Past exam of the analysis course of the University of Oxford / 2010 / B / 6 / a / ii / Solution

For $a_n=⟨x,e_n⟩$, the finite orthogonal projection $P_{N}x={\sum }_{|n|\le N}{a}_n{e}_n$ satisfiesThis is Bessel's inequality, so the coefficients are square summable. The preceding result gives a limit $s={\sum }_n{a}_n{e}_n$. For each fixed $k$, continuity of the inner product gives $⟨x-s,e_k⟩=0$. The completeness assumption on the orthonormal system forces $x-s=0$. HenceTaking squared norms also gives Parseval's identity, $\Vert x{\Vert }^2={\sum }_n|⟨x,e_n⟩{|}^2$.