Atlas of Fundamental Concepts

Vector spaces acquire topology, seminorms, norms, completeness, and inner products. Operators must then be classified with attention to boundedness and domains. Adjoint, self-adjoint, unitary, positive, projection, and trace-class operators lead to spectra and spectral measures; spectral and Stone theorems connect operators with decompositions and continuous unitary evolution. Rigged and tensor-product Hilbert spaces, partial traces, distributions, and Fock spaces supply the analytic infrastructure of quantum theory.