Let $1<p<\infty$. Let $(f\_{n})\_{n}$ be a sequence in the unit sphere $S\_{L\_{p}(\mu)}$ of $L\_{p}(\mu)$ such that $(f\_{n})\_{n}$ weakly converges to $f\in S\_{L\_{p}(\mu)}$. Does $(f\_{n})\_{n}$ necessarily converge to $f$ in norm? It seems that it is true. But I am not sure. Thank you!



Weak convergent sequences and norm convergent sequences in $L_{p}(\mu)$
Consider $g\_n=(f+f\_n)/2$ and let $f^\*$ be the supporting functional of $f$. Then $f^\*(g\_n)\to 1$, hence $||g\_n||\to 1$. By the uniform convexity of $L\_p$ for $1<p<\infty$ this implies $||f-f\_n||\to 0$.


