterization of Fredholm Operators

IV: MEASURE AND TOPOLOGY: INVARIANT MEASURES

  1. 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 Cc (X): The Riesz-Markov Theorem
    • 21.5 The Riesz Representation Theorem for the Dual of C(X): The Riesz-Kakutani Theorem
    • 21.6 Regularity Properties of Baire Measures
  1. 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

Need help? Get in touch