Product of weakly converging sequences

Question

Suppose $f_n, g_n \in L^p([0,1])$ and that $$ f_n \to f \quad \text{weakly in } L^p$$ $$ g_n \to g \quad \text{weakly in } L^p$$ $$ f_ng_n \to h \quad \text{weakly in } L^p $$ for all $p \in [1,\infty]$. Is it true that $h = fg$?

Answer 1

No. Consider for example $f_n(x)=\sin(nx)$. Then $f_n$ converges weakly in $L^p[0,2\pi]$ to $0$ but $f_n\cdot f_n=\sin^2(nx)=\frac{1}{2}(1-\cos (2nx))$ converges weakly to $\frac{1}{2}\neq 0\cdot 0$.