Density property between $L^1$ and $H^1$

I would like to know if the following property is true

Given the sobolev space $H^1(\mathbb{R}^d)$ and the Lebesgue space $L^1(\mathbb{R}^d)$, is is true that $H^1(\mathbb{R}^d)\cap H^1(\mathbb{R}^d)$ is dense in $H^1(\mathbb{R}^d)$ and in $L^1(\mathbb{R}^d)$?