Applied Functional Analysis

Preface
Acknowledgments
Contents
About the Author
1 Operator Theory
	1.1 Quick Review of Hilbert Space
		1.1.1 Lebesgue Spaces
		1.1.2 Convergence Theorems
		1.1.3 Complete Space
		1.1.4 Hilbert Space
		1.1.5 Fundamental Mapping Theorems on Banach Spaces
	1.2 The Adjoint of Operator
		1.2.1 Bounded Linear Operators
		1.2.2 Definition of Adjoint
		1.2.3 Adjoint Operator on Hilbert Spaces
		1.2.4 Self-adjoint Operators
	1.3 Compact Operators
		1.3.1 Definition and Properties of Compact Operators
		1.3.2 The Integral Operator
		1.3.3 Finite-Rank Operators
	1.4 Hilbert–Schmidt Operator
		1.4.1 Definition of Hilbert–Schmidt Operator
		1.4.2 Basic Properties of HS Operators
		1.4.3 Relations with Compact and Finite-Rank Operators
		1.4.4 The Fredholm Operator
		1.4.5 Characterization of HS Operators
	1.5 Eigenvalues of Operators
		1.5.1 Spectral Analysis
		1.5.2 Definition of Eigenvalues and Eigenfunctions
		1.5.3 Eigenvalues of Self-adjoint Operators
		1.5.4 Eigenvalues of Compact Operators
	1.6 Spectral Analysis of Operators
		1.6.1 Resolvent and Regular Values
		1.6.2 Bounded Below Mapping
		1.6.3 Spectrum of Bounded Operator
		1.6.4 Spectral Mapping Theorem
		1.6.5 Spectrum of Compact Operators
	1.7 Spectral Theory of Self-adjoint Compact Operators
		1.7.1 Eigenvalues of Compact Self-adjoint Operators
		1.7.2 Invariant Subspaces
		1.7.3 Hilbert–Schmidt Theorem
		1.7.4 Spectral Theorem For Self-adjoint Compact Operators
	1.8 Fredholm Alternative
		1.8.1 Resolvent of Compact Operators
		1.8.2 Fundamental Principle
		1.8.3 Fredholm Equations
		1.8.4 Volterra Equations
	1.9 Unbounded Operators
		1.9.1 Introduction
		1.9.2 Closed Operator
		1.9.3 Basics Properties of Unbounded Operators
		1.9.4 Toeplitz Theorem
		1.9.5 Adjoint of Unbounded Operators
		1.9.6 Deficiency Spaces of Unbounded Operators
		1.9.7 Symmetry of Unbounded Operators
		1.9.8 Spectral Properties of Unbounded Operators
	1.10 Differential Operators
		1.10.1 Green\'s Function and Dirac Delta
		1.10.2 Laplacian Operator
		1.10.3 Sturm–Liouville Operator
		1.10.4 Momentum Operator
	1.11 Problems
2 Distribution Theory
	2.1 The Notion of Distribution
		2.1.1 Motivation For Distributions
		2.1.2 Test Functions
		2.1.3 Definition of Distribution
	2.2 Regular Distribution
		2.2.1 Locally Integrable Functions
		2.2.2 Notion of Regular Distribution
		2.2.3 The Dual Space mathcalD
		2.2.4 Basic Properties of Regular Distributions
	2.3 Singular Distributions
		2.3.1 Notion of Singular Distribution
		2.3.2 Dirac Delta Distribution
		2.3.3 Delta Sequence
		2.3.4 Gaussian Delta Sequence
	2.4 Differentiation of Distributions
		2.4.1 Notion of Distributional Derivative
		2.4.2 Calculus Rules
		2.4.3 Examples of Distributional Derivatives
		2.4.4 Properties of δ
	2.5 The Fourier Transform Problem
		2.5.1 Introduction
		2.5.2 Fourier Transform on mathbbRn
		2.5.3 Existence of Fourier Transform
		2.5.4 Plancherel Theorem
	2.6 Schwartz Space
		2.6.1 Rapidly Decreasing Functions
		2.6.2 Definition of Schwartz Space
		2.6.3 Derivatives of Schwartz Functions
		2.6.4 Isomorphism of Fourier Transform on Schwartz Spaces
	2.7 Tempered Distributions
