Let $f_n$ be a sequence in a $L^p$ space that converges weakly to $f$. Here $1<p<\infty$.
Also assume that $\| f_n\| _p$ converges to $\| f\| _p$.
Then how do I show that $\| f_n-f\|_p $ goes to $0$? It was easy for $p=2$ but I am stuck at arbitrary cases. Could anyone help me?
The following inequality for $f$ and $g$ in the unit ball of $L^p$ $$ \lVert f - g \rVert_{p}^{p} \leq C_{p} \left[ \lVert f \rVert_{p}^{p} + \lVert g \rVert_{p}^{p} -2 \left\lVert \frac{f + g}{2} \right\rVert_{p}^{p} \right]^{\min q(p)}, $$ where $q(p)=\min \{1, p/2 \}$, was shown here.
In our problem, we can assume that $\lVert f_n\rVert_p\leqslant 1$ for all $n$.
Applying the mentioned inequality with $g=f_n$ gives $$ \lVert f -f_n \rVert_{p}^{p/q(p)}\leq C_{p} \left(\lVert f \rVert_{p}^{p} + \lVert f_n \rVert_{p}^{p} -2 \lVert \frac{f +f_n}{2} \rVert_{p}^{p}\right). $$ Take the $\limsup_{n\to +\infty}$ and use the assumption and the fact that $h_n=f+f_n\to 2f$ weakly in $L^p$ and $$ \lVert h\rVert_p^p\leqslant \liminf_{n\to +\infty}\lVert h_n\rVert_p^p. $$
