Boundedness and Strong convergence

Question

$f_n\rightarrow f$ in $L^2(0,1)$,
$\{ f,f_1,f_2,\ldots \}\subset H^1(0,1)$,
$||f_n||_{H^1(0,1)}\leq M,\ \forall n\geq 1 $,
Is is true that $f_n\rightharpoonup f$ in $H^1(0,1)$? If not, then what is a counterexample?

Answer 1

Have you seen the "double subsequence trick"?

Hint: if we do not have $f_n\rightharpoonup f$, then there is a subsequence $f_{n_k}$ converging weakly in $H^1$ to some function $g \ne f$. (Use Alaoglu.) We still have $f_{n_k} \to f$ in $L^2$. Now show that this leads to a contradiction. It will help to consider a linear functional on $H^1$ which takes different values on $f$ and $g$.