Let $H$ a Hilbert space and $B=\{e_\alpha\}_{\alpha\in\Lambda}$ a orthonormal system in H. Prove $B$ is orthonormal maximal in $H$ iff for each $x\in H$ such that $\langle x,e_\alpha\rangle=0\quad\forall\alpha\in\Lambda$ we have that $x=0.$
My attempt:
$(\implies)$ I proved this using SOM theorem.
$(\impliedby)$ Suppose $B$ it's orthonormal maximal, then exists a set orthonormal $\bar{B}$ more larger than $B$ in other words, $B\subset\bar{B}$.
Let $x\in \bar{B}-B$ then $x$ is orthonormal.
Here I'm stuck.
Suppose that $\langle x,e_{\alpha}\rangle=0$ for all $\alpha\in\Lambda$ implies $x=0$. Assume that $B$ is strictly contained in some another orthonormal set $\bar B$. Take $x\in \bar B\setminus B$. So $\|x\|=1$ and $\langle x,e_{\alpha}\rangle=0$ for all $\alpha\in\Lambda$ by orthonormality. Then we get $x=0$ by assumption and hence $\|x\|=0$.