Euclidean space/Distance to subspace/Inner product/Fact/Proof

Proof

Because of fact, we have

$d(v,U)=d(v,p_U(v)),$

and, according to fact, we have

$p_U(v)=\sum _{i=1}^m⟨v,u_i⟩u_i.$

The vectors $p_U(v)$ and $v-p_U(v)$ are orthogonal to each other. Therefore, using the Pythagorean theorem, we have

$\begin{array}{cc}d(v,U{)}^2& =\Vert v-p_U(v){\Vert }^2\\ & =\Vert v{\Vert }^2-\Vert p_U(v){\Vert }^2\\ & =\Vert v{\Vert }^2-\Vert \sum _{i=1}^m⟨v,u_i⟩u_i{\Vert }^2\\ & =\Vert v{\Vert }^2-\sum _{i=1}^m{⟨v,u_i⟩}^2.\end{array}$

To fact