Usually interchanging the order of summation requires $\sum_n\sum_m|a_{n,m}|<\infty$. Unfortunately, I don't have this condition on my hands. Only conditions I have are:$$\sum_n\left|\sum_m a_{n,m}\right|<\infty\quad \sum_m\left|\sum_n a_{n,m}\right|<\infty$$Would these be sufficient for this specific rearrangement of the order of summation (switching the $n$- and $m$-sum)?I tried using the dominated convergence theoreom (notation from the wikipedia): if we set $f_n(m):=\sum_{k=1}^n a_{k,m}$ we obtain our claim by applying the theorem, after proving that there exists $g$ such that $|f_n|<g$ for all $n\in\mathbb{N}$. As summing $f_n(m)$ over $m$ is possible, $|f_n(m)|$ is bounded in $m$, i.e. $f_n$ is a bounded function. I'm not sure if there is a way to proceed (or a possible counterexample).
Thank you!
