Hello I have a question: In perturbation method perturbed eigenstates expanded by unpurturbed eigenstates, but we know when the system purturbed its hilbert space altered and hence its basis changed, then we can't state this basis spans the new hilbert space and we can't expand purturbed eigenstates by them. How this problem resolved? Please don't answer this question to say that it is approximately true since in textbooks this is done in exact manner.

In perturbation theory perturbed eigenstates expanded by unperturbed eigenstates, but we know when the system perturbed its Hilbert space altered and hence its basis changed, then we can't state this basis spans the new Hilbert space and we can't expand perturbed eigenstates by them. How this problem resolved? Please don't answer this question to say that it is approximately true since in textbooks this is done in exact manner.