$$\frac{d}{d x}\left\{p(x) \frac{d y}{d x}\right\} + \big(q(x) + \lambda r(x)\big)y = f(x)$$
It is standard form of nonhomogeneous Sturm - Liouville .
We first solve the homogeneous part and find eigenvalues and corresponding eigenfunctions if there are any. Further, we say that eigenfunctions are complete that is they generate the "functions space", every function can be expressed as power series with eigenfunctions (like writting vectors with bases).
My question is: What is that eigenfunction space? How general is it? For example, how we are so sure that $f(x)$ at the nonhomogeneous part is in this eigenfunction space?