Let $\left\{e_1, \ldots, e_n\right\}$ be a finite orthonormal system in a inner product space $(E, \langle\cdot, \cdot\rangle)$ and let $F=\operatorname {span}\left\{e_1, \ldots, e_n\right\}$. Prove that, if $v\in F^\perp$ and $v\in F$ then $v=0$.
My attempt
We can write $\|v\|^2 = \langle v, v\rangle =0$, which holds because we can consider $v\in F$ in the left side entry, and $v\in F^{\perp}$ in the right side entry of the inner product. But $\|v\|=0$ if and only if $v=0$. What do you think? Is my procedure correct?