TY  - BOOK
AU  - XIAO,JIE
TI  - INTEGRAL AND FUNCTIONAL ANALYSIS (UPDATED EDITION)
T2  - Mathematics Research Developments Ser
SN  - 9781536196177
AV  - QA431 .I584 2021
KW  - Integral equations
KW  - Metric spaces
KW  - Functional analysis
KW  - Electronic Books
N1  - 2; 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 and Weak* 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; 2; b
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