Having trouble with this problem. Any ideas?
Let $\Omega$ be a measure space. Let $f_n$ be a sequence in $L^p(\Omega)$ with $1 and let $f \in L^p(\Omega)$. Suppose that $f_n \rightharpoonup f \text{ weakly in } \sigma(L^p,L^{p'})$ and $\|f_n\|_p \to \|f\|_p.$ Prove that $\|f_n-f\|_p \to 0$. Also, can you come up with a counter-example for the $L^1$ case?
while it is not for $p\in\{1,\infty\}$?
– 2012-06-26