I was reading Introduction to quantum mechanics by David J. Griffiths and came across following paragraph:
" 3. The eigenvectors of a hermitian transformation span the space.
As we have seen, this is equivalent to the statement that any hermitian matrix can be diagonalized. This rather technical fact is, in a sense, the mathematical support on which much of a quantum mechanics leans. It turns out to be a thinner reed then one might have hoped, because the proof does not carry over to infinite-dimensional spaces."
If much of a quantum mechanics leans on it, but the proof does not carry over to infinite-dimensional spaces, then hermitian transformations with infinite dimensionality are spurious.
But there is infinite set of separable solutions for e.g. particle in a box. So Hamiltionan for that system has spectrum with infinite number of eigenvectors and is of infinite dimensionality.
If we can't prove that this infinite set of eigenvectors span the space then how can we use completness all the time?
Am I missing something here? Any missconceptions?
I'd appriciate any help.