This book offers an essential introduction to the theory of Hilbert space, a fundamental tool for non-relativistic quantum mechanics. Linear, topological, metric, and normed spaces are all addressed in detail, in a rigorous but reader-friendly fashion. The rationale for providing an introduction to the theory of Hilbert space, rather than a detailed study of Hilbert space theory itself, lies in the strenuous mathematics demands that even the simplest physical cases entail. Graduate courses in physics rarely offer enough time to cover the theory of Hilbert space and operators, as well as distribution theory, with sufficient mathematical rigor. Accordingly, compromises must be found between full rigor and the practical use of the instruments. Based on one of the authors's lectures on functional analysis for graduate students in physics, the book will equip readers to approach Hilbert space and, subsequently, rigged Hilbert space, with a more practical attitude. It also includesa brief introduction to topological groups, and to other mathematical structures akin to Hilbert space. Exercises and solved problems accompany the main text, offering readers opportunities to deepen their understanding. The topics and their presentation have been chosen with the goal of quickly, yet rigorously and effectively, preparing readers for the intricacies of Hilbert space. Consequently, some topics, e.g., the Lebesgue integral, are treated in a somewhat unorthodox manner. The book is ideally suited for use in upper undergraduate and lower graduate courses, both in Physics and in Mathematics.

Autorentext

Prof. Carlo Alabiso obtained his degree in Physics at Milan University, Italy, and then taught for more than 40 years at Parma University, Parma, Italy (with a period spent as a research fellow at the Stanford Linear Accelerator Center and at CERN, Geneva), until his retirement in 2011. His teaching encompassed topics in Quantum Mechanics, special relativity, field theory, elementary particle physics, mathematical physics, and functional analysis. His research fields include mathematical physics (Padé approximants), elementary particle physics (symmetries and quark models), and statistical physic (ergodic problems), and he has published articles in a wide range of national and international journals, as well as the previous Springer book (with Alessandro Ciesa), Problemi di Meccanica Quantistica non Relativistica.

Dr Ittay Weiss received his PhD in Mathematics from Utrecht University, the Netherlands, in 2007. He has published extensively on algebraic topology, general topology, metric space theory, and category theory, and has taught undergraduate and postgraduate mathematics in the Netherlands, Fiji, and England. He is currently a Senior Lecturer in Mathematics at the School of Mathematics and Physics, University of Portsmouth, UK. Born in Israel, his first encounter with advanced mathematics was at the age of 16 while he was following computer science courses at the Israeli Open University. So profound was his fascination with the beauty and utility of mathematics that, despite the digital economic boom at the time, he enrolled, as soon as he could, for the BSc program in Mathematics at the Hebrew University and continued to pursue his MSc immediately afterwards, receiving both degrees cum laude. Convinced that the distinction between pure and applied mathematics is illusory and that the abstract and the concrete form a symbiosis of endless mutual nourishment, he finds great joy in digging deep into the mathematical foundations of applied topics. The communication of mathematics is close to his heart. Occasionally he finds an angle he particularly likes, which he might explore in an article in the online journal The Conversation.

