Linear operators in Hilbert spaces play a fundamental role in the formulation of quantum theory. With complete proofs and numerous exercises, this book offers a self-contained presentation of the most important tools and methods from Hilbert space theory, with a particular focus on the spectral theory of self-adjoint operators. It also goes further by describing several applications in quantum mechanics, including the analysis of Schrödinger operators and quantum scattering theory. In addition, the book discusses Mourre’s conjugate operator method and its consequences for scattering theory. Based on a one-year course for advanced undergraduates, this text provides a fundamental treatment of the basic ideas of applied Hilbert space theory.