Suppose $(f_n)$ and $f$ are in $L^2$ and $\int_E f_ng \rightarrow \int_E fg,\,$ for $g \in L^2(E)$. If $\|\,f_n\|_2 \rightarrow \|\,f\|_2$, then show that $f_n \rightarrow f$ in $L^2$. (All functions are real valued).
I think that I have to add and substract something inside $\|\,f_n-f\|_2$ and use some inequality (Holder's/Triangular) and someway make the inner product $(f_n,g)$ appear but I still haven't found a way. Any tips? Thanks in advance.
Let $\langle\, f,g\rangle=\int_E fg\,d\mu$.
We have that $$ \|\,f_n-f\|^2=\langle\, f_n-f,f_n-f\rangle=\langle\, f_n,f_n\rangle-2\langle\, f_n,f\rangle+\langle\,f,f\rangle. $$ Hence, as $\langle\, f_n,f\rangle\to \langle\, f,f\rangle$, we have that if $\langle\, f_n,f_n\rangle\to\langle\, f,f\rangle$, then $\|\,f_n-f\|\to 0$.