Real Analysis

PART I: LEBESGUE INTEGRATION FOR FUNCTIONS OF A SINGLE REAL VARIABLE

 

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

 

2. Lebesgue Measure

2.1 Introduction

2.2 Lebesgue Outer Measure

2.3 The σ-algebra of Lebesgue Measurable Sets

2.4 Outer and Inner Approximation of Lebesgue Measurable Sets

2.5 Countable Additivity and Continuity of Lebesgue Measure

2.6 Nonmeasurable Sets

2.7 The Cantor Set and the Cantor-Lebesgue Function

 

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

 

4. Lebesgue Integration

4.1 The Riemann Integral

4.2 The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure

4.3 The Lebesgue Integral of a Measurable Nonnegative Function

4.4 The General Lebesgue Integral

4.5 Countable Additivity and Continuity of Integraion

4.6 Uniform Integrability: The Vitali Convergence Theorem

 

5. Lebesgue Integration: Further Topics

5.1 Uniform Integrability and Tightness: A General Vitali Convergence Theorem

5.2 Convergence in measure

5.3 Characterizations of Riemann and Lebesgue Integrability

 

6. 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 Convex Functions

 

7. 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: The Riesz-Fischer Theorem

7.4 Approximation and Separability

 

8. The LΡ Spaces: Duality and Weak Convergence

8.1 The Dual Space of LΡ

8.2 Weak Sequential Convergence in LΡ

8.3 Weak Sequential Compactness

8.4 The Minimization of Convex Functionals

 

PART II: ABSTRACT SPACES: METRIC, TOPOLOGICAL, AND HILBERT

 

9. Metric Spaces: General Properties

9.1 Examples of Metric Spaces

9.2 Open Sets, Closed Sets and Convergent Sequences

9.3 Continuous Mappings Between Metric Spaces

9.4 Complete Metric Spaces

9.5 Compact Metric Spaces

9.6 Separable Metric Spaces

 

10. Metric Spaces: Three Fundamental Theorems

10.1 The Arzelà-Ascoli Theorem

10.2 The Baire Category Theorem

10.3 The Banach Contraction Principle

 

11. Topological Spaces: General Properties

11.1 Open Sets, Closed Sets, Bases and Subbases

11.2 The Separation Properties

11.3 Countability and Separability

11.4 Continuous Mappings Between Topological Spaces

11.5 Compact Topological Spaces

11.6 Connected Topological Spaces

 

12. Topological Spaces: Three Fundamental Theorems

12.1 Urysohn's Lemma and the Tietze Extension Theorem

12.2 The Tychonoff Product Theorem

12.3 The Stone-Weierstrass Theorem

 

13. Continuous Linear Operators Between Banach Spaces

13.1 Normed Linear Spaces

13.2 Linear Operators

13.3 Compactness Lost: Infinite Dimensional Normed Linear Spaces

13.4 The Open Mapping and Closed Graph Theorems

13.5 The Uniform Boundedness Principle

 

14. Duality for Normed Linear Spaces

14.1 Linear Functionals, Bounded Linear Functionals and Weak Topologies

14.2 The Hahn-Banach Theorem

14.3 Reflexive Banach Spaces and Weak Sequential Convergence

14.4 Locally Convex Topological Vector Spaces

14.5 The Separation of Convex Sets and Mazur's Theorem

14.6 The Krein-Milman Theorem

 

15. Compactness Regained: The Weak Topology

15.1 Alaoglu's Extension of Helley's Theorem

15.2 Reflexivity and Weak Compactness: Kakutani's Theorem

15.3 Compactness and Weak Sequential Compactness: The Eberlein-Šmulian Theorem

15.4 Metrizability of Weak Topologies

 

16. Continuous Linear Operators on Hilbert Spaces

16.1 The Inner Product and Orthogonality

16.2 The Dual Space and Weak Sequential Convergence

16.3 Bessel's Inequality and Orthonormal Bases

16.4 Adjoints and Symmetry for Linear Operators

16.5 Compact Operators

16.6 The Hilbert Schmidt Theorem

16.7 The Riesz-Schauder Theorem: Characterization of Fredholm Operators

 

PART III: MEASURE AND INTEGRATION: GENERAL THEORY

 

17. General Measure Spaces: Their Properties and Construction

17.1 Measures and Measurable Sets

17.2 Signed Measures: The Hahn and Jordan Decompositions

17.3 The Carathéodory Measure Induced by an Outer Measure

17.4 The Construction of Outer Measures

17.5 The Carathéodory-Hahn Theorem: The Extension of a Premeasure to a Measure

 

18. Integration Over General Measure Spaces

18.1 Measurable Functions

18.2 Integration of Nonnegative Measurable Functions

18.3 Integration of General Measurable Functions

18.4 The Radon-Nikodym Theorem

18.5 The Saks Metric Space: The Vitali-Hahn-Saks Theorem

 

19. General LΡ Spaces: Completeness, Duality and Weak Convergence

19.1 The Completeness of LΡ ( Χ, μ), 1 ≤ Ρ ≤ ∞

19.2 The Riesz Representation theorem for the Dual of LΡ ( Χ, μ), 1 ≤ Ρ ≤ ∞

19.3 The Kantorovitch Representation Theorem for the Dual of L∞ (Χ, μ)

19.4 Weak Sequential Convergence in LΡ (X, μ), 1 < Ρ < 1

19.5 Weak Sequential Compactness in L1 (X, μ): The Dunford-Pettis Theorem

 

20. The Construction of Particular Measures

20.1 Product Measures: The Theorems of Fubini and Tonelli

20.2 Lebesgue Measure on Euclidean Space Rn

20.3 Cumulative Distribution Functions and Borel Measures on R

20.4 Carathéodory Outer Measures and hausdorff Measures on a Metric Space

 

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 Cc (X): The Riesz-Markov Theorem

21.5 The Riesz Representation Theorem for the Dual of C(X)

21.6 Regularity Properties of Baire Measures

 

22. Invariant Measures

22.1 Topological Groups: The General Linear Group

22.2 Fixed Points of Representations: Kakutani's Theorem

22.3 Invariant Borel Measures on Compact Groups: von Neumann's Theorem

22.4 Measure Preserving Transformations and Ergodicity: the Bogoliubov-Krilov Theorem