I recently asked a similar question but it was from the point of view of axiomatic quantum field theory, so I'd like to ask it in a more general form. Given a quantum theory (either quantum mechanics or quantum field theory), why do we require that the eigenstates of an observable form a basis (or at least a dense set) for the Hilbert space of states?
In quantum field theory for example we instead require that every state in the Hilbert space can be written as some combination of the fields or functions of the fields acting on the vacuum vector. I have also seen this condition described as "we don't want the Hilbert space to be too large" or that the theory can be described by only the fields of interest.
Why is not having completeness or having "too big" of a Hilbert space an issue? Just to be clear, I'm looking for a more physical answer and not something related to the spectral theorem.
