Real Analysis, 5th edition

  • Halsey Royden, 
  • Patrick Fitzpatrick

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Real Analysis covers the essentials of real analysis for graduate students. It explores the classical theory of functions of a real variable, measure and integration theory, and some of the more important and elementary topics in general topology and normed linear space theory. Content is divided into three parts: classical theory of functions, including the classical Banach spaces; general topology and the theory of general Banach spaces; and abstract treatment of measure and integration. It assumes a general background in undergraduate mathematics and familiarity with the material covered in an undergraduate course on the fundamental concepts of analysis.

Published by Pearson (July 7th 2023) - Copyright © 2023

ISBN-13: 9780136853473

Subject: Advanced Math

Category: Real Analysis

Table of contents


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


  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)


  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-Šmulian 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


  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



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