 Dirac’s formal framework for quantum mechanics was very useful and influential despite its lack of mathematical rigor. It was used extensively by physicists and it inspired some powerful mathematical developments in functional analysis. Eventually, mathematicians developed a suitable framework for placing Dirac’s formal framework on a firm mathematical foundation, which is known as a rigged Hilbert space (and is also referred to as a Gelfand Triplet). This came about as follows. A rigorous definition of the \(\delta\) function became possible in distribution theory, which was developed by Schwartz from the mid-1940s to the early 1950s. Distribution theory inspired Gelfand and collaborators during the mid-to-late 1950s to formulate the notion of a rigged Hilbert space, the firm foundation for Dirac’s formal framework. This development was facilitated by Grothendiek’s notion of a nuclear space, which he introduced in the mid-1950s. The rigged Hilbert space formulation of quantum mechanics was then developed independently by Böhm and by Roberts in 1966. Since then, it has been extended to a variety of different contexts in the quantum domain including decay phenomena and the arrow of time. The mathematical developments of Schwartz, Gelfand, and others had a substantial effect on QFT as well. Distribution theory was taken forward by Wightman in developing the axiomatic approach to QFT from the mid-1950s to the mid-1960s. In the late 1960s,  the axiomatic approach was explicitly put into the rigged Hilbert space framework by Bogoliubov and co-workers.