I want to solve the following exercise, and I thankfully welcome some hints. Note that this is not homework.
Problem: Let $1 < p,q < \infty$ be conjugate exponents. Assume $f_k \rightarrow f$ in $(L^p, \| \cdot \|_p)$ and that $g_k \rightarrow g$ weakly in $(L^q, \| \cdot \|_q)$. Show that $f_k g_k \rightarrow fg$ weakly in $(L^1, \| \cdot \|_1)$.
I realize that normed convergence implies weak convergence and that $(L^q)^\ast$ can be identified with $L^p$. What I need to show is that
$$|\phi(f_k g_k) - \phi(f g)| \rightarrow 0, \ \ \forall \phi \in (L^1)^\ast$$
which follows if I can show $\| f_k g_k - f g \|_1 \rightarrow 0$, since
$$|\phi(f_k g_k) - \phi(f g)| \leq \|\phi \| \| f_k g_k - f g \|_1.$$
Thank you in advance!