Reference request for the convergences of different modes of double series: $\sum_{i,j}x_{i,j},\sum_{i}\sum_{j}x_{i, j},\sum_{j}\sum_{i}x_{i, j}$

Question

I am looking for some reference text which goes over the convergence of different double series $\sum_{i,j}x_{i,j}$, $\sum_{i}\sum_{j}x_{i, j}$ and $\sum_{j}\sum_{i}x_{i, j}$. I have seen multiple mathexchange/blog posts which discuss partly the concept of double series and conclude (without a proof) that if the iterated double series $\sum_{j}\sum_{i}x_{i, j}$ converges absolutely, then all three modes $\sum_{i,j}x_{i,j}$, $\sum_{i}\sum_{j}x_{i, j}$ and $\sum_{j}\sum_{i}x_{i, j}$ are equal. What I am looking is a reference that I could 1.) cite at a later point when I undoubtedly forget the precise argument 2.) a reference which also discusses these sort of convergence and equality in a more general context, e.g. with topological vector spaces.