Functional Analysis, Spectral Theory, and Applications

Preface
	Leitfaden
Contents
Chapter 1 Motivation
	1.1 From Even and Odd Functions to Group Representations
	1.2 Partial Differential Equations and the Laplace Operator
		1.2.1 The Heat Equation
		1.2.2 The Wave Equation
		1.2.3 The Mantegna Fresco
	1.3 What is Spectral Theory?
	1.4 The Prime Number Theorem
	1.5 Further Topics
Chapter 2 Norms and Banach Spaces
	2.1 Norms and Semi-Norms
		2.1.1 Normed Vector Spaces
		2.1.2 Semi-Norms and Quotient Norms
		2.1.3 Isometries are Affine
		2.1.4 A Comment on Notation
	2.2 Banach Spaces
		2.2.1 Proofs of Completeness
		2.2.2 The Completion of a Normed Vector Space
		2.2.3 Non-Compactness of the Unit Ball
	2.3 The Space of Continuous Functions
		2.3.1 The Arzela–Ascoli Theorem
		2.3.2 The Stone–Weierstrass Theorem
		2.3.3 Equidistribution of a Sequence
		2.3.4 Continuous Functions in Lp Spaces
	2.4 Bounded Operators and Functionals
		2.4.1 The Norm of Continuous Functionals on C0(X)
		2.4.2 Banach Algebras
	2.5 Ordinary Differential Equations
		2.5.1 The Volterra Equation
		2.52 The Sturm–Liouville Equation
	2.6 Further Topics
Chapter 3 Hilbert Spaces, Fourier Series, Unitary Representations
	3.1 Hilbert Spaces
		3.1.1 Definitions and Elementary Properties
		3.1.2  Sets in Uniformly Convex Spaces
		3.1.3 An Application to Measure Theory
	3.2 Orthonormal Bases and Gram–Schmidt
		3.2.1 The Non-Separable Case
	3.3 Fourier Series on Compact Abelian Groups
	3.4 Fourier Series on Td
		3.4.1 Convolution on the Torus
		3.4.2  Dirichlet and Fejér Kernels
		3.4.3 Differentiability and Fourier Series
	3.5 Group Actions and Representations
		3.5.1 Group Actions and Unitary Representations
		3.5.2 Unitary Representations of Compact Abelian Groups
		3.5.3 The Strong (Riemann) Integral
		3.5.4 The Weak (Lebesgue) Integral
		3.5.5 Proof of the Weight Decomposition
		3.5.6 Convolution
	3.6 Further Topics
Chapter 4 Uniform Boundedness and the Open Mapping Theorem
	4.1 Uniform Boundedness
		4.1.1 Uniform Boundedness and Fourier Series
	4.2 The Open Mapping and Closed Graph Theorems
		2.2.1 Baire Category
		4.2.2 Proof of the Open Mapping Theorem
		4.2.3 Consequences: Bounded Inverses and Closed Graphs
	4.3 Further Topics
Chapter 5 Sobolev Spaces and Dirichlet's Boundary Problem
	5.1 Sobolev Spaces and Embedding on the Torus
		5.1.1 L2 Sobolev Spaces on Td
		5.1.2 The Sobolev Embedding Theorem on Td
	5.2 Sobolev Spaces on Open Sets
		5.2.1 Examples
		5.2.2 Restriction Operators and Traces
		5.2.3 Sobolev Embedding in the Interior
	5.3 Dirichlet's Boundary Value Problem and Elliptic Regularity
		5.3.1 The Semi-Inner Product
		5.3.2 Elliptic Regularity for the Laplace Operator
		5.3.3 Dirichlet's Boundary Value Problem
	5.4 Further Topics
Chapter 6 Compact Self-Adjoint Operators, Laplace Eigenfunctions
	6.1 Compact Operators
		6.1.1 Integral Operators are Often Compact
	6.2 Spectral Theory of Self-Adjoint Compact Operators
		6.2.1 The Adjoint Operator
