Abstract
In this paper the notion of Dirac basis will be introduced. It is the continuous pendant of the discrete basis for Hilbert spaces. The introduction of this new notion is closely related to the theory of generalized functions. Here De Graaf's theory will be employed. It is based on the triplet S X, A ⊂X⊂ T X, A where X is a Hilbert space. In a well specified way any member of T X, A can be expanded with respect to a Dirac basis. Both the introduction of Dirac bases and a new interpretation of Dirac's bracket notion will lead to a mathematical rigorization of various aspects of Dirac's formalism for quantum mechanics. This rigorization goes much beyond earlier proposals.
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