Integral and Functional Analysis
by Jie Xiao
- Length: 398 pages
- Edition: 1
- Language: English
- Publisher: Nova Science Pub Inc
- Publication Date: 2021-05-05
- ISBN-10: 1536192805
- ISBN-13: 9781536192803
- Sales Rank: #0 (See Top 100 Books)
This textbook is based on three closely related courses: 1) Integration and Metric Spaces; 2) Lebesgue Integration; 3) Functional Analysis. Although the contents have been used for joint undergraduate and graduate courses, this textbook is designed primarily for senior undergraduate students. The prerequisites of this textbook are deliberately modest, and it is assumed that the students have some familiarity with calculus and linear algebra plus the basic (direct, indirect) proof methods.
INTEGRAL AND FUNCTIONALANALYSIS(UPDATED EDITION) INTEGRAL AND FUNCTIONALANALYSIS(UPDATED EDITION) Contents Preface Acknowledgments Chapter 1Preliminaries 1.1 Sets, Relations, Functions, Cardinals and Ordinals 1.2 Reals, Some Basic Theorems and Sequence Limits Problems Chapter 2Riemann Integrals 2.1 Definitions, Examples and Basic Properties 2.2 Algebraic Operations and the Darboux Criterion 2.3 Fundamental Theorem of Calculus 2.4 Improper Integrals Problems Chapter 3Riemann-Stieltjes Integrals 3.1 Functions of Bounded Variation 3.2 Definition and Basic Properties 3.3 Nonexistence and Existence for Integrals 3.4 Evaluations of Integrals 3.5 Improper Situations Problems Chapter 4Lebesgue-Radon-StieltjesIntegrals 4.1 Foundational Material 4.2 Essential Properties 4.3 Convergence Theorems 4.4 Extension via Measurability 4.5 Double, Iterated and Generic Integrals Problems Chapter 5Absolute Continuitiesin Lebesgue Integrals 5.1 Lebesgue’s Outer Measure and Vitali’s Covering 5.2 Derivatives of Increasing Functions 5.3 Absolutely Continuous Functions 5.4 Cantor’s Ternary Set and Singular Function 5.5 Lebesgue’s Points Problems Chapter 6Metric Spaces 6.1 Metrizable Topology and Connectedness 6.2 Completeness 6.3 Compactness, Density and Separability Problems Chapter 7Continuous Mappings 7.1 Criteria for Continuity 7.2 Continuous Mappings over Compactor ConnectedMetric Spaces 7.3 Sequences of Mappings 7.4 Contractions 7.5 Structures of Metric Spaces Problems Chapter 8Normed Linear Spaces 8.1 Linear Spaces, Norms and Quotient Spaces 8.2 Finite Dimensional Spaces 8.3 Bounded Linear Operators 8.4 Linear Functionals via Hahn-Banach Extension Problems Chapter 9Banach Spaces via Operatorsand Functionals 9.1 Definition and Beginning Examples 9.2 Uniform Boundedness - Open Map - Closed Graph 9.3 Dual Banach Spaces by Examples 9.4 Weak andWeak* Topologies 9.5 Compact and Dual Operators Problems Chapter 10Hilbert Spaces and TheirOperators 10.1 Definition, Examples and Basic Properties 10.2 Orthogonality, Orthogonal Complementand Duality 10.3 Orthonormal Sets and Bases 10.4 Five Special Bounded Operators 10.5 Compact Operators via Spectrum Problems Hints or Solutions 1 Preliminaries 3 Riemann-Stieltjes Integrals 4 Lebesgue-Radon-Stieltjes Integrals 5 Absolute Continuities in Lebesgue Integrals 6 Metric Spaces 7 Continuous Mappings 8 Normed Linear Spaces 9 Banach Spaces via Operators and Functionals 10 Hilbert Spaces and Their Operators 2 Riemann Integrals References About the Author Index Blank Page Blank Page
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