Let $\mu$ be the Lebesgue measure on $[0,1]$ on the borelians, and consider the Volterra operator $V:L^1[0,1]\to C[0,1]$ given by
$$
Vf(t)=\int\_0^t f d \mu
$$
So, I want to show the following

> 
> **Proposition:** The Volterra operator sends weak compact sets into norm compact sets. 
> 
> 
> 

I've already proved that this operator sends uniformly integrable sets into uniformly bounded and equicontinuous sets of $C[0,1]$, i.e, relatively compact sets in $C[0,1]$. I know that uniformly integrable sets are the same that relatively $\omega$-compact sets. I don't get how to finish the problem with this.

Any help would be welcome. Thanks in advanced.