Limit and Integral sign in $L^2$.

Question

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 $v \in L^2 (\Bbb R^n)$?

Answer 1

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