Suppose that we have a collection of (non-negative) real numbers $\{a_{k,l}:k,l\in\mathbb{N}\}$, then what does the series/sum/limit $\sum_{k,l\geq1}a_{k,l}$ actually mean? For example:
- Does it mean that the sequence $(\sum_{l=1}^{L}a_{k,l})_{L\geq1}$ converges (with limit $:=\sum_{l=1}^{\infty}a_{k,l}$) for every $k\geq1$ and $(\sum_{k=1}^{K}\sum_{l=1}^{\infty}a_{k,l})_{K\geq1}$ converges?
- Does it mean that the sequence $(\sum_{l=1}^{L}a_{k,l})_{K\geq1}$ converge for every $l\geq1$ and $(\sum_{l=1}^{L}\sum_{k=1}^{\infty}a_{k,l})_{L\geq1}$ converges?
- Does it mean that the sequence $(\sum_{k=1}^{N}\sum_{l=1}^{N}a_{k,l})_{N\geq1}$ converges?
- Let $\mathscr{F}$ be the collection of all finite subsets of $\mathbb{N}^{2}$. Then inclusion $\subset$ defines an order relation on $\mathscr{F}$. Does it mean that the net $(\sum_{(k,l)\in F}a_{k,l})_{F\in\mathscr{F}}$ converges?
- Etc.
It really confuses me a lot. Any help would be greatly appreciated! Thanks in advance.
If it converges, then all of the definitions above should be equivalent. This sum can be thought of as a double sum such that $$\sum_{k=1}^\infty\sum_{l=1}^\infty a_{k,l}=\sum_{l=1}^\infty\sum_{k=1}^\infty a_{k,l}$$ which I would normally interpret as falling under one of the first two definitions you give. You could also order $\mathbb{N}^2$ to make this a double sum.
But I think the most common definition would be that it is defined as $$lim_{||(M,N)||\to\infty}\sum_{k=1}^M\sum_{l=1}^N a_{k,l}$$ for some choice of "norm" on $\mathbb{N}^2$, e.g. $||(M,N)||=M+N$.