MHB Why is (x,e_i) a zero sequence?

[mathmari]

Hey!

An infinite orthonormal system $\{e_1, e_2, ... \} \subset H$ is closed in $H$ iff $\forall x \in H$
$$||x||^2=\sum_{i=1}^{n}{|(x,e_i)|^2}$$

From the summability of the right part of the relation above, we conclude to that the sequence $(x,e_i)$ is a zero sequence.

Could you explain me how we conlude to that?

 

[Evgeny.Makarov]

Do you mean why $(x,e_i)\to0$ as $i\to\infty$? This follows from the nth term test.

 

[mathmari]

Do you mean why $(x,e_i)\to0$ as $i\to\infty$? This follows from the nth term test.

Aha! I got it! Thank you! (Smirk)