		6.2.2 The Spectral Theorem
		6.2.3 Proof of the Spectral Theorem
		6.2.4 Variational Characterization of Eigenvalues
	6.3 Trace-Class Operators
	6.4 Eigenfunctions for the Laplace Operator
		6.4.1 Right Inverse and Compactness on the Torus
		6.4.2 A Self-Adjoint Compact Right Inverse
		6.4.3 Eigenfunctions on a Drum
		6.4.4 Weyl's Law
	6.5 Further Topics
Chapter 7 Dual Spaces
	7.1 The Hahn–Banach Theorem and its Consequences
		7.1.1 The Hahn–Banach Lemma and Theorem
		7.1.2 Consequences of the Hahn–Banach Theorem
		7.1.3 The Bidual
		7.1.4 An Application of the Spanning Criterion
	7.2 Banach Limits, Amenable Groups, Banach–Tarski
		7.2.1 Banach Limits
		7.2.2 Amenable Groups
		7.2.3 The Banach–Tarski Paradox
	7.3 The Duals of Lp(X)
		7.3.1 The Dual of L1(X)
		7.3.2 The Dual of Lp(X) for p>1
		7.3.3 Riesz–Thorin Interpolation
	7.4 Riesz Representation: The Dual of C(X)
		7.4.1 Uniqueness
		7.4.2 Totally Disconnected Compact Spaces
		7.4.3 Compact Spaces
		7.4.4 Locally Compact -Compact Metric Spaces
		7.4.5 Continuous Linear Functionals on C0(X)
	7.5 Further Topics
Chapter 8 Locally Convex Vector Spaces
	8.1 Weak Topologies and the Banach–Alaoglu Theorem
		8.1.1 Weak* Compactness of the Unit Ball
		8.1.2 More Properties of the Weak and Weak* Topologies
		8.1.3 Analytic Functions and the Weak Topology
	8.2 Applications of Weak* Compactness
		8.2.1 Equidistribution
		8.2.2 Elliptic Regularity for the Laplace Operator
		8.2.3 Elliptic Regularity at the Boundary
	8.3 Topologies on the space of bounded operators
	8.4 Locally Convex Vector Spaces
	8.5 Distributions as Generalized Functions
	8.6 Convex Sets
		8.6.1 Extreme Points and the Krein–Milman Theorem
		8.6.2 Choquet's Theorem
	8.7 Further Topics
Chapter 9 Unitary Operators and Flows, Fourier Transform
	9.1 Spectral Theory of Unitary Operators
		9.1.1 Herglotz's Theorem for Positive-Definite Sequences
		9.1.2 Cyclic Representations and the Spectral Theorem
		9.1.3 Spectral Measures
		9.1.4 Functional Calculus for Unitary Operators
		9.1.5 An Application of Spectral Theory to Dynamics
	9.2 The Fourier Transform
		9.2.1 The Fourier Transform on L1(Rd)
		9.2.2 The Fourier Transform on L2(Rd)
		9.2.3 The Fourier Transform, Smoothness, Schwartz Space
		9.2.4 The Uncertainty Principle
	9.3 Spectral Theory of Unitary Flows
		9.3.1 Positive-Definite Functions; Cyclic Representations
		9.3.2 The Case G=Rd
		9.3.3 Stone's Theorem
	9.4 Further Topics
Chapter 10 Locally Compact Groups, Amenability, Property (T)
	10.1 Haar Measure
	10.2 Amenable Groups
		10.2.1 Definitions and Main Theorem
		10.2.2 Proof of Theorem 10.15
		10.2.3 A More Uniform Følner Set
		10.2.4 Further Equivalences and Properties
	10.3 Property (T)
		10.3.1 Definitions and First Properties
		10.3.2 Main Theorems
		10.3.3 Proof of Každan's Property (T), Connected Case
		10.3.4 Proof of Každan's Property (T), Discrete Case
		10.3.5 Iwasawa Decomposition and Geometry of Numbers
	10.4 Highly Connected Networks: Expanders
