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Real Analysis, 4th updated printing ( Pearson Modern Classic) 요약정보 및 구매

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지은이 H.L. Royden & P.M. Fitzpatrick
발행년도 2017-02-23
페이지 528
ISBN 9780134689494
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판매가격 69,000원
포인트 0점
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  • Real Analysis, 4th updated printing ( Pearson Modern Classic)
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관련상품

  • Real Analysis, 4th Edition, covers the basic material that every graduate student should know in 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. This text assumes a general background in undergraduate mathematics and familiarity with the material covered in an undergraduate course on the fundamental concepts of analysis. Patrick Fitzpatrick of the University of Maryland―College Park spearheaded this revision of Halsey Royden’s classic text.

  • 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

  • H.L. Royden & P. M. Fitzpatrick 

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  • Real Analysis, 4th updated printing ( Pearson Modern Classic)
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