Weak convergence of orthonormal sequences in Hilbert space

[kisengue]

So, I've found the result that orthonormal sequences in Hilbert spaces always converge weakly to zero. I've only found wikipedia's "small proof" of this statement, though I have found the statement itself in many places, textbooks and such.

I've come to understand that this property follows from the Bessel inequality, and I've worked out many of the details, so I feel that I understand the Bessel inequality itself quite well. What I don't get is how the inequality gives us the weak convergence - the proof on wikipedia only states that "Therefore, $$|\langle e_n, x \rangle |^2 \rightarrow 0$$" after stating the Bessel inequality. It doesn't make sense to me - how is this information gleaned?

 

[Fredrik]

Can you at least link to the Wikipedia page? Are you asking how to see that $|\langle e_n, x \rangle |^2 \rightarrow 0$, or how to see that this means that $e_n\to 0$ with respect to the weak topology?

 

[micromass]

You do know that if the series

$$\sum_{n=0}^{+\infty}{a_n}$$

converges, that $a_n\rightarrow 0$??

 

[kisengue]

The proof (sorry for not linking it immediately). Fredrik, I'm asking the first of those two - the second I understand.

Micromass: I didn't think of that... but of course. Of course. Damn it. Now I get it, I think.