Functional Analysis for the Applied Mathematician (Textbooks in Mathematics)

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Title Page
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Contents
Preface
Author Biographies
Synopsis
CHAPTER 1: Preliminaries
	1.1. ELEMENTARY OR POINT-SET TOPOLOGY
	1.2. LEBESGUE MEASURE AND INTEGRATION
	1.3. HOLOMORPHIC FUNCTIONS AND COMPLEX CONTOUR INTEGRATION
	1.4. EXERCISES
CHAPTER 2: Normed Linear Spaces and Banach Spaces
	2.1. BASIC CONCEPTS AND DEFINITIONS
	2.2. LINEAR MAPS AND THE DUAL SPACE
	2.3. SOME IMPORTANT EXAMPLES
		2.3.1. Finite-dimensional spaces
		2.3.2. The spaces ℓp
		2.3.3. The Lebesgue spaces Lp(Ω)
	2.4. HAHN-BANACH THEOREMS
	2.5. APPLICATIONS OF THE HAHN-BANACH THEOREM
	2.6. THE OPEN MAPPING THEOREM
	2.7. THE UNIFORM BOUNDEDNESS PRINCIPLE
	2.8. THE EMBEDDING OF X INTO ITS DOUBLE DUAL X∗∗
	2.9. COMPACTNESS AND WEAK CONVERGENCE IN AN NLS
		2.9.1. The norm or strong topology
		2.9.2. The weak and weak-∗ topologies
	2.10. THE DUAL OF AN OPERATOR
	2.11. EXERCISES
CHAPTER 3: Hilbert Spaces
	3.1. BASIC PROPERTIES OF INNER-PRODUCTS
	3.2. BEST APPROXIMATION AND ORTHOGONAL PROJECTION
	3.3. DUALITY IN HILBERT SPACES
	3.4. ORTHONORMAL SUBSETS AND BASES
	3.5. WEAK CONVERGENCE IN A HILBERT SPACE
	3.6. EXERCISES
CHAPTER 4: Spectral Theory and Compact Operators
	4.1. DEFINITIONS OF THE RESOLVENT AND SPECTRUM
	4.2. BASIC SPECTRAL THEORY IN BANACH SPACES
	4.3. COMPACT LINEAR OPERATORS ON A BANACH SPACE
	4.4. BOUNDED SELF-ADJOINT LINEAR OPERATORS ON A HILBERT SPACE
	4.5. COMPACT SELF-ADJOINT LINEAR OPERATORS ON A HILBERT SPACE
	4.6. THE ASCOLI-ARZELÀ THEOREM
	4.7. STURM-LIOUVILLE THEORY
		4.7.1. Sturm-Liouville problems and Green’s functions
		4.7.2. Spectral properties of the solution operator
		4.7.3. Some applications
	4.8. EXERCISES
CHAPTER 5: Distributions
	5.1. THE NOTION OF GENERALIZED FUNCTIONS
	5.2. TEST FUNCTIONS
	5.3. DISTRIBUTIONS
	5.4. OPERATIONS WITH DISTRIBUTIONS
		5.4.1. Multiplication by a smooth function
		5.4.2. Differentiation
		5.4.3. Translations and dilations of Rd
		5.4.4. Convolutions
	5.5. CONVERGENCE OF DISTRIBUTIONS AND APPROXIMATIONS TO THE IDENTITY
	5.6. SOME APPLICATIONS TO DIFFERENTIAL EQUATIONS
		5.6.1. Ordinary differential equations
		5.6.2. Partial differential equations and fundamental solutions
	5.7. LOCAL STRUCTURE OF D′
	5.8. EXERCISES
CHAPTER 6: The Fourier Transform
	6.1. MOTIVATION FOR FOURIER ANALYSIS
	6.2. THE L1(Rd) THEORY
	6.3. THE SCHWARTZ SPACE THEORY
	6.4. THE L2(Rd) THEORY
	6.5. THE S′ THEORY
	6.6. SOME APPLICATIONS
		6.6.1. The heat equation
		6.6.2. The Schrödinger equation
		6.6.3. Signal processing and translation invariance
	6.7. EXERCISES
CHAPTER 7: Sobolev Spaces
	7.1. DEFINITIONS AND BASIC PROPERTIES
	7.2. EXTENSIONS FROM Ω TO Rd
	7.3. THE SOBOLEV EMBEDDING THEOREM
	7.4. COMPACTNESS
	7.5. THE Hs SOBOLEV SPACES
	7.6. TRACE THEOREMS
	7.7. THE Ws,p(Ω) SOBOLEV SPACES
	7.8. EXERCISES
CHAPTER 8: Boundary Value Problems
	8.1. SECOND ORDER LINEAR ELLIPTIC PDES
		8.1.1. Practical examples
		8.1.2. Boundary conditions (BCs)
	8.2. VARIATIONAL PROBLEMS AND MINIMIZATION OF ENERGY
	8.3. THE CLOSED RANGE THEOREM AND LINEAR OPERATORS BOUNDED BELOW
	8.4. THE BABUŠKA-LAX-MILGRAM THEOREM
	8.5. APPLICATION TO LINEAR ELLIPTIC PDES
		8.5.1. The general Dirichlet problem
		8.5.2. The Neumann problem with lowest order term
		8.5.3. The Neumann problem with no zeroth order term
		8.5.4. Elliptic regularity
	8.6. GALERKIN METHODS
	8.7. GREEN’S FUNCTIONS
	8.8. EXERCISES
CHAPTER 9: Differential Calculus in Banach Spaces
	9.1. DIFFERENTIATION
		9.1.1. The chain rule
		9.1.2. The Mean-Value Theorem
		9.1.3. Partial differentiation
	9.2. FIXED POINTS AND CONTRACTIVE MAPS
	9.3. NONLINEAR EQUATIONS
		9.3.1. Newton methods
		9.3.2. The Inverse Function Theorem
		9.3.3. The Implicit Function Theorem
	9.4. HIGHER DERIVATIVES
	9.5. EXTREMA
	9.6. EXERCISES
CHAPTER 10: The Calculus of Variations
	10.1. THE EULER-LAGRANGE EQUATIONS
	10.2. CONSTRAINED EXTREMA AND LAGRANGE MULTIPLIERS
	10.3. LOWER SEMICONTINUITY AND EXISTENCE OF MINIMA
	10.4. EXERCISES
Bibliography
Index