Let $H$ be a real or complex [Hilbert space](/page/Hilbert%20Space), and let $P \in \mathcal{L}(H)$. Then $P$ is an [orthogonal projection](/theorems/437) operator, meaning that
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\begin{align*}
P^2=P
\end{align*}
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and
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\begin{align*}
(Px,y)_H=(x,Py)_H
\end{align*}
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for all $x,y\in H$, if and only if there exists a closed linear subspace $M\subset H$ such that
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\begin{align*}
P=P_M,
\end{align*}
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where $P_M:H\to M$ denotes the orthogonal projection onto $M$. In that case, for this subspace $M$,