Let $H$ be a real [Hilbert space](/page/Hilbert%20Space), let $n \in \mathbb{N}$, and let $\varphi_1,\dots,\varphi_n \in H$. Define the finite-dimensional subspace
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\begin{align*}
V := \operatorname{span}\{\varphi_1,\dots,\varphi_n\} \subset H.
\end{align*}
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Let $x \in H$, and suppose that $p \in V$ is a best approximation to $x$ from $V$, meaning
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\begin{align*}
\|x-p\|_H \leq \|x-v\|_H \quad \text{for every } v \in V.
\end{align*}
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If
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\begin{align*}
p = \sum_{j=1}^n a_j \varphi_j
\end{align*}
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for real coefficients $a_1,\dots,a_n \in \mathbb{R}$, then the coefficients satisfy the normal equations