The relation between the eigenspaces of $H_0$ and $H_0+H_1 $ For the unperturbed Hamiltonian $H_0$, we have a set of eigenfunctions $\{ f_i \}$. For the perturbed Hamiltonian $H_1$, we have another set of eigenfunctions $\{g_i \}$. 
Sometimes, it is not clear to me that the $f$'s and the $g$'s span the same Hilbert space. 
For example, let $H_0 = \frac{1}{2} p^2 + \frac{1}{2}x^2 $, and $H_1 = g \delta (x)$. It is not difficult to solve the eigenstates of $H_0+H_1$, at least numerically. But the question is, is it for sure that these eigenstates can be expanded by the harmonic oscillator eigenfunctions? 
 A: First of all, your example is flawed, because the "perturbation" you consider consists of a distribution, i.e. lies outside of the Hilbert space $L^2(\mathbb R)$. In this sense, one could formulate your problem in Hilbert space only (this is the domain of application of the famous Kato-Rellich theorem (*)), in which the full Hamiltonian is defined and is under certain conditions self-adjoint as is the non-perturbed one. In this case, both Hamiltonians have their seld-adjointness domains within the same Hilbert space. 
However, since you bring in eigenvectors, then it is automatically natural to work with rigged Hilbert spaces, case in which both $H_0$ and $H_0 + H_1$ have the same space of distributions in which their generalized eigenvectors exist and are well-defined. 
Example: the free particle Hamiltonian in 3D $\frac{\bf{p}^2}{2m}$ has the set of eigenvectors in the same $\left(C_0^{\infty}\left(\mathbb R^3\right)\right)'$ space as the Schroedinger H-atom (virtual particle) Hamiltonian $\frac{\bf{p}^2}{2m} + \frac{1}{r}$. You can check for yourself that there is a particular way in which these two sets of distributions are related one to another.  
(*) http://people.math.gatech.edu/~loss/14SPRINGTEA/katorellich.pdf [I hope this link won't rot over the years to come] 
