How to make sense of a sequence of sequences

Question

Say for each fixed $n\in \mathbb N$, $\{f_k(n)\}_{k\in\mathbb N}$ converges in some normed space.

How do we make sense of the convergence of $\{\{f_k(n)\}_{k\in \mathbb N}\}_{n\in \mathbb N}$?

(Sorry if this is a standard construction/consideration in some areas. But I don't think I have seen it in my studies. If it is, then I'll be happy to look into suggested references.)

Answer 1

Let $(X,\|\cdot\|)$ be the normed space. Then for each fixed $n\in \mathbb N$, $f(n)=\{f_k(n)\}_{k\in\mathbb N}$ is a point of $X^\Bbb N$. Then $\{f(n)\} _{n\in \mathbb N}=\{\{f_k(n)\}_{k\in \mathbb N}\}_{n\in \mathbb N}$ is a sequence of points of $X^\Bbb N$. Thus to $\{f(n)\} _{n\in \mathbb N}$ are applicaple the standard notions of convergence.

Namely, a uniform convergence

$$\exists f\in\Bbb X^\Bbb N\, \forall\varepsilon>0\, \exists M\in\Bbb N\, \forall m>M\, \forall n\in\Bbb N\, (\|f(n)-f_m(M)\|<\varepsilon),$$

and a pointwise convergence

$$\exists f\in\Bbb X^\Bbb N\, \forall n\in\Bbb N\, \forall\varepsilon>0\, \exists M\in\Bbb N\, \forall m>M\, (\|f(n)-f_m(M)\|<\varepsilon).$$