Let (p, q in[1, infty]) be conjugate, i.e. (p^{-1}+q^{-1}=1), and assume that (left(u_{k}ight)_{k in mathbb{N}} subset mathcal{L}^{p})

Question:

Let $p, q \in[1, \infty]$ be conjugate, i.e. $p^{-1}+q^{-1}=1$, and assume that $\left(u_{k}ight)_{k \in \mathbb{N}} \subset \mathcal{L}^{p}$ and $\left(w_{k}ight)_{k \in \mathbb{N}} \subset \mathcal{L}^{q}$ are sequences with limits $u$ and $w$ in $\mathcal{L}^{p}$-sense, resp. $\mathcal{L}^{q}$-sense. Show that $u_{k} w_{k}$ converges in $\mathcal{L}^{1}$ to the function $u w$

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