What does $\sup _{n \in \mathbb{N}} \sum_{j, k=0}^{n}\left|x_{j k}\right|<\infty$ mean in the context of double series?

Question

I'm reading the section Double Series in my analysis textbook.

I could not understand how this notation make sense $$\sup _{n \in \mathbb{N}} \sum_{j, k=0}^{n}\left|x_{j k}\right|<\infty$$

What does it mean that $(j,k)$ runs from $j, k=0$ to $n$?

Answer 1

It means that $(j,k)$ travels through $$(0,0),(0,1),\cdots,(0,n),(1,0),(1,1),\cdots,(1,n),\cdots,(n,0),(n,1),\cdots,(n,n).$$

Answer 2

It's saying that if you have two family of index $I_{1},I_{2}$ that give a partion of $\mathbb{N} \times \mathbb{N}$ you can always think that $I_{1} \times I_{2} = \{I_{1} \times \{j\}\}_{j \in I_{2}}$ or vice-versa

(That's the idea of raw and columuns because it represents in a certain way a matrix)

Anothery way to write if the sums on the ''columns'' and ''rows'' if the latter are convergent is :

$$\sum\limits_{(i,j) \in I_{1} \times I_{2}} a_{ij} = \sum\limits_{i \in I_{1}}\hspace{0.1cm}\sum\limits_{j\in I_{2}}a_{ij}$$

The idea behind the $sup$ is that you could alway define given $I$ set of index, as in your case $\mathbb{N}$, $$\sum\limits_{i \in I}a_{i} := sup \{\sum\limits_{j \in J}a_{j}, J \subseteq \mathcal{F}(I)\}$$

Where $\mathcal{F}(I):= \bigcup\limits_{k \in \mathbb{N}} \mathcal{P}_{k}(I) = \{J \subseteq I, \lvert J \rvert < +\infty\} \subseteq \mathcal{P}(I)$

With $\mathcal{P}(I)$ the set of all subsets of $I$.