Compare also the central conceit of QM /Koopmania.
Take a classical nonlinear finite-dimensional system X described by a say a PDE. This is a dynamical system with evolution operator X → X.
Now look at the space H(X) of C/R-valued functions on the phase space of X. After completion we obtain an Hilbert space H.
Now the evolution operator on X induces a map on H= H(X).
We have now turned a finite-dimensional nonlinear problem into an infinite-dimensional linear problem.
Compare also the central conceit of QM /Koopmania. Take a classical nonlinear finite-dimensional system X described by a say a PDE. This is a dynamical system with evolution operator X → X. Now look at the space H(X) of C/R-valued functions on the phase space of X. After completion we obtain an Hilbert space H. Now the evolution operator on X induces a map on H= H(X). We have now turned a finite-dimensional nonlinear problem into an infinite-dimensional linear problem.