Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2009 / Algebra / 2 / a / Solution

For any $w\in W$, the decomposition $v-w=Tv+(Pv-w)$ is orthogonal. The Pythagorean identity givesEquality is attained at $w=Pv$. Thus $\boxed{\underset{w\in W}{\inf }\Vert v-w\Vert =\Vert Tv\Vert }$, and the minimizer in $W$ is uniquely $Pv$.