In Griffiths 2nd Edition Introduction to Quantum Mechanics, it is stated that "Although none of the eigenfunctions of p lives in Hilbert space, a certain family of them (those with real eigenvalues) resides in the nearby suburbs, with a kind of quasi-normalizability." However, my understanding is that observables are represented by hermitian operators.
So my question is:
For an operator to yield an observable, the wave function should be square integrable and thus belong to a Hilbert space (a complete inner-product space). So if the eigenfunctions of the momentum operator do not belong to a Hilbert space, how is the operator Hermitian?