Can someone help me, I am stuck in this easy problem
Let $I=(0,1)$, $1<p<\infty$ and $\frac{1}{p}+\frac{1}{q}=1$. If $f_n$ converges weakly to $f \in L^p(I)$ and $g_n$ converges (strongly) to $g \in L^q(I)$, then $$f_ng_n \rightharpoonup fg \text{ (converges weakly in) } L^1(I)$$
Also I am very curious with the statement "$g_n$ converges strongly to $g$" can be change with "$g_n \rightharpoonup g$"