Let $O={u_1,...,u_k}$ be an orthonormal set in $V$. Prove that $O$ is an orthonormal basis if and only if Parseval's identity holds for all $v,w \in V$ i.e if and only if
$$\langle v,w\rangle=\sum_{i=1}^{k} \langle v,u_i\rangle\bar{\langle w,u_i\rangle}$$
(where $\bar{\langle w,u_i\rangle}$ is the conjugate of $\langle w,u_i\rangle$).
I've shown that if $O$ is an orthonormal basis then the identity holds but I'm having trouble with the converse. I thought maybe an argument by contradiction (assume $O$ does not span $S$ and show that the identity does not hold then assume $O$ is not linearly independent and find a contradiction) might work but I'm not sure where to start. Any thoughts?