Weak convergence and $\|f_{n}\|\rightarrow\|f\|$ implies strong convergence

Question

Let $\left\{ f_{n}\right\} _{n=1}^{\infty}$ be a sequence in $L^{p}\left([0,1]\right)$ for $p\geq1$. Suppose that there exists $f\in L^{p}\left([0,1]\right)$ satisfying $\lim_{n\rightarrow\infty}\int_{0}^{1}f_{n}\left(x\right)g(x)\ dx=\int_{0}^{1}f(x)g(x)\ dx$ for any $g\in L^{2}\left([0,1]\right)$. Prove that if $\lim_{n\rightarrow\infty}\|f_{n}\|_{p}=\|f\|_{p}$, then $\lim_{n\rightarrow\infty}\|f_{n}-f\|_{p}=0$.

I don't know how to approach to this question. Weak convergence does NOT imply the subsequence convergence. And I expanded $|f_{n}-f|^{p}$ but nothing I can obtain. Can anyone help me?

Answer 1

Partial answer, for $p=2$.

$$ \|\,f_n-f\|^2=(\,f_n-f,f_n-f)=(\,f_n,f_n-f)-(\,f,f_n-f) $$ and clearly $(\,f,f_n-f)\to 0$. Also $$ (\,f_n,f_n-f)=(\,f_n,f_n)-(f_n,f)\to \|\,f\|^2-\|\,f\|^2=0, $$ and finally, $\|\,f_n-f\|\to 0$.