Originally Posted by

**Jose27** I don't know if this is useful (I personally think I'm in the right direction, but there are some things missing) so I'm posting it here to see if anyone can do something with it:

Since $\displaystyle \Vert f_n \Vert \leq M$ it follows that there exists $\displaystyle (f_{n_k}) \subset (f_n)$ such that $\displaystyle f_{n_k} \rightarrow h \in L^2(\mathbb{R} )$ weakly i.e. $\displaystyle \int \ f_{n_k}g \rightarrow \int \ hg$ for all $\displaystyle g \in L^2 (\mathbb{R} )$. I'm stuck trying to prove $\displaystyle f=h$ and that in fact $\displaystyle f_n \rightarrow h$ weakly.

I know it's not much, but at least it looks like what we're trying to prove. Hope it helps, I'll keep trying to figure it out.