Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 8 / 1 / ii / Solution

Apply part (i) to the continuous difference $h=f-g$. Every Fourier coefficient of $h$ vanishes, so every Fourier partial sum is zero. The orthogonal projection convergence from part (i) therefore gives $\Vert h{\Vert }_2=0$.

If $h(t_0)\ne 0$, continuity gives an interval on which $|h|$ is bounded below by a positive number. That interval would contribute positively to $\Vert h{\Vert }_2^2$, a contradiction. HenceThe continuity hypothesis upgrades equality almost everywhere to pointwise equality.