How to solve the following problem:
Let $f_n, f \in L^2(\mathbb{R}^d)$ for all $n \geq 1$ be such that $\|f_n\|_2 \to \|f\|_2$ as $n \to \infty$. Suppose, moreover, that $$\int f_{n}g \to \int fg $$ for all $g \in L^2(\mathbb{R}^d)$. Then $f_n$ converges to $f$ in $L^2$-norm.