N.4 Are Hermitian operators really like that?

A mathematician might choose to phrase the problem of Hermitian operators having or not having eigenvalues and eigenfunctions in a suitable space of permissible functions and then find, with some justification, that some operators in quantum mechanics, like the position or momentum operators do not have any permissible eigenfunctions. Let alone a complete set. The approach of this text is to simply follow the formalism anyway, and then fix the problems that arise as they arise.

More generally, what this book tells you about operators is absolutely true for systems with a finite number of variables, but gets mathematically suspect for infinite systems. The functional analysis required to do better is well beyond the scope of this book and the abstract mathematics a typical engineer would ever want to have a look at.

In any case, when problems are discretized to a finite one for numerical solution, the problem no longer exists. Or rather, it has been reduced to figuring out how the numerical solution approaches the exact solution in the limit that the problem size becomes infinite.