Supose that $N \geq 3$, $N \in \mathbb{N}$. I know from Brezis, Functional Analysis, Theorem 9.9 that exists a constant $c > 0$ such that $$ (1)\qquad |u|_{L^{2^*}(\mathbb{R}^N)} \leq c \, \|\nabla u \|_{L^{2}(\mathbb{R}^N)},\; \forall u \in W^{1,2}(\mathbb{R}^N). $$ Defining $D^{1,2}(\mathbb{R}^N) := \{u \in L^{2^*}(\mathbb{R}^N) : |\nabla u| \in L^2(\mathbb{R}^N)\}$, from $(1)$ we see that $W^{1,2}(\mathbb{R}^N) \subset D^{1,2}(\mathbb{R}^N)$. I have heard that the application $$ \|u\|_{D^{1,2}(\mathbb{R}^N)} := |\nabla u |_{L^2(\mathbb{R}^N)}, $$ is a norm in $D^{1,2}(\mathbb{R}^N)$. However I didn't find any reference with this proof. The difficult part to prove that $\|\cdot\|_{D^{1,2}(\mathbb{R}^N)}$ is a norm, is to prove that $u = 0$, whenever $\|u\|_{D^{1,2}(\mathbb{R}^N)} = 0$. For this, I would need an inequality like (1), for the bigger space $D^{1,2}(\|u\|_{D^{1,2}(\mathbb{R}^N)})$.
I would thank any help or reference.