Inhalt

1. Introduction and preliminaries 1.1. Hilbert Space Theory - A Quick Overview.1.1.1. The Real Numbers - Where it All begins.1.1.2. Linear Spaces.1.1.3. Topological Spaces.1.1.4. Metric Spaces.1.1.5. Normed Spaces and Banach Spaces.1.1.6. Topological Groups.1.2. Preliminaries.1.2.1. Sets.1.2.2. Common Sets.1.2.3. Relations Between Sets.1.2.4. Families of Sets; Union and Intersection.1.2.5. Set Difference, Complementation, and De Morgan's Laws.1.2.6. Finite Cartesian Products.1.2.7. Functions.1.2.8. Arbitrary Cartesian Products.1.2.9. Direct and Inverse Images.1.2.10. Indicator Functions.1.2.11. Cardinality.1.2.12. The Cantor-Shroeder-Berenstein Theorem.1.2.13. Countable Arithmetic.1.2.14. Relations.1.2.15. Equivalence Relations.1.2.16. Ordered Sets.1.2.17. Zorn's Lemma.1.2.18. A Typical Application of Zorn's Lemma.1.2.19. The Real Numbers. 2. Linear Spaces2.1. Linear Spaces - Elementary Properties and Examples.2.1.1. Elementary Properties of Linear Spaces.2.1.2. Examples of Linear Spaces.2.2. The Dimension of a Linear Space. 2.2.1. Linear Independence, Spanning Sets, and Bases.2.2.2. Existence of Bases.2.2.3. Existence of Dimension.2.3. Linear Operators.2.3.1. Examples of Linear Operators.2.3.2. Algebra of Operators.2.3.3. Isomorphism.2.4. Subspaces, Products, and Quotients.2.4.1. Subspaces.2.4.2. Kernels and Images.2.4.3. Products and Quotients.2.4.4. Complementary Subspaces.2.5. Inner Product Spaces and Normed Spaces.2.5.1. Inner Product Spaces.2.5.2. The Cauchy-Schwarz Inequality.2.5.3. Normed Spaces.2.5.4. The Family of `p Spaces.2.5.5. The Family of Pre-Lp Spaces. 3. Topological Spaces3.1. Topology - Definition and Elementary Results.3.1.1. Definition and Motivation.3.1.2. More Examples.3.1.3. Elementary Observations.3.1.4. Closed Sets.3.1.5. Bases and Subbases.3.2. Subspaces, Point-Set Relationships, and Countability Axioms.3.2.1. Subspaces and Point-Set Relationships.3.2.2. Sequences and Convergence.3.2.3. Second Countable and First Countable Spaces.3.3. Constructing Topologies.3.3.1. Generating Topologies.3.3.2. Coproducts, Products, and Quotients. 3.4. Separation and Connectedness.3.4.1. The Huasdorff Separation Property.3.4.2. Path-Connectedness and Connected Spaces.3.5. Compactness. 4. Metric Spaces4.1. Metric Spaces - Definition and Examples.4.2. Topology and Convergence in a Metric Space.4.2.1. The Induces Topology.4.2.2. Convergence in a Metric Space.4.3. Non-Expanding Functions and Uniform Continuity.4.4. Complete Metric Spaces.4.4.1. Complete Metric Spaces.4.4.2. Banach's Fixed Point Theorem.4.4.3. Baire's Theorem.4.4.4. Completion of a Metric Space.4.5. Compactness and Boundedness. 5. The Lebesgue Integral Following Mikusiniski 6. Banach Spaces 6.1. Semi-Norms, Norms, and Banach Spaces.6.1.1. Semi-Norms and Norms.6.1.2. Banach Spaces.6.1.3. Bounded Operators.6.1.4. The Open Mapping Theorem.6.1.5. Banach Spaces of Linear and Bounded Operators.6.2. Fixed Point Techniques in Banach Spaces.6.2.1. Systems of Linear Equations.6.2.2. Cauchy's Problem and the Volterra Equation.6.2.3. Fredholm Equations.6.3. Inverse Operators.6.3.1. Existence of Bounded Inverses.6.3.2. Fixed Point Techniques Revisited.6.4. Dual Spaces.6.4.1. Duals of Classical Spaces.6.4.2. The Hahn-Banach Theorem.6.5. Unbounded Operators and Locally Convex Spaces.6.5.1. Closed Operators.6.5.2. Locally Convex Spaces. 7. Hilbert Spaces 7.1. Definition and Examples.7.2. Orthonormal Bases.7.3. Orthogonal Projections.7.4. Riesz Representation Theorem.7.5. Fourier Series.7.6. Self-adjoint operators.7.7. The Schroedinger Equation and the Heisenberg Uncertainty Principle. 8. A Survery of mathematical structures related to Hilbert space theory8.1. Topological Groups.8.1.1. Groups and Homomorphisms.8.1.2. Topological Groups and Homomorphisms.8.1.3. Topological Subgroups.8.1.4. Quotient Groups.8.1.5. Uniformities.8.2. Topological Vector Spaces.8.3. Sobolev Spaces.8.4. C algebras.8.5. Distributions.8.6. Dagger compact categories.9. Solved Problems