We introduce $f ∈ V$ (where $V = C([0,1],\mathbb R)$), and we put $a_n = <f, e_n>$ (where $e_n$ is an orthonormal sequence in $V$).
Assuming $g(x) = \sum_{n=0}^∞ a_n e_n(x)$ is continuous, we have to show that $g = f$. The sum here is with respect to the norm $|| · ||$ generated by $<·,·>$ and hence $\lim\limits_{N \to ∞}\||g(x) - \sum_{n=0}^N a_n e_n(x)|| = 0$
I am not sure how to approach the problem, from what I think is I have to show that the sum converges to $f ∈ V$ in the complete inner product space $V$, and thus $f = g$.