"A Hilbert Space Problem Book" by Paul R. Halmos is a graduate-level textbook that aims to teach mathematics through problem-solving. The book is structured into three main parts: problems, hints, and solutions. It covers a wide range of topics in Hilbert space theory, from standard textbook material to more advanced and boundary areas. The problems are designed to challenge the reader's thinking, encouraging them to explore related questions, generalizations, and special cases. The book emphasizes the importance of active learning, where readers are encouraged to attempt proofs and think critically about the material. It also includes historical remarks and references to further information. The content is divided into several chapters, each focusing on specific aspects of Hilbert space theory, such as vectors and spaces, weak topology, analytic functions, infinite matrices, boundedness and invertibility, multiplication operators, operator matrices, properties of spectra, examples of spectra, spectral radius, norm topology, strong and weak topologies, partial isometries, unilateral shift, compact operators, subnormal operators, numerical range, unitary dilations, commutators of operators, and Toeplitz operators. The book is intended for readers who have a foundational knowledge of Hilbert space theory and are looking to deepen their understanding through problem-solving."A Hilbert Space Problem Book" by Paul R. Halmos is a graduate-level textbook that aims to teach mathematics through problem-solving. The book is structured into three main parts: problems, hints, and solutions. It covers a wide range of topics in Hilbert space theory, from standard textbook material to more advanced and boundary areas. The problems are designed to challenge the reader's thinking, encouraging them to explore related questions, generalizations, and special cases. The book emphasizes the importance of active learning, where readers are encouraged to attempt proofs and think critically about the material. It also includes historical remarks and references to further information. The content is divided into several chapters, each focusing on specific aspects of Hilbert space theory, such as vectors and spaces, weak topology, analytic functions, infinite matrices, boundedness and invertibility, multiplication operators, operator matrices, properties of spectra, examples of spectra, spectral radius, norm topology, strong and weak topologies, partial isometries, unilateral shift, compact operators, subnormal operators, numerical range, unitary dilations, commutators of operators, and Toeplitz operators. The book is intended for readers who have a foundational knowledge of Hilbert space theory and are looking to deepen their understanding through problem-solving.