Convergence of linear functionals

Let $(\ell_n)_{n\in\mathbb{N}}$ be a sequence of linear functionals in $\mathrm{BV}^*$, namely the dual of the space of functions with bounded variation. Suppose that $\ell_n$ converges to $\ell$ as $n \to \infty$ in the strong topology on $\mathrm{BV}^*$. Now consider a sequence $(f_n)_{n\in\mathbb{N}}$ such that for every $n\in\mathbb{N}$, $f_n\in \mathrm{BV}$.

I am interested in the following question. Is it true that$$\lim_{n\to \infty}\ell_n(f_n)=\lim_{n\to \infty} \ell(f_n).$$I think this is okay since $\ell_n$ converges strongly to $\ell$, but I am not entirely sure since the argument of the functionals also depend on the limiting parameter.