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Table of Contents
Intro
Preface
Contents
1 Topology
1.1 Basic Notions and Facts
1.2 Continuous Functions: Nets
1.3 Separation and Countability Properties
1.4 Weak, Product, and Quotient Topologies
1.5 Compact and Locally Compact Spaces
1.6 Connectedness
1.7 Polish, Souslin, and Baire Spaces
1.8 Semicontinuous Functions
1.9 Remarks
1.10 Problems
2 Measure Theory
2.1 Algebras of Sets and Measures
2.2 Measurable Functions
2.3 Polish, Souslin, and Borel Spaces
2.4 Integration
2.5 Signed Measures and the Lebesgue-Radon-Nikodym Theorem
2.6 Lp-Spaces
2.7 Modes of Convergence: Uniform Integrability
2.8 Measures and Topology
2.9 Remarks
2.10 Problems
3 Banach Space Theory
3.1 Introduction
3.2 Locally Convex Spaces: Banach Spaces
3.3 Hahn-Banach Theorem-Separation Theorems
3.4 Three Basic Theorems
3.5 Weak and Weak* Topologies
3.6 Separable and Reflexive Normed Spaces
3.7 Dual Operators -Compact Operators- Projections
3.8 Hilbert Spaces
3.9 Unbounded Linear Operators
3.10 Remarks
3.11 Problems
4 Function Spaces
4.1 Lebesgue Spaces
4.2 Variable Exponent Lebesgue Spaces
4.3 Sobolev Spaces
4.4 Lebesgue-Bochner Spaces
4.5 Spaces of Measures
4.6 Remarks
4.7 Problems
5 Multivalued Analysis
5.1 Continuity of Multifunctions
5.2 Measurability of Multifunctions
5.3 Continuous and Measurable Selections
5.4 Decomposable Sets
5.5 Set-Valued Integral
5.6 Caratheodory Multifunctions
5.7 Remarks
5.8 Problems
6 Smooth and Nonsmooth Calculus
6.1 Differential Calculus in Normed Spaces
6.2 Convex Functions-Subdifferential Theory
6.3 Convex Functions-Duality Theory
6.4 Infimal Convolution-Regularization-Coercivity
6.5 Locally Lipschitz Functions
6.6 Generalizations
6.7 Remarks
6.8 Problems
7 Nonlinear Operators
7.1 Compact and Fredholm Maps
7.2 Monotone Operators
7.3 Operators of Monotone Type
7.4 Remarks
7.5 Problems
8 Variational Analysis
8.1 Convergence of Sets
8.2 Variational Convergence of Functions
8.3 G-Convergence of Operators
8.4 Variational Principles
8.5 Remarks
8.6 Problems
References
Index
Preface
Contents
1 Topology
1.1 Basic Notions and Facts
1.2 Continuous Functions: Nets
1.3 Separation and Countability Properties
1.4 Weak, Product, and Quotient Topologies
1.5 Compact and Locally Compact Spaces
1.6 Connectedness
1.7 Polish, Souslin, and Baire Spaces
1.8 Semicontinuous Functions
1.9 Remarks
1.10 Problems
2 Measure Theory
2.1 Algebras of Sets and Measures
2.2 Measurable Functions
2.3 Polish, Souslin, and Borel Spaces
2.4 Integration
2.5 Signed Measures and the Lebesgue-Radon-Nikodym Theorem
2.6 Lp-Spaces
2.7 Modes of Convergence: Uniform Integrability
2.8 Measures and Topology
2.9 Remarks
2.10 Problems
3 Banach Space Theory
3.1 Introduction
3.2 Locally Convex Spaces: Banach Spaces
3.3 Hahn-Banach Theorem-Separation Theorems
3.4 Three Basic Theorems
3.5 Weak and Weak* Topologies
3.6 Separable and Reflexive Normed Spaces
3.7 Dual Operators -Compact Operators- Projections
3.8 Hilbert Spaces
3.9 Unbounded Linear Operators
3.10 Remarks
3.11 Problems
4 Function Spaces
4.1 Lebesgue Spaces
4.2 Variable Exponent Lebesgue Spaces
4.3 Sobolev Spaces
4.4 Lebesgue-Bochner Spaces
4.5 Spaces of Measures
4.6 Remarks
4.7 Problems
5 Multivalued Analysis
5.1 Continuity of Multifunctions
5.2 Measurability of Multifunctions
5.3 Continuous and Measurable Selections
5.4 Decomposable Sets
5.5 Set-Valued Integral
5.6 Caratheodory Multifunctions
5.7 Remarks
5.8 Problems
6 Smooth and Nonsmooth Calculus
6.1 Differential Calculus in Normed Spaces
6.2 Convex Functions-Subdifferential Theory
6.3 Convex Functions-Duality Theory
6.4 Infimal Convolution-Regularization-Coercivity
6.5 Locally Lipschitz Functions
6.6 Generalizations
6.7 Remarks
6.8 Problems
7 Nonlinear Operators
7.1 Compact and Fredholm Maps
7.2 Monotone Operators
7.3 Operators of Monotone Type
7.4 Remarks
7.5 Problems
8 Variational Analysis
8.1 Convergence of Sets
8.2 Variational Convergence of Functions
8.3 G-Convergence of Operators
8.4 Variational Principles
8.5 Remarks
8.6 Problems
References
Index