If ${x_n}$ is weakly convergent to $x$ in $L^p(0,1)$ space, and if $\lim_{n}\|x_n\|=\|x\|$, then if $\lim_{n}x_n=x$?

Question

If ${x_n}$ is weakly convergent to $x$ in $L^p(0,1)$ space, and if $\lim_{n}\|x_n\|=\|x\|$, then if $\lim_{n}x_n=x$?

This question is from problem 6.1.4 in Avner Friedman's "Foundations of Modern Analysis". The answer seems to be no since in $L_p(0,1)$ the parallelogram law is not fullfilled. But I cannot give a proof of this question and find a counter example.