Euclidean space/Distance to subspace/Orthogonal projection/Fact

Let ${\displaystyle {}V}$ be a Euclidean vector space, ${\displaystyle {}U\subseteq V}$ a linear subspace, and ${\displaystyle {}v\in V}$.

Then ${\displaystyle {}p_{U}(v)}$ is the point on ${\displaystyle {}U}$ that has, among all points of ${\displaystyle {}U}$, the minimal distance to ${\displaystyle {}v}$.

In particular, we have

${\displaystyle {}d(v,U)=d(v,p_{U}(v))\,.}$