mathmari
Gold Member
MHB
- 4,984
- 7
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?
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?