Exercise 24, § 5

Let $p, q$ be two conjugate exponents. If $B$ (resp. $C$) is a bounded subset of $L^p$ (resp. $L^q$), show that the mapping of $B \times C$ into $L^1$, deduced from $(f, g) \mapsto fg$ by passage to quotients, is not necessarily continuous when $L^p, L^q$ and $L^1$ are equipped with the topologies $\sigma(L^p, L^q)$, $\sigma(L^q, L^p)$ and $\sigma(L^1, L^\infty)$, respectively (cf. Exer 10).