 Equality in the projected Cauchy-Schwarz inequality

For an orthogonal projection $P$ onto $W$ in an inner product space, apply the Cauchy-Schwarz inequality to $Pu,Pv$ and then use $\Vert Pu\Vert \le \Vert u\Vert$, $\Vert Pv\Vert \le \Vert v\Vert$. If either original vector is zero, equality holds. Otherwise equality holds exactly when both vectors belong to $W$ and are linearly dependent. Necessity follows because equality in the product of the two norm bounds forces equality in each, while equality in Cauchy-Schwarz inequality forces dependence. This retains the zero cases that a division by the original norms would lose.