I know how to show that coordinate convergence implies norm convergence in finite euclidian space.
Also I know how how to show that the same thing does not work in infinite space.
For example:
Consider $\ell^2:=\{(x_1,x_2,\dots):\sum_{n\geq 0}x_n^2<\infty\}$ with norm $\Vert(x_1,x_2,\dots)\Vert=\sqrt{\sum_{n\geq 0}x_n^2}$. Now consider two sequences $(x_n)$ and $(y_n^{(k)})$ where $x_n=0$ for all $n$ and $y_n^{(k)}=\mathbb{1}[n=k]$. Notice that we have $(x_n),(y_n^{(k)})\in\ell^2$ for all $k$. Furthermore, $y_n^{(k)}\to x_n$ as $k\to\infty$ since $y_n^{(k)}=0$ whenever k>n. However, for every k, $\Vert (x_n)-(y_n^{(k)})\Vert=1$, so $(y_n^{(k)})$ does not converge to $(x_n)$ in the norm.
Question 1. But I dont understand it. Can someone show what would go wrong in this example if we replace infinite space with finite space. Might be then I could see difference.
Question 2. Does coordinate convergence and so called point-wise convergence the same things?