$\langle u, v\rangle=\langle q,v \rangle$ where $u,q,v$ are elements of the inner product space, not fixed. Does this imply $q=u$?
Attempt: $\langle 0,v \rangle = 0$ therefore $\langle u,v \rangle = \langle q,v\rangle$ iff. $u=q$ since $\langle u-q,v \rangle =0$ implies $u-q=0$
why my idea fails to complete the proof:
$u-q$ can just be an orthogonal vector to $v$