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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.

Hilbert space --- Hilbert, Espaces de --- Mathematical Methods in Physics. --- Algebraic Topology. --- Spaces, Metric --- Physics. --- Functional analysis. --- Algebraic topology. --- Functional Analysis. --- Hilbert space. --- Metric spaces. --- Topological spaces. --- Generalized spaces --- Set theory --- Topology --- Spaces, Topological --- Banach spaces --- Hyperspace --- Inner product spaces --- Mathematical physics. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Physical mathematics --- Physics --- Mathematics --- Hilbert, Espaces de. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics

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The book is a graduate text on unbounded self-adjoint operators on Hilbert space and their spectral theory with the emphasis on applications in mathematical physics (especially, Schrödinger  operators) and analysis (Dirichlet and Neumann Laplacians, Sturm-Liouville operators, Hamburger moment problem) . Among others, a number of advanced special topics   are treated on a text book level  accompanied by numerous illustrating examples and exercises. The main themes of the book are the following: - Spectral integrals and  spectral decompositions of self-adjoint and normal operators - Perturbations of self-adjointness and of spectra of self-adjoint operators - Forms and operators - Self-adjoint extension theory :boundary triplets, Krein-Birman-Vishik theory of positive self-adjoint extension.

Selfadjoint operators. --- Hilbert space. --- Banach spaces --- Hyperspace --- Inner product spaces --- Operators, Selfadjoint --- Self-adjoint operators --- Linear operators --- Selfadjoint operators --- Hilbert space --- Operator theory --- Spectral theory (Mathematics) --- Opérateurs auto-adjoints --- Hilbert, Espaces de --- Opérateurs, Théorie des --- Théorie spectrale (mathématiques) --- Functional analysis. --- Mathematical physics. --- Operator theory. --- Functional Analysis. --- Mathematical Methods in Physics. --- Operator Theory. --- Mathematical Physics. --- Theoretical, Mathematical and Computational Physics. --- Functional analysis --- Physical mathematics --- Physics --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Mathematics --- Opérateurs auto-adjoints. --- Hilbert, Espaces de. --- Opérateurs, Théorie des. --- Mathematics. --- Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Opérateurs auto-adjoints. --- Opérateurs, Théorie des. --- Théorie spectrale (mathématiques)