I have this problem: let $f_n$ converge weakly to $f$ in $L^2[0,1]$ and let $F_n(x)=\int_0^xf_n(t) \, \textrm{d}t,$ $F(x)=\int_0^xf(t) \, \textrm{d}t.$ Then $F_n,F$ are continuous and $F_n$ converges uniformly to $F$.
Writing $F_n(x)=\int_0^1 f_n(t) \mathbb{1}_{[0,x]} \, \textrm{d}t$ and applying the Lebesgue dominated convergence theorem, the continuity of $F_n$ should be proved and analogously of $F$. But I don't know about the uniform convergence and how to use the weak convergence hypothesis..