Hilbert-space techniques are used for both quantum theory and classical physics, with the Koopman–von Neumann formulation and the method of “classical” wave functions being important examples. In 1931, Bernard Koopman published a paper outlining a method for identifying functions representing observables as vectors in Hilbert space, and in 1932, John von Neumann published follow-up papers further developing this method.
Wikipedia details Von Neumann expanded Koopman’s 1931 Hilbert-Space work
Hilbert-space techniques are used for both quantum theory and classical physics, with the Koopman–von Neumann formulation and the method of “classical” wave functions being important examples. In 1931, Bernard Koopman published a paper outlining a method for identifying functions representing observables as vectors in Hilbert space, and in 1932, John von Neumann published follow-up papers further developing this method.