		2.7.1 Definition of Tempered Distribution
		2.7.2 Functions of Slow Growth
		2.7.3 Examples of Tempered Distributions
	2.8 Fourier Transform of Tempered Distribution
		2.8.1 Motivation
		2.8.2 Definition
		2.8.3 Derivative of F.T. of Tempered Distribution
	2.9 Inversion Formula of The Fourier Transform
		2.9.1 Fourier Transform of Gaussian Function
		2.9.2 Fourier Transform of Delta Distribution
		2.9.3 Fourier Transform of Sign Function
	2.10 Convolution of Distribution
		2.10.1 Derivatives of Convolutions
		2.10.2 Convolution in Schwartz Space
		2.10.3 Definition of Convolution of Distributions
		2.10.4 Fundamental Property of Convolutions
		2.10.5 Fourier Transform of Convolution
	2.11 Problems
3 Theory of Sobolev Spaces
	3.1 Weak Derivative
		3.1.1 Notion of Weak Derivative
		3.1.2 Basic Properties of Weak Derivatives
		3.1.3 Pointwise Versus Weak Derivatives
		3.1.4 Weak Derivatives and Fourier Transform
	3.2 Regularization and Smoothening
		3.2.1 The Concept of Mollification
		3.2.2 Mollifiers
		3.2.3 Cut-Off Function
		3.2.4 Partition of Unity
		3.2.5 Fundamental Lemma of Calculus of Variations
	3.3 Density of Schwartz Space
		3.3.1 Convergence of Approximating Sequence
		3.3.2 Approximations of mathcalS and Lp
		3.3.3 Generalized Plancherel Theorem
	3.4 Construction of Sobolev Spaces
		3.4.1 Completion of Schwartz Spaces
		3.4.2 Definition of Sobolev Space
		3.4.3 Fractional Sobolev Space
	3.5 Basic Properties of Sobolev Spaces
		3.5.1 Convergence in Sobolev Spaces
		3.5.2 Completeness and Reflexivity of Sobolev Spaces
		3.5.3 Local Sobolev Spaces
		3.5.4 Leibnitz Rule
		3.5.5 Mollification with Sobolev Function
		3.5.6 W0k,p(Ω)
	3.6 W1,p(Ω)
		3.6.1 Absolute Continuity Characterization
		3.6.2 Inclusions
		3.6.3 Chain Rule
		3.6.4 Dual Space of W1,p(Ω)
	3.7 Approximation of Sobolev Spaces
		3.7.1 Local Approximation
		3.7.2 Global Approximation
		3.7.3 Consequences of Meyers–Serrin Theorem
	3.8 Extensions
		3.8.1 Motivation
		3.8.2 The Zero Extension
		3.8.3 Coordinate Transformations
		3.8.4 Extension Operator
	3.9 Sobolev Inequalities
		3.9.1 Sobolev Exponent
		3.9.2 Fundamental Inequalities
		3.9.3 Gagliardo–Nirenberg–Sobolev Inequality
		3.9.4 Poincare Inequality
		3.9.5 Estimate for W1,p
		3.9.6 The Case p=n
		3.9.7 Holder Spaces
		3.9.8 The Case p>n
		3.9.9 General Sobolev Inequalities
	3.10 Embedding Theorems
		3.10.1 Compact Embedding
		3.10.2 Rellich–Kondrachov Theorem
		3.10.3 High Order Sobolev Estimates
		3.10.4 Sobolev Embedding Theorem
		3.10.5 Embedding of Fractional Sobolev Spaces
	3.11 Problems
4 Elliptic Theory
	4.1 Elliptic Partial Differential Equations
		4.1.1 Elliptic Operator
		4.1.2 Uniformly Elliptic Operator
		4.1.3 Elliptic PDEs
	4.2 Weak Solution
		4.2.1 Motivation for Weak Solutions
		4.2.2 Weak Formulation of Elliptic BVP
		4.2.3 Classical Versus Strong Versus Weak Solutions
	4.3 Poincare Equivalent Norm
		4.3.1 Poincare Inequality on H01
		4.3.2 Equivalent Norm on H01
		4.3.3 Poincare–Wirtinger Inequality
		4.3.4 Quotient Sobolev Space
	4.4 Elliptic Estimates
		4.4.1 Bilinear Forms
		4.4.2 Elliptic Bilinear Mapping
		4.4.3 Garding\'s Inequality
	4.5 Symmetric Elliptic Operators