		10.4.1 Constructing an Explicit Expander Family
	10.5 Further Topics
Chapter 11 Banach Algebras and the Spectrum
	11.1 The Spectrum and Spectral Radius
		11.1.1 The Geometric Series and its Consequences
		11.1.2 Using Cauchy Integration
	11.2 C*-algebras
	11.3 Commutative Banach Algebras and their Gelfand Duals
		11.3.1 Commutative Unital Banach Algebras
		11.3.2 Commutative Banach Algebras without a Unit
		11.3.3 The Gelfand Transform
		11.3.4 fand Transform for Commutative C*-algebras
	11.4 Locally Compact Abelian Groups
		11.4.1 The Pontryagin Dual
	11.5 Further Topics
Chapter 12 Spectral Theory and Functional Calculus
	12.1 Definitions and Basic Lemmas
		12.1.1 Decomposing the Spectrum
		12.1.2 The Numerical Range
		12.1.3 The Essential Spectrum
	12.2 The Spectrum of a Tree
		12.2.1 The Correct Upper Bound for the Summing Operator
		12.2.2 The Spectrum of S
		12.2.3 No Eigenvectors on the Tree
	12.3 Main Goals: The Spectral Theorem and Functional Calculus
	12.4 Self-Adjoint Operators
		12.4.1 Continuous Functional Calculus
		12.4.2 Corollaries to the Continuous Functional Calculus
		12.4.3 Spectral Measures
		12.4.4 The Spectral Theorem for Self-Adjoint Operators
		12.4.5 Consequences for Unitary Representations
	12.5 Commuting Normal Operators
	12.6 Spectral Measures and the Measurable Functional Calculus
		12.6.1 Non-Diagonal Spectral Measures
		12.6.2 The Measurable Functional Calculus
	12.7 Projection-Valued Measures
	12.8 Locally Compact Abelian Groups and Pontryagin Duality
		12.8.1 The Spectral Theorem for Unitary Representations
		12.8.2 Characters Separate Points
		12.8.3 The Plancherel Formula
		12.8.4 Pontryagin Duality
	12.9 Further Topics
Chapter 13 Self-Adjoint and Symmetric Operators
	13.1 Examples and Definitions
	13.2 Operators of the Form T*T
	13.3 Self-Adjoint Operators
	13.4 Symmetric Operators
		13.4.1 The Friedrichs Extension
		13.4.2 Cayley Transform and Deficiency Indices
	13.5 Further Topics
Chapter 14 The Prime Number Theorem
	14.1 Two Reformulations
	14.2 The Selberg Symmetry Formula and Banach Algebra Norm
		14.2.1 Dirichlet Convolution and Möbius Inversion
		14.2.2 The Selberg Symmetry Formula
		14.2.3 Measure-Theoretic Reformulation
		14.2.4 A Density Function and the Continuity Bound
		14.2.5 Mertens' Theorem
		14.2.6 Completing the Proof
	14.3 Non-Trivial Spectrum of the Banach Algebra
	14.4 Trivial Spectrum of the Banach Algebra
	14.5 Primes in Arithmetic Progressions
		14.5.1 Non-Vanishing of Dirichlet L-function at 1
Appendix A: Set Theory and Topology
	A.1 Set Theory and the Axiom of Choice
	A.2 Basic Definitions in Topology
	A.3 Inducing Topologies
	A.4 Compact Sets and Tychonoff's Theorem
	A.5 Normal Spaces
Appendix B: Measure Theory
	B.1 Basic Definitions and Measurability
	B.2 Properties of the Integral
	B.3 The p-Norm
	B.4 Near-Continuity of Measurable Functions
	B.5 Signed Measures
Hints for Selected Problems
Notes
References
	Notation
	General Index