Fundamentals of Functional Analysis [1 ed.] 9819930286, 9789819930289, 9789819930296, 9789819930319

Table of contents :
Preface
Acknowledgments
Contents
About the Author
1 Normed Spaces
1.1 Motivation
1.1.1 Gate to Infinite-dimensional Spaces
1.1.2 Why We Need Norms
1.2 Function Spaces
1.2.1 The Space of Continuous Functions
1.2.2 The Space of n–Continuously Differentiable Functions
1.2.3 The Space of All Continuous Functions with Compact Support
1.2.4 The Space of Continuous Functions Vanishing at Infinity
1.2.5 The Space of Functions of Bounded Variation
1.2.6 The Space of Absolutely Continuous Functions
1.2.7 The Space of Riemann Integrable Functions
1.2.8 The Space of Riemann–Stieltjes Integrable Functions
1.3 The Notion of Norm
1.3.1 Definition and Examples
1.3.2 Continuity of Norms
1.3.3 Norms Versus Metrics
1.4 Definition and Examples of Normed Spaces
1.4.1 Definition of Normed Spaces
1.4.2 Convergence in Normed Spaces
1.4.3 Examples of Normed Spaces
1.5 Lp Spaces
1.5.1 Lp Spaces
1.5.2 Linfty Spaces
1.5.3 ellpand ellinftySpaces
1.5.4 Basic Inequalities
1.5.5 Measure Theory and Integration: Quick Review
1.6 Topology of Normed Spaces
1.6.1 Balls and Sphere
1.6.2 Open and Closed Sets
1.6.3 Quotient Spaces
1.6.4 Distance to Closed Sets
1.6.5 Riesz's Lemma
1.6.6 Compactness
1.6.7 Separability
1.7 Equivalent Norms
1.7.1 Equivalence of Norms
1.7.2 Topological Interpretation
1.7.3 Norms in Finite-Dimensional Spaces
1.8 Problems
2 Linear Functionals
2.1 Linear Mappings
2.1.1 Definitions: Operators and Functionals
2.1.2 Continuous Operators
2.1.3 Identity and Shift Operators
2.1.4 Differential and Integral Operators
2.1.5 Integral Functional
2.2 Inverse of Operator
2.2.1 Injective and Surjective Operators
2.2.2 Inverses and Kernels
2.2.3 Isometry and Isomorphism
2.3 Dual Spaces
2.3.1 The Notion of Dual Space
2.3.2 Dual Space of a Finite-dimensional Spaces
2.3.3 Annihilators
2.3.4 Dual Space of Euclidean Space
2.3.5 Dual Space of ellp
2.3.6 Dual Space of c0
2.4 Riesz Representation Theorems
2.4.1 Introduction
2.4.2 Riesz Representation Theorem for Lp Spaces
2.4.3 Dual Space of Lp
2.4.4 Riesz Representation Theorem for C[a,b]
2.4.5 Dual Space of C[a,b]
2.5 Extension Theorems
2.5.1 Introduction
2.5.2 Extensions and Seminorms
2.5.3 Fundamental Extension Theorem
2.5.4 The Extension Theorem in Finite-Dimensional Spaces
2.6 Hahn–Banach Theorem
2.6.1 Historical Remarks
2.6.2 Hausdorff Maximal Principle
2.6.3 HBT for Vector Spaces
2.6.4 HBT with Seminorms
2.6.5 Hahn–Banachaut]Hahn, H. Theorem for Normed Spaces (HBT)
2.7 Consequences of Hahn–Banach Theorem
2.7.1 The Existence of Bounded Functional
2.7.2 Richness of Dual Spaces
2.7.3 Banach Separation Theorem
2.7.4 Separability of Normed Spaces
2.7.5 Proof of Riesz Representation Theorem for C[a,b]
2.8 Problems
3 Locally Convex Spaces
3.1 Hyperplanes
3.1.1 Introduction and Motivation
3.1.2 Separating Hyperplanes
3.1.3 Convex Sets
3.2 Minkowski Functional
3.2.1 Definition
3.2.2 Properties of Minkowski Functional
3.3 Topological Vector Spaces
3.4 Constructing Locally Convex Spaces
3.4.1 TVS with Seminorms
3.4.2 Locally Convex Spaces
3.4.3 Metrizability
3.5 Separation Theorems
3.5.1 Mazur Separation Theorem
3.5.2 Bourgin Separation Theorem
3.5.3 Generalized Banach Separation Theorem
3.5.4 Eidelheit Separation Theorem
3.5.5 Tuckey—Klee Separation Theorem
