How can I argue that $L\{(x_1,x_2,...,x_n,...)\} = \{x_1, x_2/2, x_3/3 ... x_n/n ...\}$, $L: \ell_2 \rightarrow \ell_2$ is bounded operator?
I think that it's intuitively since $\ell_2$ is sequences s.t.
$$\sum_{i=1}^{\infty} |x_i|^2 < \infty$$
Then
$$\|x\|=\sqrt{\sum_{i=1}^{\infty} |x_i|^2} < \infty$$ intuitively, but the operator boundedness requires showing $M> 0$ s.t.
$$\| Lx\|_Y \leq M \|x\|_X$$
and I don't know how such $M$ could be picked.
Particularly, how can I know that
$$\sqrt{\sum_{i=1}^{\infty} |x_i/i|^2} \leq \sqrt{\sum_{i=1}^{\infty} |x_i|^2}$$