Real Analysis

Chapter Preliminaries: Preliminaries on Sets, Mappings, and Relations

• • Unions and Intersections of Sets
• • Mappings Between Sets
• • Equivalence Relations, the Axiom of Choice and Zorn's Lemma

Chapter 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

Chapter 2: 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

Chapter 3: 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

Chapter 4: 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

Chapter 5: 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

Chapter 6: Differentation 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

Chapter 7: 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 Riesz-Fischer Theorem
• • 7.4 Approximation and Separability

Chapter 8: 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

Chapter 9: 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

Chapter 10: 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 Outer-measures and Hausdorff Measures
• • 10.5 Signed Measures: the Hahn and Jordan Decompositions

Chapter 11: 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

Chapter 12: 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)

Chapter 13: 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

Chapter 14: 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

Chapter 15: 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

Chapter 16: 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

Chapter 17: 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

Chapter 18: 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

Chapter 19: 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

Chapter 20: 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

Chapter 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*): The Riesz-Kakutani Theorem
• • 21.6 Regularity Properties of Baire Measures

Chapter 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