Past exam of the options II course of the University of Oxford / 2008 printed variant / P / P3 / b / Solution

Use the usual weighted $L^2$ interpretation, identifying functions equal almost everywhere; on raw pointwise functions the displayed integral is only a seminorm. Assume $q\in {\Pi }_k\subset V$. If $f-q$ is orthogonal to ${\Pi }_k$, then for every $p\in {\Pi }_k$,Conversely, suppose $q$ minimizes the error. For $r\in {\Pi }_k$ and every real $t$, compare $q$ with $q+tr$:If the linear coefficient were nonzero, a sufficiently small $t$ with the appropriate sign would make the expression negative. Thus $⟨f-q,r⟩=0$ for every $r$, proving best approximation is equivalent to orthogonality of the residual.