Real Vector Projections
Definition
The projection of $x \in \R ^n$ onto the subset $S \subset \R ^n$ is the point in $S$ closest to $x$.
Notation
We sometimes the projection of $x$ onto $S$ by
$\proj_S(x)$.
We have
\[
\proj_S(x) = \argmin_{z \in S}{\norm{x - z}}
\]