Let $H$ be a Hilbert space over $\mathbb{C}$. Let $\phi_{j} \in H$ and let $c_j$ be scalars. Prove that there exists a representation $f=\sum_{j}c_{j}\langle f,\phi_{j}\rangle\phi_{j}, \forall f\in H$ if and only if $\|f\|^2=\sum_{j}c_{j}|\langle f,\phi_{j}\rangle|^2, \forall f\in H$
My queries: For $\implies$: $\|f\|^2= \langle f,f \rangle = \sum_{i,j} c_{i} \overline{c_{j}} \langle f,\phi_{i} \rangle \overline{\langle f,\phi_{j} \rangle}\langle \phi_{i}, \phi_{j}\rangle$. It can not be simplified further due to $\phi_{j}$ are may not be orthogonal. Even if one assumes orthogonality it would become $\|f\|^2=\sum_{j}|c_{j}|^2|\langle f,\phi_{j}\rangle|^2\|\phi_{j}\|^2$ which is still not the desired form. Is there any way to apply the fact that the representation for $f$ is valid for "all" $f$ in some way to deduce the desired equality?
For sufficiency I intend to apply the polarization identity to $\langle f,\phi_{j}\rangle$ then substitute it into the desired formula for $f$, but the polarization identity implied $\langle f,\phi_{j} \rangle =0$ which is not true in general, is there any point I have missed?