Functional Analysis

Contents
Preface
Notation and Conventions
1 Banach Spaces
	1.1 Banach Spaces
	1.2 Bounded Operators
	1.3 Finite-Dimensional Spaces
	1.4 Compactness
	1.5 Integration in Banach Spaces
	Problems
2 The Classical Banach Spaces
	2.1 Sequence Spaces
	2.2 Spaces of Continuous Functions
	2.3 Spaces of Integrable Functions
	2.4 Spaces of Measures
	2.5 Banach Lattices
	Problems
3 Hilbert Spaces
	3.1 Hilbert Spaces
	3.2 Orthogonal Complements
	3.3 The Riesz Representation Theorem
	3.4 Orthonormal Systems
	3.5 Examples
	Problems
4 Duality
	4.1 Duals of the Classical Banach Spaces
	4.2 The Hahn–Banach Extension Theorem
	4.3 Adjoint Operators
	4.4 The Hahn–Banach Separation Theorem
	4.5 The Krein–Milman Theorem
	4.6 The Weak andWeak∗ Topologies
	4.7 The Banach–Alaoglu Theorem
	Problems
5 Bounded Operators
	5.1 The Uniform Boundedness Theorem
	5.2 The Open Mapping Theorem
	5.3 The Closed Graph Theorem
	5.4 The Closed Range Theorem
	5.5 The Fourier Transform
	5.6 The Hilbert Transform
	5.7 Interpolation
	Problems
6 Spectral Theory
	6.1 Spectrum and Resolvent
	6.2 The holomorphic Functional Calculus
	Problems
7 Compact Operators
	7.1 Compact Operators
	7.2 The Riesz–Schauder Theorem
	7.3 Fredholm Theory
	Problems
8 Bounded Operators on Hilbert Spaces
	8.1 Selfadjoint, Unitary, and Normal Operators
	8.2 The Continuous Functional Calculus
	Problems
9 The Spectral Theorem for Bounded Normal Operators
	9.1 The Spectral Theorem for Compact Normal Operators
	9.2 Projection-Valued Measures
	9.3 The Bounded Functional Calculus
	9.4 The Spectral Theorem for Bounded Normal Operators
	9.5 The Von Neumann Bicommutant Theorem
	9.6 Application to Orthogonal Polynomials
	Problems
10 The Spectral Theorem for Unbounded Normal Operators
	10.1 Unbounded Operators
	10.2 Unbounded Selfadjoint Operators
	10.3 Unbounded Normal Operators
	10.4 The Spectral Theorem for Unbounded Normal Operators
	Problems
11 Boundary Value Problems
	11.1 Sobolev Spaces
	11.2 The Poisson Problem −Δu = f
	11.3 The Lax–Milgram Theorem
	Problems
12 Forms
	12.1 Forms
	12.2 The Friedrichs Extension Theorem
	12.3 The Dirichlet and Neumann Laplacians
	12.4 The Poisson Problem Revisited
	12.5 Weyl’s Theorem
	Problems
13 Semigroups of Linear Operators
	13.1 C₀-Semigroups
	13.2 The Hille–Yosida Theorem
	13.3 The Abstract Cauchy Problem
	13.4 Analytic Semigroups
	13.5 Stone’s Theorem
	13.6 Examples
	13.7 Semigroups Generated by Normal Operators
	Problems
14 Trace Class Operators
	14.1 Hilbert–Schmidt Operators
	14.2 Trace Class Operators
	14.3 Trace Duality
	14.4 The Partial Trace
	14.5 Trace Formulas
	Problems
15 States and Observables
	15.1 States and Observables in Classical Mechanics
	15.2 States and Observables in Quantum Mechanics
	15.3 Positive Operator-Valued Measures
	15.4 Hidden Variables
	15.5 Symmetries
	15.6 Second Quantisation
	Problems
Appendix A Zorn’s Lemma
Appendix B Tensor Products
Appendix C Topological Spaces
Appendix D Metric Spaces
Appendix E Measure Spaces
Appendix F Integration
Appendix G Notes
	Chapter 1,2
	Chapter 3
	Chapter 4, 5
	Chapter 6, 7
	Chapter 8, 9
	Chapter 10, 11
	Chapter 12
	Chapter 13
	Chapter 14
	Chapter 15
	Appendices
References
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Index
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