Let $X$ a Banach space and $x_{ij} : \mathbb{N} \times \mathbb{N} \to X$. Suppose $\sum_i \sum_j \|x_{ij}\| < \infty$. Then it is easy to show that the following series are convergent: $$\sum_j x_{ij}, \sum_i x_{ij}, \sum_i \sum_j x_{ij}, \sum_j \sum_i x_{ij}.$$ Then I want to know if: $$\sum_i \sum_j x_{ij} = \sum_j \sum_i x_{ij}.$$
This would be a direct corollary of a Fubini-Tonelli type theorem for Bochner integrals, but I can't find a reference on that. So if someone could point me to a reference to such a theorem for Bochner integrals, that'd be ideal (or let me know if/why such a theorem doesn't exist). But also a direct proof of the above would work.
Let $f \in X^*$ arbitrary. Then: $$f(\sum_i \sum_j x_{ij}) = \sum_i \sum_j f(x_{ij}).$$ Then $|f(x_{ij})| \leq \|f\| \|x_{ij}\|$, so: $$\sum_i \sum_j |f(x_{ij})| \leq \|f\|\sum_i \sum_j \|x_{ij}\| < \infty.$$ Thus we can apply the standard result on reordering series (i.e. the corollary of Fubini-Tonelli), giving: $$f(\sum_i \sum_j x_{ij}) = \sum_i \sum_j f(x_{ij}) = \sum_j \sum_i f(x_{ij}) = f(\sum_j \sum_i x_{ij}).$$ Then since $f \in X^*$ was arbitrary, we have: $$\sum_i \sum_j x_{ij} = \sum_j \sum_i x_{ij},$$ as desired.