= Equality in the projected Cauchy-Schwarz inequality
{c}
{title2=$|\langle Pu,Pv\rangle|^2\leq\|u\|^2\|v\|^2$}
For an <orthogonal projection> $P$ onto $W$ in an <inner product space>, apply the <Cauchy-Schwarz inequality> to $Pu,Pv$ and then use $\|Pu\|\leq\|u\|$, $\|Pv\|\leq\|v\|$. 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.
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