I The Problem: An Axiomatic Basis for Quantum Mechanics -- 1 The Axiomatic Formulation of a Physical Theory -- 2 The Fundamental Domain for Quantum Mechanics -- 3 The Measurement Problem -- II Microsystems, Preparation, and Registration Procedures -- 1 The Concept of a Physical Object -- 2 Selection Procedures -- 3 Statistical Selection Procedures -- 4 Physical Systems -- III Ensembles and Effects -- 1 Combinations of Preparation and Registration Methods -- 2 Mixtures and Decompositions of Ensembles and Effects -- 3 General Laws: Preparation and Registration of Microsystems -- 4 Properties and Pseudoproperties -- 5 Ensembles and Effects in Quantum Mechanics -- 6 Decision Effects and Faces of K -- IV Coexistent Effects and Coexistent Decompositions -- 1 Coexistent Effects and Observables -- 2 Structures in the Class of Observables -- 3 Coexistent and Complementary Observables -- 4 Realizations of Observables -- 5 Coexistent Decompositions of Ensembles -- 6 Complementary Decompositions of Ensembles -- 7 Realizations of Decompositions -- 8 Objective Properties and Pseudoproperties of Microsystems -- V Transformations of Registration and Preparation Procedures. Transformations of Effects and Ensembles -- 1 Morphisms for Selection Procedures -- 2 Morphisms of Statistical Selection Procedures -- 3 Morphisms of Preparation and Registration Procedures -- 4 Morphisms of Ensembles and Effects -- 5 Isomorphisms and Automorphisms of Ensembles and Effects -- VI Representation of Groups by Means of Effect Automorphisms and Mixture Automorphisms -- 1 Homomorphic Maps of a Group 𝒢 in the Group 𝓐 of ?-continuous Effect Automorphisms -- 2 The 𝒢-invariant Structure Corresponding to a Group Representation -- 3 Properties of Representations of 𝒢 which are Dependent on the Special Structure of 𝓐(?) in Quantum Mechanics -- VII The Galileo Group -- 1 The Galileo Group as a Set of Transformations of Registration Procedures Relative to Preparation Procedures -- 2 Irreducible Representations of the Galileo Group and Their Physical Meaning -- 3 Irreducible Representations of the Rotation Group -- 4 Position and Momentum Observables -- 5 Energy and Angular Momentum Observables -- 6 Time Observable? -- 7 Spatial Reflections (Parity Transformations) -- 8 The Problem of the Space 𝓓 for Elementary Systems -- 9 The Problem of Differentiability -- VIII Composite Systems -- 1 Registrations and Effects of the Inner Structure -- 2 Composite Systems Consisting of Two Different Elementary Systems -- 3 Composite Systems Consisting of Two Identical Elementary Systems -- 4 Composite Systems Consisting of Electrons and Atomic Nuclei -- 5 The Hamiltonian Operator -- 6 Microsystems in External Fields -- 7 Criticism of the Description of Interaction in Quantum Mechanics and the Problem of the Space 𝓓 -- Appendix I -- Summary of Lattice Theory -- 1 Definition of a Lattice -- 2 Orthomodularity -- 3 Boolean Rings -- 4 Set Lattices -- Appendix II -- Remarks about Topological and Uniform Structures -- 1 Topological Spaces -- 2 Uniform Spaces -- 3 Baire Spaces -- 4 Connectedness -- Appendix III -- Banach Spaces -- 1 Linear Vector Spaces -- 2 Normed Vector Spaces and Banach Spaces -- 3 The Dual Space for a Banach Space -- 4 Weak Topologies -- 5 Linear Maps of Banach Spaces -- 6 Ordered Vector Spaces -- Appendix IV -- Operators in Hubert Space -- 1 The Hubert Space Structure Type -- 2 Orthogonal Systems and Closed Subspaces -- 3 The Banach Space of Bounded Operators -- 4 Bounded Linear Forms -- 6 Projection Operators -- 7 Isometric and Unitary Operators -- 8 Spectral Representation of Self-adjoint and Unitary Operators -- 9 The Spectrum of Compact Self-adjoint Operators -- 10 Spectral Representation of Unbounded Self-adjoint Operators -- 11 The Trace as a Bilinear Form -- 12 Gleason’s Theorem -- 13 Isomorphisms and Anti-isomorphisms -- 14 Products of Hubert Spaces -- References -- List of Frequently Used Symbols -- List of Axioms.