Real Analysis, 5th edition
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Overview
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
ISBN13: 9780136853473
Subject: Advanced Math
Category: Real Analysis
Table of contents
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
 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 RealValued Functions of a Real Variable
 Lebesgue Measure
 2.1 Introduction
 2.2 Outer Measure
 2.3 The σalgebra of Lebesgue Measurable Sets
 2.4 Finer Properties of Measurable Sets
 2.5 Countable Additivity and Continuity of Measure, and the BorelCantelli Lemma
 2.6 Vitali’s Example of a Nonmeasurable Set
 2.7 The Cantor Set and the CantorLebesgue Function
 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
 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 NonNegative Measurable Function
 4.4 The General Lebesgue Integral
 4.5 Countable Additivity and Continuity of Integration
 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
 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
 The L^{Ρ} Spaces: Completeness and Approximation
 7.1 Normed Linear Spaces
 7.2 The Inequalities of Young, Hölder and Minkowski
 7.3 L^{Ρ} is Complete: Rapidly Cauchy Sequences and The RieszFischer Theorem
 7.4 Approximation and Separability
 The L^{Ρ} 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 L^{p}, 1 ≤ p < ∞
 8.3 Weak Sequential Convergence in L^{p}
 8.4 The Minimization of Convex Functionals
II: MEASURE AND INTEGRATION: GENERAL THEORY
 General Measure Spaces: Their Properties and Construction
 9.1 Measurable Sets and Measure Spaces
 9.2 Measures Induced by an Outermeasure
 9.3 The CarathéodoryHahn Theorem
 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 R^{n}, and Cumulative Distribution Functions
 10.4 Carathéodory Outermeasures and Hausdorff Measures
 10.5 Signed Measures: the Hahn and Jordan Decompositions
 Integration Over General Measure Spaces
 11.1 Measurable Functions: the Egoroff and Lusin Theorems
 11.2 Integration of Nonnegative 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 RadonNikodym Theorem
 11.5 Product Measures: the Tonelli and Fubini Theorems
 11.6 Products of Lebesgue measure on Euclidean spaces: Cavalieri’s Principle
 General L^{p} Spaces: Completeness, Convolution, and Duality
 12.1 The Spaces L^{p}(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 L^{p}(X; μ); 1 ≤ p < ∞
 12.4 Weak Sequential Compactness in L^{p}(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
 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
 Metric Spaces: Three Fundamental Theorems and Applications
 14.1 The ArzelàAscoli Theorem
 14.2 The Banach Contraction Principle
 14.3 The Baire Category Theorem
 14.4 The Nikodym Metric Space: The VitaliHahnSaks Theorem and the DunfordPettis Theorem
 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
 Topological Spaces: Three Fundamental Theorems
 16.1 Urysohn’s Lemma and the Tietze Extension Theorem
 16.2 The Tychonoff Product Theorem
 16.3 The StoneWeierstrass Theorem
 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
 Duality for Normed Linear Spaces
 18.1 Linear Functionals, Bounded Linear Functionals, and Weak Topologies
 18.2 The HahnBanach 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 KreinMilman Theorem
 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 EberleinŠmulian Theorem
 19.4 Metrizability of Weak Topologies
 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 HilbertSchmidt Theorem
 20.7 The RieszSchauder Theorem: Characterization of Fredholm Operators
IV: MEASURE AND TOPOLOGY: INVARIANT MEASURES
 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 RieszMarkov Theorem
 21.5 The Riesz Representation Theorem for the Dual of C(X): The RieszKakutani Theorem
 21.6 Regularity Properties of Baire Measures
 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 BogoliubovKrilov Theorem
Bibliography
Index
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