A friend of mine asked me the following question (which is motivated by an image processing problem of which I am unable to say more). Let $(f\_n)\_{n\geq 0}$ be an orthonormal sequence in $L^2([0,1])$, and define $F\_n(x)=\int\_0^x f\_n(t)dt$. Is it true that $\frac{F\_n}{\Vert F\_n\Vert\_2}\to 0$ weakly in $L^2$? Thanks in advance.



A weakly null sequence?
No, this isn't true even for $f\_n(x) = \sqrt{2} \sin n\pi x$. Then $g\_n := \frac{F\_n}{\|F\_n\|} = \sqrt{\frac{2}{3}}(1-\cos n\pi x)$ and $\langle g\_n, 1\rangle = \sqrt{\frac{2}{3}} \not\to 0$.


