Integral and Functional Analysis (Updated Edition)
Published on 3. May 2021
398 pages
978-1-5361-9617-7 (ISBN)
Description
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.
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Jie Xiao
Integral and Functional Analysis
Book
05/2021
Nova Science Publishers Inc
€303.05
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Content
- Intro
- 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
