Past exam of the options I course of the University of Oxford / 2012 / P / P1 / b

Take any $q\in {\Pi }_n$. Write $f-q=(f-p_n)+(p_n-q)$. The second term belongs to ${\Pi }_n$, so the assumed orthogonality gives $⟨f-p_n,p_n-q⟩=0$. Expanding the squared norm yields the Pythagorean theorem in this inner product:Taking square roots provesEquality can hold only when $\Vert p_n-q\Vert =0$, hence $p_n=q$, so the best polynomial is unique. Completeness of the ambient space is unnecessary for this orthogonality argument.