Limit and Integral sign in $L^2$. If $\lim_{k \to \infty} \| u_k - u  \|_{L^2(\Bbb R^n)} = 0$ then how can I show that $$ \lim_{k \to \infty} \int_{\Bbb R^n} u_k v  = \int_{\Bbb R^n} uv$$ for any $v \in L^2 (\Bbb R^n)$?

$$ \left| \int u_k v - \int u v \right|  = \left| \int (u_k -u)v \right| \leqslant \int | u_k - u | |v| \leqslant \| u_k - u \|_2 \| v \|_2 \to 0$$
