Let $(f_n)$ be functions in $L^p(\Omega), 1
weakly in $L^p(\Omega)$ without first showing that $f_n \rightarrow f$ strongly in $L^p(\Omega)$? (I know that under these hypotheses we in fact get that $f_n \rightarrow f$ strongly, as explained here
If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$ ,
but there must be an easier, direct argument that $f_n \rightarrow f$ weakly.)
I know that some subsequence $(f_{n_k})$ converges weakly to some $g$ in $L^p(\Omega)$ since the $f_n's$ are bounded, but then (1) how do we pass from the subsequence to the original sequence, and (2) how do we show that $g=f$?
Thanks!