I was reading this maths stack exchange post about the difference between closed and continuous linear operators.
The example given there is that the operator defined on $l^2$ by $T(x):=(x_1,2x_2,...,nx_n,...)$ is closed but not continuous.
For this operator I would like to show that
(a) It is closed.
(b) That it is not continuous.
I'd like help in showing (a) and (b).
I noticed however that in the linked post it was not declared the domain $D(T)\subseteq l^2$ of $T$.
I took a guess and assumed that the most natural choice of domain for $T$ was one which it's image made sense in $l^2$;
$$D(T)= \{ x \in l^2 : ||T(x)||^2=\sum_{n=1}^\infty |n x_n|^2 < \infty \}$$
However I am not sure if this is was the intended domain or not..
Here is my attempt.
(a)
To show closed lets assume $x^k \to x \in l^2$ and $Tx^k \to y \in l^2$ for $(x^k)_k \subset D(T)$ a squence. For (a) it is enough to show that (i) $x \in D(T)$ and (ii) $Tx=y$.
Well for (i) I just show that $\sum_{n=1}^\infty |n x_n|^2$ is finite, by a variant of Minkowski's inequality we have
\begin{align*} \left(\sum_{n=1}^N |n x_n|^2\right)^\frac{1}{2} &\leq \left(\sum_{n=1}^N |n x_n-nx_n^k|^2\right)^\frac{1}{2} + \left(\sum_{n=1}^N |nx_n^k|^2\right)^\frac{1}{2} \newline &\leq N\left(\sum_{n=1}^N |x_n-x_n^k|^2\right)^\frac{1}{2} + \left(\sum_{n=1}^\infty|nx_n^k|^2\right)^\frac{1}{2} \newline &\leq N||x-x^k||+||Tx^k|| \end{align*} Where the last line follows since each $x^k$ is in $D(T)$. Now since $x^k \to x$ in $l^2$ we may make $||x-x^k||<N^{-1}$ for suitably large $k$. Hence taking the limit $k \to \infty$ gives
$$ \left(\sum_{n=1}^N |n x_n|^2\right)^\frac{1}{2} \leq 1+ ||y||$$ for all $N\in \mathbb{N}$.
Hence the sequence of partial sums is bounded above and strictly increasing and hence convergent. So $x \in D(T)$ this is (i).
For (ii) I am much less sure, I would like to show $||x-Tx_n|| \to 0$ but this seems very close to showing that $T$ is continuous which I know it shouldn't be as mentioned in the linked stack exchange post.
(b)
To show $T$ is not continuous I am also struggling.
Any help would be appreciated.