This is probably something quite trivial I'm not getting. I'm studying CFT (conformal field theory) through David Tong's lecture notes and on page 9 he says:
We now define the operator product expansion (OPE). It is a statement about what happens as local operators approach each other. The idea is that two local operators inserted at nearby points can be closely approximated by a string of operators at one of these points. Let's denote all the local operators of the CFT by $\mathcal{O}_i$, where $i$ runs over the set of all operators. Then the OPE is $$\mathcal{O}_i(z,\bar{z})\mathcal{O}_j(w,\bar{w})=\sum_k C_{ij}^k (z-w,\bar{z}-\bar{w})\mathcal{O}_k(w,\bar{w})\tag{4.10}$$ Here $C_{ij}^k(z-w,\bar{z}-\bar{w})$ are a set of functions which, on grounds of translational invariance, depend only on the separation bewtween the two operators. We will write a lot of operator equations of the form (4.10) and it's important to clarify exactly what they mean: they are always to be understood as statements which hold as operator insertions inside time-ordered correlation functions, $$\langle \mathcal{O}_i(z,\bar{z})\mathcal{O}_j(w,\bar{w})\cdots\rangle = \sum_k C_{ij}^k(z-w,\bar{z}-\bar{w})\langle \mathcal{O}_k(w,\bar{w})\cdots\rangle$$ where the $\cdots$ can be any other operators we choose.
My question is: why rigorously would we expect such an expansion to even exist? I mean he says that such an expansion can be written down, but what is the proof of it? Furthermore, how is the expansion to be understood?
It is a weird expansion, a sum over all operators, which I don't even know how he is parameterizing here.
So being more rigorous how can the OPE be understood and why should it exist?