For any inner-product space, $|\langle u,v\rangle|^2\le\langle u,u\rangle\langle v,v\rangle$. Equality holds iff $u,v$ are linearly dependent. For nonzero $u$, expand the nonnegative squared norm of $v-u\langle u,v\rangle/\|u\|^2$ to prove it. This works over both real and complex scalars with the stated convention.
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