Limit of double sequence

Question

Let $X$ be a Banach space and $\left\lbrace u^k_n\right \rbrace_{n,k\geq 1}\subset X$. Is it true that $u^k_n\rightarrow u$ for $n \rightarrow \infty$ implies $\lim_{n,k\to\infty}u^k_n= u$?

Answer 1

Consider a sequence in $\mathbb{R}$ given by $$u_n^k = \frac{k}{n+k}$$

Then for every $k \in \mathbb{N}$ we have $u_n^k \xrightarrow{n\to\infty} 0$ but $u_k^k = \frac12$ so the double limit cannot be $0$.

Answer 2

No, that's not even true for $X=\Bbb R$, just think of

$$ u_n^k=\begin{cases} 1, & n=k, \\ 0, & \text{else}.\end{cases}$$

This double sequence fulfills

$$ u_n^k \xrightarrow{n\to\infty} 0 \quad \text{for all $k\in\Bbb N$}$$

and

$$ u_n^k \xrightarrow{k\to\infty} 0 \quad \text{for all $n\in\Bbb N$},$$

but since $u_k^k = 1$ for all $k\in\Bbb N$, it cannot converge to 0.