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# Real Analysis by Patrick M. Fitzpatrick, H.L. Royden (Hardback, 2010)

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## About this product

### Key Features

- Author(s)H.L. Royden,Patrick M. Fitzpatrick
- PublisherPearson Education (US)
- Date of Publication15/01/2010
- Language(s)English
- FormatHardback
- ISBN-10013143747X
- ISBN-139780131437470
- GenreScience & Mathematics: Textbooks & Study Guides

### Publication Data

- Place of PublicationUpper Saddle River
- Country of PublicationUnited States
- ImprintPearson Education (US)
- Content Noteblack & white illustrations

### Dimensions

- Weight1021 g
- Width182 mm
- Height237 mm
- Spine22 mm
- Pagination544

### Editorial Details

- Edition StatementUnited States ed of 4th revised ed

### Description

- Table Of ContentsPART 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 sigma-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 LRHO Spaces: Completeness and Approximation 7.1 Normed Linear Spaces 7.2 The Inequalities of Young, Holder and Minkowski 7.3 LRHO is Complete: The Riesz-Fischer Theorem 7.4 Approximation and Separability 8. The LRHO Spaces: Duality and Weak Convergence 8.1 The Dual Space of LRHO 8.2 Weak Sequential Convergence in LRHO 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 Arzela-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 Co

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