Real Vector Projections

Definition

The projection of $x\in {\mathsf{R}}^n$ onto the subset $S\subset {\mathsf{R}}^n$ is the point in $S$ closest to $x$.

Notation

We sometimes the projection of $x$ onto $S$ by ${\text{proj}}_S(x)$. We have

$$\underset{S}{\text{proj}}(x)=\underset{z\in S}{\text{argmin}}\Vert x-z\Vert$$