Real Analysis
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This is the classic introductory graduate text. Heart of the book is measure theory and Lebesque integration.
PART I: LEBESGUE INTEGRATION FOR FUNCTIONS OF A SINGLE REAL VARIABLE1. The Real Numbers: Sets, Sequences and Functions1.1 The Field, Positivity and Completeness Axioms1.2 The Natural and Rational Numbers1.3 Countable and Uncountable Sets1.4 Open Sets, Closed Sets and Borel Sets of Real Numbers1.5 Sequences of Real Numbers1.6 Continuous Real-Valued Functions of a Real Variable2. Lebesgue Measure2.1 Introduction2.2 Lebesgue Outer Measure2.3 The σ-algebra of Lebesgue Measurable Sets2.4 Outer and Inner Approximation of Lebesgue Measurable Sets2.5 Countable Additivity and Continuity of Lebesgue Measure2.6 Nonmeasurable Sets2.7 The Cantor Set and the Cantor-Lebesgue Function3. Lebesgue Measurable Functions3.1 Sums, Products and Compositions3.2 Sequential Pointwise Limits and Simple Approximation3.3 Littlewood's Three Principles, Egoroff's Theorem and Lusin's Theorem4. Lebesgue Integration4.1 The Riemann Integral4.2 The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure4.3 The Lebesgue Integral of a Measurable Nonnegative Function4.4 The General Lebesgue Integral4.5 Countable Additivity and Continuity of Integraion4.6 Uniform Integrability: The Vitali Convergence Theorem5. Lebesgue Integration: Further Topics5.1 Uniform Integrability and Tightness: A General Vitali Convergence Theorem5.2 Convergence in measure5.3 Characterizations of Riemann and Lebesgue Integrability6. Differentiation and Integration6.1 Continuity of Monotone Functions6.2 Differentiability of Monotone Functions: Lebesgue's Theorem6.3 Functions of Bounded Variation: Jordan's Theorem6.4 Absolutely Continuous Functions6.5 Integrating Derivatives: Differentiating Indefinite Integrals6.6 Convex Functions7. The LΡ Spaces: Completeness and Approximation7.1 Normed Linear Spaces7.2 The Inequalities of Young, Hölder and Minkowski7.3 LΡ is Complete: The Riesz-Fischer Theorem7.4 Approximation and Separability8. The LΡ Spaces: Duality and Weak Convergence8.1 The Dual Space of LΡ8.2 Weak Sequential Convergence in LΡ8.3 Weak Sequential Compactness8.4 The Minimization of Convex FunctionalsPART II: ABSTRACT SPACES: METRIC, TOPOLOGICAL, AND HILBERT9. Metric Spaces: General Properties9.1 Examples of Metric Spaces9.2 Open Sets, Closed Sets and Convergent Sequences9.3 Continuous Mappings Between Metric Spaces9.4 Complete Metric Spaces9.5 Compact Metric Spaces9.6 Separable Metric Spaces10. Metric Spaces: Three Fundamental Theorems10.1 The Arzelà-Ascoli Theorem10.2 The Baire Category Theorem10.3 The Banach Contraction Principle11. Topological Spaces: General Properties11.1 Open Sets, Closed Sets, Bases and Subbases11.2 The Separation Properties11.3 Countability and Separability11.4 Continuous Mappings Between Topological Spaces11.5 Compact Topological Spaces11.6 Connected Topological Spaces12. Topological Spaces: Three Fundamental Theorems12.1 Urysohn's Lemma and the Tietze Extension Theorem12.2 The Tychonoff Product Theorem12.3 The Stone-Weierstrass Theorem13. Continuous Linear Operators Between Banach Spaces13.1 Normed Linear Spaces13.2 Linear Operators13.3 Compactness Lost: Infinite Dimensional Normed Linear Spaces13.4 The Open Mapping and Closed Graph Theorems13.5 The Uniform Boundedness Principle14. Duality for Normed Linear Spaces14.1 Linear Functionals, Bounded Linear Functionals and Weak Topologies14.2 The Hahn-Banach Theorem14.3 Reflexive Banach Spaces and Weak Sequential Convergence14.4 Locally Convex Topological Vector Spaces14.5 The Separation of Convex Sets and Mazur's Theorem14.6 The Krein-Milman Theorem15. Compactness Regained: The Weak Topology15.1 Alaoglu's Extension of Helley's Theorem15.2 Reflexivity and Weak Compactness: Kakutani's Theorem15.3 Compactness and Weak Sequential Compactness: The Eberlein-Šmulian Theorem15.4 Metrizability of Weak Topologies16. Continuous Linear Operators on Hilbert Spaces16.1 The Inner Product and Orthogonality16.2 The Dual Space and Weak Sequential Convergence16.3 Bessel's Inequality and Orthonormal Bases16.4 Adjoints and Symmetry for Linear Operators16.5 Compact Operators16.6 The Hilbert Schmidt Theorem16.7 The Riesz-Schauder Theorem: Characterization of Fredholm OperatorsPART III: MEASURE AND INTEGRATION: GENERAL THEORY17. General Measure Spaces: Their Properties and Construction17.1 Measures and Measurable Sets17.2 Signed Measures: The Hahn and Jordan Decompositions17.3 The Carathéodory Measure Induced by an Outer Measure17.4 The Construction of Outer Measures17.5 The Carathéodory-Hahn Theorem: The Extension of a Premeasure to a Measure18. Integration Over General Measure Spaces18.1 Measurable Functions18.2 Integration of Nonnegative Measurable Functions18.3 Integration of General Measurable Functions18.4 The Radon-Nikodym Theorem18.5 The Saks Metric Space: The Vitali-Hahn-Saks Theorem19. General LΡ Spaces: Completeness, Duality and Weak Convergence19.1 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 < 119.5 Weak Sequential Compactness in L1 (X, μ): The Dunford-Pettis Theorem20. The Construction of Particular Measures20.1 Product Measures: The Theorems of Fubini and Tonelli20.2 Lebesgue Measure on Euclidean Space Rn20.3 Cumulative Distribution Functions and Borel Measures on R20.4 Carathéodory Outer Measures and hausdorff Measures on a Metric Space21. Measure and Topology21.1 Locally Compact Topological Spaces21.2 Separating Sets and Extending Functions21.3 The Construction of Radon Measures21.4 The Representation of Positive Linear Functionals on Cc (X): The Riesz-Markov Theorem21.5 The Riesz Representation Theorem for the Dual of C(X)21.6 Regularity Properties of Baire Measures22. Invariant Measures22.1 Topological Groups: The General Linear Group22.2 Fixed Points of Representations: Kakutani's Theorem22.3 Invariant Borel Measures on Compact Groups: von Neumann's Theorem22.4 Measure Preserving Transformations and Ergodicity: the Bogoliubov-Krilov Theorem
