Let $\phi:\mathbb R^{3} \to \mathbb C$ such that $\|\phi\|_{H^1}=\|\nabla \phi \|_{L^2} < \infty$. Let $\{x_n\}_{n\in \mathbb N} \subset \mathbb R^3 $ such that $x_n \to \infty$ as $n\to \infty.$
We note that $\|\tau_{x_n}\phi \|_{H^1}= \|\phi\|_{H^{1}},$ where $\tau_{x_n}f(x)=f(x-x_{n})$
Let $f:\mathbb R^{3} \to \mathbb C$ such that $\|\nabla f\|_{L^2} <\infty.$
My Question: Can we say $\langle\tau_{x_n}\phi, f\rangle= \int \phi(x-x_n) f(x) dx \to 0$ as $n\to \infty$? In other, words can we say $\{\tau_{x_n}\phi \}$ converges weakly to zero?