		4.5.1 Riesz Representation Theorem for Hilbert Spaces
		4.5.2 Existence and Uniqueness Theorem—Poisson\'s Equation
		4.5.3 Existence and Uniqueness Theorem—Helmholtz Equation
		4.5.4 Ellipticity and Coercivity
		4.5.5 Existence and Uniqueness Theorem—Symmetric Uniformly Operator
	4.6 General Elliptic Operators
		4.6.1 Lax–Milgram Theorem
		4.6.2 Dirichlet Problems
		4.6.3 Neumann Problems
	4.7 Spectral Properties of Elliptic Operators
		4.7.1 Resolvent of Elliptic Operators
		4.7.2 Fredholm Alternative for Elliptic Operators
		4.7.3 Spectral Theorem for Elliptic Operators
	4.8 Self-adjoint Elliptic Operators
		4.8.1 The Adjoint of Elliptic Bilinear
		4.8.2 Eigenvalue Problem of Elliptic Operators
		4.8.3 Spectral Theorem of Elliptic Operator
	4.9 Regularity for the Poisson Equation
		4.9.1 Weyl\'s Lemma
		4.9.2 Difference Quotients
		4.9.3 Caccioppoli\'s Inequality
		4.9.4 Interior Regularity for Poisson Equation
	4.10 Regularity for General Elliptic Equations
		4.10.1 Interior Regularity
		4.10.2 Higher Order Interior Regularity
		4.10.3 Interior Smoothness
		4.10.4 Boundary Regularity
	4.11 Problems
5 Calculus of Variations
	5.1 Minimization Problem
		5.1.1 Definition of Minimization Problem
		5.1.2 Lower Semicontinuity
		5.1.3 Minimization Problems in Finite-Dimensional Spaces
		5.1.4 Convexity
		5.1.5 Minimization in Infinite-Dimensional Space
	5.2 Weak Topology
		5.2.1 Notion of Weak Topology
		5.2.2 Weak Convergence
		5.2.3 Weakly Closed Sets
		5.2.4 Reflexive Spaces
		5.2.5 Weakly Lower Semicontinuity
	5.3 Direct Method
		5.3.1 Direct Verses Indirect Methods
		5.3.2 Minimizing Sequence
		5.3.3 Procedure of Direct Method
		5.3.4 Coercivity
		5.3.5 The Main Theorem on the Existence of Minimizers
	5.4 The Dirichlet Problem
		5.4.1 Variational Integral
		5.4.2 Dirichlet Principle
		5.4.3 Weierstrass Counterexample
	5.5 Dirichlet Principle in Sobolev Spaces
		5.5.1 Minimizer of the Dirichlet Integral in H01
		5.5.2 Minimizer of the Dirichlet Integral in H1
		5.5.3 Dirichlet Principle
		5.5.4 Dirichlet Principle with Neumann Condition
		5.5.5 Dirichlet Principle with Neumann B.C. in Sobolev Spaces
	5.6 Gateaux Derivatives of Functionals
		5.6.1 Introduction
		5.6.2 Historical Remark
		5.6.3 Gateaux Derivative
		5.6.4 Basic Properties of G-Derivative
		5.6.5 G-Differentiability and Continuity
		5.6.6 Frechet Derivative
		5.6.7 G-Differentiability and Convexity
		5.6.8 Higher Gateaux Derivative
		5.6.9 Minimality Condition
	5.7 Poisson Variational Integral
		5.7.1 Gateaux Derivative of Poisson Integral
		5.7.2 Symmetric Elliptic PDEs
		5.7.3 Dirichlet Principle of Symmetric Elliptic PDEs
	5.8 Euler–Lagrange Equation
		5.8.1 Lagrangian Integral
		5.8.2 First Variation
		5.8.3 Necessary Condition for Minimiality I
		5.8.4 Euler–Lagrange Equation
		5.8.5 Second Variation
		5.8.6 Legendre Condition
	5.9 Dirichlet Principle for Euler–Lagrange Equation
		5.9.1 The Lagrangian Functional
		5.9.2 Gateaux Derivative of the Lagrangian Integral
		5.9.3 Dirichlet Principle for Euler-Lagrange Equation
	5.10 Variational Problem of Euler–Lagrange Equation
		5.10.1 p-Convex Lagrangian Functional
		5.10.2 Existence of Minimizer
	5.11 Problems
Appendix  References
Index