$\newcommand{\span}{\operatorname{span}}$ Define $e_{0,0}\equiv 1$, and for all $n\in \mathbb{N}$ $$e_{n,k}=\begin{cases} 2^{n/2} &\text{if } \frac{k-1}{2^n}\leq x\lt \frac{k-\frac{1}{2}}{2^n}\\ -2^{n/2}&\text{if } \frac{k-\frac{1}{2}}{2^n}\leq x\lt \frac{k}{2^n}\\ 0 &\text{otherwise} \end{cases}$$ for $k=1,\ldots,2^n$. Let $$H:=\{e_{n,k}:n,k\in \mathbb{N}\}.$$
I want to prove that $H$ is a Hilbert's base for $L^2[0,1]$ with the usal inner product. In order to prove this we must show that $H$ is orthonormal and that $\span(H)$ is dense in $L^2[0,1]$. In other links I could understand how to test the orthogonality but do not know how to try the following exercise:
Let $f\in H^{\bot}$, i.e. $f$ is such that for all $n\in \mathbb{N}$ $$\int_0^1 f(x)e_{n,k}(x)dx=0,$$ for $k=1,\ldots,2^n$. Show that for all $n\in \mathbb{N}$ $$\int_0^1f\cdot 1_{[0,k/2^{n})}=0,$$ $k=1,\ldots,2^n$. Conclude that $f\equiv 0$.
