Need help considering series like these: $\sum_{n=1}^\infty\langle x,e_n\rangle e_n$

Question

I'm working in a Hilbert space $H$ with ONB $(e_n)$ and I have $\alpha=(\alpha_n)\in\ell^\infty$. I have an operator that looks like this: $$T_\alpha x=\sum_{n=1}^{\infty}\alpha_n\langle x;{e_n}\rangle e_n.$$ Now I'm supposed to show numerous thing about $T_\alpha$, but I got stuck on some very basic questions. For instance: "Show that $T_\alpha \in B(H)$." I tried this:

$$\Vert T_\alpha x\Vert=\Vert\sum\alpha_n\langle x;e_n\rangle e_n\Vert\leq\sum\Vert\alpha_n\langle x;e_n\rangle e_n\Vert\leq\Vert\alpha\Vert\underline{\sum\Vert\langle x;e_n\rangle e_n\Vert}$$

Now the underlined term is where I get stuck. I know that $x=\sum\langle x;e_n\rangle e_n$, but this only gives me $\Vert\sum\langle x;e_n\rangle e_n\Vert\leq\sum\Vert\langle x;e_n\rangle e_n\Vert$, which is not the kind of inequality that I'm after.

I feel like I am missing some basic properties of the series $\sum\langle x;e_n\rangle e_n$. (I feel like they're basic, because this is only question a). Could anyone give me some tips?

Answer 1

Suggestion: Work with squares. Show that if $a$ and $b$ are orthogonal, then $\|a+b\|^2=\|a\|^2+\|b\|^2$. Extend this to series of pairwise orthogonal vectors like $\langle x, e_n\rangle e_n$.

$$\|a_1 e_1+a_2e_2+\cdots\|^2 = \langle a_1 e_1+a_2 e_2+\cdots,a_1e_1+a_2e_2+\cdots\rangle=|a_1|^2+|a_2|^2+\cdots.$$