3.5.6 Historical Remarks
3.6 Extreme Points
3.6.1 Definition
3.6.2 Examples of Extreme Points
3.6.3 Faces
3.7 Krein-Milman Theorem
3.7.1 Convex Hull
3.7.2 Statement and Proof of KMT
3.7.3 Applications of KMT
3.8 Problems
4 Banach Spaces
4.1 Completeness
4.1.1 The Idea of Convergence
4.1.2 Cauchy Sequences
4.1.3 Complete Spaces
4.1.4 Absolutely Summable Series
4.1.5 Examples of Complete Spaces
4.2 Definition and Examples of Banach Spaces
4.2.1 Notion of Banach Space
4.2.2 Continuous Functions Over Compact Sets
4.2.3 Continuous Functions Over σ-Compact Sets
4.2.4 Cn(K) & Cinfty(K) Spaces
4.2.5 Completeness of ellp Space
4.2.6 Completeness of Lp Space
4.2.7 Incomplete Spaces
4.3 Properties of Banach Spaces
4.3.1 Closedness and Completeness
4.3.2 Operators Between Banach Spaces
4.3.3 Isomorphisms of Banach Spaces
4.3.4 Direct Sum of Banach Spaces
4.3.5 Completion of Incomplete Spaces
4.4 Baire Category Theorem
4.4.1 First and Second Category
4.4.2 Weak Version of the Theorem
4.4.3 Strong Version of the Theorem
4.4.4 Consequences of BCT
4.5 Open Mapping Theorem
4.5.1 Open Mapping
4.5.2 The Open Mapping Theorem
4.5.3 Applications
4.5.4 Bounded Inverse Theorem
4.6 Closed Graph Theorem
4.6.1 Closed Operators
4.6.2 The Closed Graph Theorem
4.6.3 Application
4.7 Uniform Bounded Principle
4.7.1 Pointwise and Uniformly Boundedness
4.7.2 Uniform Bounded Principle (Weak Version)
4.7.3 Historical Remarks
4.7.4 Uniform Bounded Principle (Strong Version)
4.7.5 Banach–Steinhaus Theorem
4.7.6 Totally Boundedness
4.7.7 Applications of UBP
4.8 Problems
5 Hilbert Spaces
5.1 Inner Product
5.1.1 Inner Product
5.1.2 Cauchy–Schwartz Inequality
5.1.3 Parallelogram and Polarization Laws
5.2 Inner Product Spaces and Hilbert Spaces
5.2.1 Definitions
5.2.2 Euclidean Space
5.2.3 L2 and ell2 Space
5.2.4 Counterexamples
5.3 Functionals on Hilbert Spaces
5.3.1 Riesz Representation Theorem for Hilbert Spaces
5.3.2 Hahn–Banach Theorem for Hilbert Spaces
5.4 Orthogonality in Hilbert Space
5.4.1 Orthogonality
5.4.2 Best Approximation
5.4.3 Projections in Hilbert Spaces
5.4.4 Orthonormal Sets
5.5 Orthonormal Basis
5.5.1 Total Sets
5.5.2 Orthonormal Bases
5.5.3 Parseval Identity
5.5.4 Fourier Coefficients
5.6 Nonseparable Hilbert Spaces
5.6.1 Uncountable Orthonormal Bases
5.6.2 Hilbert Dimension
5.7 Isomorphisms Between Hilbert Spaces
5.7.1 Hilbert Isomorphisms
5.7.2 Classification of Hilbert Spaces
5.7.3 Self Duality
5.8 Problems
6 Topology on Banach Spaces
6.1 Weak Convergence
6.1.1 The Weak Limit
6.1.2 Weak Convergence in Lp
6.1.3 Radon–Riesz Theorem
6.1.4 Radon–Riesz Property
6.1.5 Schur's Property
6.2 Weak Topology
6.2.1 The Idea of Weak Topology
6.2.2 Weak Sets
6.2.3 Weakly Closed Sets and Mazur's Theorem
6.2.4 Weak Compactness
6.2.5 Link to Hausdorff Topology
6.2.6 Notion of Nets
6.3 Weak* Topology
6.3.1 The Emebedding Map
6.3.2 The Bidual
6.3.3 Weak* Convergence
6.3.4 Properties of Weak* Topology
6.3.5 Link to Hausdorff Topology
6.3.6 Link to Locally Convex Topology
6.3.7 Weak* Sets
6.3.8 Compactness in the Product Spaces
6.3.9 Banach–Alaoglu Theorem
6.3.10 Goldstine Theorem
6.3.11 Weak* Heine–Borel Theorem
6.3.12 Krein–Milman Theorem on Weak Topology
6.3.13 Existence of Completion Space
6.4 Reflexive Spaces
6.4.1 Definition and Basic Properties
6.4.2 Reflexivity and Weak Compactness
6.4.3 Properties of Reflexive Spaces
6.4.4 Helly's Theorem
