 Mathematicians worked very hard to provide a rigorous foundation for Dirac’s formal framework. One key element was Schwartz’s (1945; 1950–1951) theory of distributions. Another key element, the notion of a nuclear space, was developed by Grothendieck (1955). This notion made possible the generalized-eigenvector decomposition theorem for self-adjoint operators in rigged Hilbert space – for the theorem see Gelfand and Vilenken (1964, pp. 119–127), and for a brief historical account of the convoluted path leading to it see Berezanskii (1968, pp. 756–760). The decomposition principle provides a rigorous way to handle observables such as position and momentum in the manner in which they are presented in Dirac’s formal framework. These mathematical developments culminated in the early 1960s with Gelfand and Vilenkin’s characterization of a structure that they referred to as a rigged Hilbert space (Gelfand and Vilenkin 1964, pp. 103–127). It is unfortunate that their chosen name for this mathematical structure is doubly misleading. First, there is a natural inclination to regard it as denoting a type of Hilbert space, one that is rigged in some sense, but this inclination must be resisted. Second, the term rigged has an unfortunate connotation of illegitimacy, as in the terms rigged election or rigged roulette table, and this connotation must be dismissed as prejudicial. There is nothing illegitimate about a rigged Hilbert space from the standpoint of mathematical rigor (or any other relevant standpoint). A more appropriate analogy may be drawn using the notion of a rigged ship: the term rigged in this context means fully equipped. But this analogy has its limitations since a rigged ship is a fully equipped ship, but (as the first point indicates) a rigged Hilbert space is not a Hilbert space, though it is generated from a Hilbert space in the manner now to be described.