1. Introduction and preliminaries1.1. 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 Equati
About the Author: 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.
