\begin{definition}[Definition:Orthogonal Projection]
Let $H$ be a Hilbert space.
Let $K$ be a closed linear subspace of $H$.
Then the '''orthogonal projection''' on $K$ is the map $P_K: H \to H$ defined by
:$k = \map {P_K} h \iff k \in K$ and $\map d {h, k} = \map d {h, K}$
where the latter $d$ signifies distance to a set.
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That $P_K$ is well-defined follows from Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space.
The name orthogonal projection stems from the fact that $\paren {h - \map {P_K} h} \perp K$.
This and other properties of $P_K$ are collected in Properties of Orthogonal Projection.
\end{definition}