So this question isn't formulated as well as I would like, but here goes.
In solving PDEs, we often apply a technique of separation of variables to simplify the PDE into multiple eigenvalue ODEs and then add them together to get a general solution. For example, the Schrodinger equation, i.e., we compute the general solution $\Psi(t,x)$ of $i \partial_t \Psi = H\Psi$ by setting $\Psi(t,x) = \phi(t) \psi(x)$ and solving $i\partial_t \phi = E\phi$ and $H\psi =E\psi$ separately. Then the general solution is then of the form $\Psi(t,x) =\sum_n \psi_n(x) e^{-iE_n t}$ where $n$ goes over all eigenvalues $E_n$
The reason we are able to do this, in the sense that a function of $t,x$ is the infinite sum of the PRODUCT of functions in $t$ and $x$, seems to be related to the fact that the tensor product $L^2(\mathbb{R})\otimes L^2(\mathbb{R})$ is isomorphic to $L^2(\mathbb{R^2})$. However, there are a lot of subtleties, e.g., differentiation is densely defined and not bounded, that makes it difficult for me to understand rigorously why separation of variables is true. Are there any good references that talk about this?
