I would like to know how to show that the functions $r_n(t)=\operatorname{sgn}\big(\sin(2^n \pi t)\big)$ (where $\operatorname{sgn}$ is the sign function) form an orthonormal system but not an orthonormal basis from $L_2([0,1])$.
Progress: I pick two different variables $i,k$ and show that $\langle r_i (t),r_k (t)\rangle =0$, so I can take the integral $\int_0^1 r_i (t) r_k (t)\,dt$, but I do not know how to show that this is 0.