Past exam of the analysis course of the University of Oxford / 2011 / 5 / a / Solution

The orthogonal complement isIt is a linear subspace and is closed, since it is an intersection of kernels of the continuous linear functionals $w\mapsto ⟨w,y⟩$. Continuity follows from the Cauchy-Schwarz inequality.

The minimizing point and orthogonality proved below write every $x$ as $x=y_0+(x-y_0)\in Y+Y^{\perp }$. If $w\in Y\cap Y^{\perp }$, then $\Vert w{\Vert }^2=⟨w,w⟩=0$, so the intersection is zero. ConsequentlyIn particular the minimizing point is unique: an alternative orthogonal decomposition would have a difference in this zero intersection.