Textbook in PDF format Both quantum mechanics and statistical mechanics share the notable property that an understanding of the theory lies rather deep, yet a mastery of the associated calculations is quite straightforward. Moreover, it is perfectly reasonable to do research in either subject without usually considering the difficult problems underlying the the­ ory. It is unfortunate that this circumstance fosters the idea that such problems as there may be are irrelevant to anything important. The problems we have in mind do not admit of easy solutions. A necessary pre­requisite to progress is a careful and rigorous formulation. This will at least separate the essential from the inessential difficulties. It will also make it difficult to believe that there are no problems. These notes have as their goal a mathematically precise formulation of elementary quantum mechanics. Introduction. Basic Quantum Mechanics. Topological Algebras. The Algebra of Observables. The States of the System. Dynamics and Symmetries. Quantum Measurements