Prove that this sequence is complete. Suppose $\{\varphi_n\}$ is an orthonormal sequence in $L^2[a,b]$ that satisfies $$\sum_{n=1}^\infty \left|\int_a^x \varphi_n(t)\,dt\right|^2=x-a,$$ for all $x \in [a,b]$. Conclude that $\{\varphi_n\}$ is complete in $L^2[a,b]$. Now, my initial thought is showing that Plancherel's equality will hold. My assumption it already holds for these type of indicator functions $f(x)=\chi_{[a,x]}$. My idea is to use the density of these functions in $L^2[a,b]$ to somehow extend it to conclude that Plancherel's equality holds for all functions in $L^2[a,b]$ and this would finish the proof. However, I am unsure if this is the right way to do this.
Any help is appreciated.
Krull.

The problem with your argument is functions of the form $\chi_{[a,x]}$ are not dense in $L^{2}$. Linear combinations of these functions are dense but it is difficult to extend the given equaiton to linear combinations.
Here are some hints: If we show that $\chi_{[a,x]}=\sum \langle {\chi_{[a,x]}}, \phi_n\rangle \phi_n$ in $L^{2}$ sense the we can conclude that $\chi_{[a,x]}$ belongs to te closed subspace spanned by $(\phi_n)$. Now we can take linear combinations and limits to show that $L^{2}$ is the closed linear span of $(\phi_n)$, completing the proof.
To show that $\chi_{[a,x]}=\sum \langle {\chi_{[a,x]}}, \phi_n\rangle \phi_n$ simply expand $\|\chi_{[a,x]}-\sum\limits_{k=1}^{n} \langle {\chi_{[a,x]}}, \phi_k\rangle \phi_k\|^{2}$ and use orthogonality as well as the given hypothesis.
[$\|\chi_{[a,x]}-\sum\limits_{k=1}^{n} \langle {\chi_{[a,x]}}, \phi_k\rangle \phi_k\|^{2}=\|\chi_{[a,x]}\|^{2}-2 \langle {\chi_{[a,x]}}, \sum\limits_{k=1}^{n} \langle {\chi_{[a,x]}} \rangle \phi_k+\|\sum\limits_{k=1}^{n} {\chi_{[a,x]}}\phi_k\|^{2}$. Using orthonormality this reduces to $(x-a)-2\sum\limits_{k=1}^{n}|\int_a^{x}\phi_k(t)dt|^{2}+\sum\limits_{k=1}^{n}|\int_a^{x}\phi_k(t)dt|^{2}$ and this tends to $(x-a)-2(x-a)+(x-a)=0$ by hypothesis].