6.4.5 Consequences of Reflexivity
6.5 Weakly Compactness
6.5.1 The Sequentially Compactness Problem
6.5.2 Weak Compactness and Metrizibility
6.5.3 Eberlein–Smulian Theorem
6.5.4 James Theorems
6.6 Bases in Banach Spaces
6.6.1 Hamel Basis
6.6.2 Schauder Basis
6.6.3 Basic Sequences
6.6.4 Basis and Separability
6.6.5 Bases in Banach Spaces
6.6.6 Biorthogonal System
6.6.7 Applications to Banach Spaces
6.7 Weak Basis
6.7.1 Notion of Weak Basis
6.7.2 Weak Bases in Banach Spaces
6.7.3 Weak* Basis for Dual Spaces
6.7.4 Dual Projection Maps on Dual Spaces
6.8 Problems
7 Operators on Banach Spaces
7.1 Adjoint of Operator
7.1.1 The Idea of Adjoint
7.1.2 The Notion of Adjoint
7.1.3 Definition of Adjoint
7.1.4 Basic Properties of Adjoints
7.2 Adjoint Operators on Hilbert Spaces
7.2.1 Definition
7.2.2 Existence Theorem
7.2.3 Adjoint of Unbounded Operators
7.2.4 Basic Properties
7.2.5 Range and Null Spaces
7.2.6 Examples of Adjoint Operators
7.3 Classes of Operators
7.3.1 Self-adjoint Operators
7.3.2 Characterization of Self-adjoint Operators
7.3.3 Fredholm Operator
7.3.4 Decomposition of Bounded Operator
7.3.5 Normal Operators
7.3.6 Unitary and Positive Operators
7.4 Compact Operators
7.4.1 Characterization of Compact Operators
7.4.2 Compactness and Adjointness
7.4.3 Examples
7.4.4 Compactness on Finite Dimensions
7.5 Completely Continuous Operator
7.5.1 Definition
7.5.2 Completely Continuity and Compactness
7.5.3 Example of A Sequence That is Not Completely Continuous
7.5.4 Example of A Completely Continuous But Not Compact Operator
7.6 Operators of Finite Rank
7.6.1 The Notion of Finite Rank
7.6.2 Example of a Compact But Not Finite-Rank Operator
7.6.3 The Rank of the Adjoint
7.7 Approximation of Compact Operators
7.7.1 Sequence of Compact Operators
7.7.2 Approximation By Finite-Rank Operators
7.7.3 Constructing a Sequence of Finite-Rank Operators
7.7.4 Compactness of the Adjoint
7.7.5 Historical Remark: The Approximation Problem
7.8 Problems
8 Geometry of Banach Spaces
8.1 Convexity
8.1.1 Introduction
8.1.2 Strictly Convexity
8.2 Strict Convexity in p-Norm
8.2.1 Convex Functions
8.2.2 p-Characterization of Stricly Convex Spaces
8.2.3 Best Approximation in Stricly Convex Spaces
8.2.4 Taylor–Foguel Theorem
8.3 Uniform Convexity
8.3.1 Motivation for Uniform Convexity
8.3.2 Necessary and Sufficient Conditions
8.3.3 Milman–Pettis Theorem
8.3.4 Glimpse of Super-Reflexivity
8.4 Uniform Convexity in p-Norm
8.4.1 A Different Look at Uniform Convexity
8.4.2 Best Approximation in Uniformly Convex Spaces
8.4.3 Clarkson Inequalities
8.4.4 Uniform Convexity of
8.4.5 Radon–Riesz Theorem
8.4.6 Historical Remarks
8.5 Smoothness
8.5.1 Supporting Hyperplanes
8.5.2 Duality Between Strict Convexity and Smoothness
8.5.3 Uniformly Smooth Spaces
8.5.4 Modulus of Smoothness
8.5.5 Lindenstrauss Theorem
8.5.6 Duality Between Uniform Convexity and Uniform Smoothness
8.6 Differentiability
8.6.1 Gateaux Differentiability
8.6.2 Basic Properties of Gateaux Derivative
8.6.3 Frechet Derivative
8.7 Renorming Banach Spaces
8.7.1 The Idea of Renorming
8.7.2 Dual Norm
8.7.3 Smulian Criterions for Differentiability
8.7.4 Kadets Theorem
8.7.5 Locally Uniformly Conves Spaces
8.7.6 LUC Norm Theorems
8.8 Problems
Appendix Answer Key
1.8
2.8
3.8
4.8
5.8
6.8
7.8
8.8
Appendix References
Index
Author Index

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