General knowledge regarding double convergence of sequences (and series).

Question

While proving a subspace of $\textit{L}^p(\mathbb{R}^n)$ is dense I stumbled upon a family of functions that was double indexed; this doesnt fall in our usual definition of sequence, it is merely a set of functions $\{f_{i,j}\}_{i,j\in \mathbb{N}}$. The study of a collection of this type wasn't necessary for my objective, but i found a few papers online regarding double convergence and how it is defined.

As a first thing i would like to know if anybody knows about this part of analysis, becouse although it sound interesting, it's probably not all that popular.

To give some context to people who don't know what this is, ill give a first definition of convergence:

Given a measure space $(X,\textit{A},\mu)$ and given $E\in \textit{A}$ with $|E|<+\infty$ ($\star$ is this limitation necessary?$\star$), then we say that $\{f_{n,m}\}$ converges in the Pringsheim’s sense if for every $\epsilon > 0$ there exists $N∈\mathbb{N}$ such that $|f_{m,n}(x_0) −f|< \epsilon$ whenever $j$, $k ≥N$.

And all the other kind of convergences can be defined as well (a.e., in measure,...)

I now ask, what properties differenciate them from the normal sequences and convergence?

By just thinking of examples i am pretty sure a major difference is that if a double sequence converges, it still can have subsequences converging elswhere, for example:

$$f_{n,m}(x)=\frac{1}{n}+\frac{1}{m} \Longrightarrow \lim_{m\to +\infty}f_{n,m}=\frac{1}{n} \mbox{ and } \lim_{n\to +\infty}f_{n,m}=\frac{1}{m}$$ so when we fix one of the two between $n$ and $m$ we obtain a normal converging sequence: $$\forall \epsilon > 0\mbox{, } \exists N_1\in \mathbb{N} \mbox{ such that } \forall n\ge N_1 \Rightarrow |f_{n,m}-\frac{1}{m}|\le \frac{\epsilon}{2}$$ $$\forall \epsilon > 0\mbox{, } \exists N_2\in \mathbb{N} \mbox{ such that } \forall m\ge N_2 \Rightarrow \frac{1}{m}\le \frac{\epsilon}{2}$$ so that $$\forall \epsilon > 0\mbox{, } \exists N:=\max\{N_1,N_2\}\in \mathbb{N} \mbox{ such that } \forall n,m\ge N \Rightarrow |f_{n,m}-0|=|f_{n,m}-\frac{1}{m}+\frac{1}{m}-0| \le$$ $$\le|f_{n,m}-\frac{1}{m}|+\frac{1}{m}<\epsilon$$ We have obtained the "double convergence of the double sequence", but if i get the subsequence $\{f_{1,m}\}_{m\in \mathbb{N}}$, this converges in the normal definition to $1$; actually you could find infinite sunsequences, $\forall n\in \mathbb{N}$, $f_{n,m}\longrightarrow \frac{1}{n}$, for $m\rightarrow +\infty$.

As you might have seen, i am just looking for insight on what to think of this topic, maybe also knowing what interesting results one can reach.

Also I'm sorry if my english isn't that good.

Answer 1

Focusing on double sequences $(x_{mn})$ in $\mathbb{R}$ (or $\mathbb{R}^p$ or $\mathbb{C}$) many results are common to ordinary sequences, e.g., the Cauchy criterion, limits of sums and products of sequences , etc.

Perhaps the first unique and important question that is raised is the existence and equality of the double and iterated limits. That is, when do we have

$$\lim_{m,n \to \infty}x_{mn} = \lim_{m \to \infty}\lim_{n \to \infty}x_{mn} = \lim_{n \to \infty}\lim_{m \to \infty}x_{mn},$$

with the limit on the LHS defined in the Pringsheim sense. As with integrals and series, the question of when the order of limiting operations can be switched arises frequently in applications.

With double sequences we can see a variety of possibilities:

(1) $\,\,x_{mn} = (-1)^{m+n} \left(\frac{1}{m} + \frac{1}{n} \right)$ where $\underset{m,n \to \infty} \lim x_{mn} = 0$ but the iterated limits do not exist.

(2) $\,\,x_{mn} = \begin{cases}1, & m \neq n \\ 0, & m=n \end{cases}$ where the double limit does not exist but both iterated limits exist and are equal with $\underset{m \to \infty}\lim \underset{n \to \infty}\lim x_{mn} = \underset{n \to \infty}\lim \underset{m \to \infty}\lim x_{mn}=1.$

(3) $\,\,x_{mn} = \frac{m-n}{m+n}$ where the iterated limits exist but are unequal with $\underset{m \to \infty}\lim \underset{n \to \infty}\lim x_{mn} =-1$ and $\underset{n \to \infty}\lim \underset{m \to \infty}\lim x_{mn} = 1$.

Does the double limit exist in case (3)? The answer is no and a consequence of the following "fundamental" theorem of double sequences.

If the double limit $\underset{m,n \to \infty}\lim x_{mn} =x$ exists, and for each $m \in \mathbb{N}$ the limit $ \underset{n \to \infty} \lim x_{mn} = y_m$ exists, then the iterated limit exists and $\underset{m,n \to \infty} \lim x_{mn} = \underset{m \to \infty}\lim y_m = x$.

Now we can see that if both collections $\{y_m= \underset{n \to \infty} \lim x_{mn}\,|\, m \in \mathbb{N}\}$ and $\{z_n= \underset{m \to \infty} \lim x_{mn}\,|\, n \in \mathbb{N}\}$ exist along with the double limit, then we must have

$$\lim_{m,n \to \infty}x_{mn} = \lim_{m \to \infty}\lim_{n \to \infty}x_{mn} = \lim_{n \to \infty}\lim_{m \to \infty}x_{mn},$$

and if

$$\lim_{m \to \infty}\lim_{n \to \infty}x_{mn} \neq \lim_{n \to \infty}\lim_{m \to \infty}x_{mn},$$

then the double limit cannot exist.