I'm a little rusty on this so I'm sorry if this has an obvious answer. My question is the following: let $\ell^p$ be the normed vector space of sequences $x = (x_n)_{n \in \mathbb{N}}$ such that $$ \|x\|_{p}\doteq\left(\sum_{n}\left|x_{n}\right|^{p}\right)^{1 / p} < \infty $$ Is it true that a sequence $(y_n)_{n \in \mathbb{N}}$ of elements of $\ell^p$ (i.e a sequence of sequences, with $y_1 = (y^1_{1}, y^1_2, \cdots), y_2 = (y^2_1, y^2_2, \cdots), \cdots$) converges to some $y = (y_1, \cdots) \in \ell^p$ if, and only if,
$$\lim_{n \to \infty}y^n_i = y_ i$$
for every $i \in \mathbb{N}$? Does either implication hold? I know this is true if the norm topology coincides with the product topology on $\mathbb{R}^{\mathbb{N}}$, but it's clear to me if that's true either. I'd appreciate any help! Thanks in advance.
The forward implication is true but not the reverse (for $p<\infty$). Let $x$ be a sequence in $\ell^p$. You can let $y^n:=x+z^n$ where $$z^n_i:=i^{-1/p}/(\log n)^\frac1{2p}\text{ for }i\leq n\text{ and }z^n_i:=0\text{ for }i>n.$$ It is easy to see that each $y^n$ is in $\ell^p$ because we have only changed finitely-many terms. However $$\|x-y^n\|_{\ell^p}=\|z^n\|_{\ell^p}=(\log n)^{-\frac1{2p}}\left(\sum_{i=1}^n i^{-1}\right)^{1/p}\sim (\log n)^{1/2p}\to\infty.$$
To see that the forward direction is true for $p\neq\infty$, we estimate $$|y_i-y_i^n|=\left(|y_i-y_i^n|^p\right)^{1/p}\leq\left(\sum_j|y_j-y_j^n|^p\right)^{1/p}=\|y-y^n\|_{\ell^p}.$$ The $p=\infty$ case is of course trivial.