One of the quantum mechanics postulates states that a composite system can be described with the tensor product of the component systems. I've read some rationalization about this fact in some post
Should it be obvious that independent quantum states are composed by taking the tensor product?
As I understand the answers, they refer to non interacting parts of the system or at least they propose the plausibility of associate a single particle state with a part of the system.
I was thinking about the case of a many electron system, e.g. an atom. I can not see any kind of separability (except because a mean field approach) of the wave function. For example, in an two electron system (neglecting the spin part) it is said that we have
$$ \psi(x_1,x_2) = \sum_{i,j} d_{ij}\,\phi_i(x_1) \phi_j(x_2)$$
(Please correct me if I am wrong, but I think that the validity of the equation above is because the knowledge that the state of the one particle system is in $L^2(\Omega_i,\mu_i)$, and the two particle system must be in $L^2(\Omega_1 \otimes \Omega_2, \mu_1 \otimes \mu_2)$. Knowing that $\{\phi_i\}$ is a complete orthonormal set in the one particle system space $L^2(\Omega_i,\mu_i)$, $\{\phi_i(x_1)\phi_j(x_2) \}$ is a complete orthonormal set in $L^2(\Omega_1 \otimes \Omega_2, \mu_1 \otimes \mu_2)$).
I have not found by intuition any physical sense to the one electron functions $\{\phi_i\}$ (because electron-electron repulsion), as I see it, it is not even necessary that they are solutions of the one electron problem so:
Question 1: Leaving aside spin contribution, why we need tensor product if not as a method to recover the vectors in the many particle space in a simple way (that also is helpful to impose symmetry restrictions)?
Additional doubt: Is the equation above valid in a more general space of functions, or it is restricted to square-integrable functions?