 Dirac’s attempt to prove the equivalence of matrix mechanics and wave mechanics made essential use of the \(\delta\) function, as indicated above. The \(\delta\) function was used by physicists before Dirac, but it became a standard tool in many areas of physics only after Dirac very effectively put it to use in quantum mechanics. It then became widely known by way of his textbook (Dirac 1930), which was based on a series of lectures on quantum mechanics given by Dirac at Cambridge University. This textbook saw three later editions: the second in 1935, the third in 1947, and the fourth in 1958. The fourth edition has been reprinted many times. Its staying power is due, in part, to another innovation that was introduced by Dirac in the third edition, his bra-ket formalism. He first published this formalism in (Dirac 1939), but the formalism did not become widely used until after the publication of the third edition of his book. There is no question that these tools, first the \(\delta\) function and then the bra-ket notation, were extremely effective for physicists practicing and teaching quantum mechanics both with regards to setting up equations and to the performance of calculations. Most quantum mechanics textbooks use \(\delta\) functions and plane waves, which are key elements of Dirac’s formal framework, but they are not included in von Neumann’s rigorous mathematical framework for quantum mechanics. Working physicists as well as teachers and students of quantum mechanics often use Dirac’s framework because of its simplicity, elegance, power, and relative ease of use. Thus, from the standpoint of pragmatics, Dirac’s framework is much preferred over von Neumann’s. The notion of a rigged Hilbert space placed Dirac’s framework on a firm mathematical foundation.