In details. I know that "the states of a quantum mechanical system are vectors in a certain Hilbert space, the observables are hermitian operators on that space, the symmetries of the system are unitary operators, and measurements are orthogonal projections" But this concept was developed by Von Neumann. Before he developed Hilbert Space. What are their counterpart in the pure Schroedinger equation up to Born interpretation of the amplitude square as the probability that electron can be found there?

Please answer more in words or conceptual and not with dense mathematical equations. Thanks.