A Primer on Hilbert Space Theory: Linear Spaces, Topological Spaces, Metric Spaces, Normed Spaces, and Topological Groups (UNITEXT for Physics)

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 includes a 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. Preface to the Second Edition Preface to the First Edition About This Book Contents List of Symbols 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 Classical Sequence and Function Spaces 1.1.6 Banach Spaces 1.1.7 Topological Groups 1.1.8 Hilbert Spaces 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-Shröder-Bernstein 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 1.2.20 The Complex Numbers References 2 Linear Spaces 2.1 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 ellp Spaces 2.5.5 The Family of Pre-Lp Spaces References 3 Topological Spaces 3.1 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 Hausdorff Separation Property 3.4.2 Path-Connected and Connected Spaces 3.5 Compactness References 4 Metric Spaces 4.1 Metric Spaces—Definition and Examples 4.2 Topology and Convergence in a Metric Space 4.2.1 The Induced Topology 4.2.2 Convergence in Metric Spaces 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 References 5 A Non-classical View of the Classical Spaces 5.1 Metric and Topological Considerations 5.1.1 The Induced Topology and Equivalent Norms 5.1.2 Nets and Dense Sets 5.1.3 Point-to-Set Distance and Closed Sets 5.1.4 The Operator Norm 5.2 Finite-Dimensional Spaces 5.2.1 All Norms are Equivalent 5.2.2 Finite-Dimensional Sequence and Function Spaces 5.3 Countable-Dimensional Sequence and Function Spaces 5.3.1 The Space c00 as a Colimit 5.3.2 The Space mathbbP as a Colimit 5.3.3 Norm Extension 5.3.4 Operator Extension and Boundedness 5.4 The Classical Sequence and Function Spaces 5.4.1 From Finite Dimensions to the Classical Spaces 5.4.2 Summation and Integration 5.5 Separability and Duality 5.5.1 Separability 5.5.2 Duality 5.5.3 Non-separability References 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 Duality 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 References 7 Topological Groups 7.1 Groups and Homomorphisms 7.2 Topological Groups and Homomorphism 7.3 Topological Subgroups 7.4 Quotient Groups 7.5 Uniformities References 8 Hilbert Spaces 8.1 The Closest Point Property 8.2 Orthogonal Complements 8.3 Bases: Hamel, Schauder, and Hilbert 8.4 Fourier Series 8.5 The Riesz Representation Theorem References 9 Solved Problems 9.1 Linear Spaces 9.2 Topological Spaces 9.3 Metric Spaces 9.4 Normed Spaces and Banach Spaces 9.5 Topological Groups Appendix Solutions Index This book is an 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 an introduction to the theory of Hilbert space, rather than a detailed study of Hilbert space theory itself, resides in the very high mathematical difficulty of even the simplest physical case. Within an ordinary graduate course in physics there is insufficient time to cover the theory of Hilbert spaces and operators, as well as distribution theory, with sufficient mathematical rigor. Compromises must be found between full rigor and practical use of the instruments. The book is based on the author's lessons on functional analysis for graduate students in physics. It will equip the reader to approach Hilbert space and, subsequently, rigged Hilbert space, with a more practical attitude. With respect to the original lectures, the mathematical flavor in all subjects has been enriched. Moreover, a brief introduction to topological groups has been added in addition to exercises and solved problems throughout the text. With these improvements, the book can be used in upper undergraduate and lower graduate courses, both in Physics and in Mathematics.