The Dirac Delta function is a generalized function, or distribution, introduced by P.A.M. Dirac: it is used to model the density of an idealized point mass, as a function equal to zero everywhere except for zero, and whose integral over the real line is equal to one.
As a distribution, the Dirac delta function is a linear function that maps every function to its value at zero
[Hilbert Space] was established systematically by von Neumann, summarized in his book. The book was based on three papers, published in 1927. Of these three papers, the first introduced the notion of abstract Hilbert space, and presented the "eigenvalue problem", of self-adjoint operators having a continuous part in their spectrum, in a mathematically rigorous form, without making use of Dirac's delta function. The complete, analysis of the spectral theorem was worked out in a following paper in 1930.