Consider the Einstein summation notation $A^{ij}B_{ij}$. I know that this summation takes over a each pair of indices $i,j$ and there is no confusion if they are tensors and the indexing sets are finite sets.
It seems to me when the addition (the summation) is not commutative or the indexing sets are infinite set. Then in general, $$\sum_i\sum_{j} A^{ij}B_{ij} \neq \sum_{j}\sum_{i} A^{ij}B_{ij}$$ does't hold for infinite sums.
Edited: (to focus on one problem)
Is Einstein summation notation defined only for tensors and finite sums?