Functional Analysis

Description

This comprehensive introduction to functional analysis covers both the abstract theory and applications to spectral theory, the theory of partial differential equations, and quantum mechanics. It starts with the basic results of the subject and progresses towards a treatment of several advanced topics not commonly found in functional analysis textbooks, including Fredholm theory, form methods, boundary value problems, semigroup theory, trace formulas, and a mathematical treatment of states and observables in quantum mechanics. The book is accessible to graduate students with basic knowledge of topology, real and complex analysis, and measure theory. With carefully written out proofs, more than 300 problems, and appendices covering the prerequisites, this self-contained volume can be used as a text for various courses at the graduate level and as a reference text for researchers in the field. 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 B CD EFG HJ KLM NOPR S TUVWYZ Index ab c de fgh ijkl mno p qrs t uvw