Let $H$ be a [Hilbert space](/page/Hilbert%20Space), let $V \subset H$ be a linear subspace, and let $f \in H$. An element $v^* \in V$ is the best approximation to $f$ from $V$ if and only if
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\begin{align*}
(f-v^*,v)_H=0 \quad \text{for all } v \in V.
\end{align*}