I'm looking for a counterexample in a Banach space. I've seen the counterexample at Sum of Closed Operators Closable?, but I don't understand why $A$ and $B$ are closed. Could someone expand on this or provide a simpler counterexample?
EDIT: Here's what I'm thinking: If $u_n\to u$ (i.e. $(u_{n,1},\ldots,u_{n,k},\ldots)\to(u_1\ldots,u_k,\ldots)$)
$\|Au\|^2=|\textstyle\sum_{k=1}^\infty ku_k|^2+\displaystyle\sum_{k=2}^\infty k^4|u_k|^2$
$\qquad\quad\leq|\textstyle\sum_{k=1}^\infty ku_k|^2+\displaystyle\sum_{k=2}^\infty k^4(|u_k-u_{n,k}|+|u_{n,k}|)^2$
$\qquad\quad=|\textstyle\sum_{k=1}^\infty ku_k|^2+\displaystyle\sum_{k=2}^\infty k^4|u_k-u_{n,k}|^2+\sum_{k=2}^\infty k^4|u_{n,k}|^2+2\sum_{k=2}^\infty k^4|u_k-u_{n,k}||u_{n,k}|$
The middle (large) sum in the bottom expression converges by hypothesis, and the third will converge if the first two (large) sums do (by C-S). I don't know why the first (large) sum converges though. This is why I was asking about uniform convergence. Or have I already used too crude an inequality? Furthermore, how do we get the first term (small sum) to converge?
(When I say large or small sum, I'm talking about the size of the summation symbol, just to distinguish them.)