A mathematical introduction to Diracs formalism This monograph contains a functional analytic introduction to Diracs formalism. The first part presents some new mathematical notions in the setting of triples of Hilbert spaces mentioning the concept of Dirac basis. The second part introduces a conceptually new theory of generalized functions integrating the notions of the first part. The last part of the book is devoted to a mathematical interpretation of the main features of Diracs formalism. It involves a pairing between distributional bras and kets continuum expansions and continuum matrices