We defined in class $H_{1}^{0}$ as the Hilbert space obtained as the closure in $W^{1}(G) = \{ u \in L^2(G) : \exists \partial_{j} u \in L^2(G), 1 \leq j \leq n \}$ of $\mathcal{D}(G) = \{ g: \Omega \rightarrow \mathbb{R}, C^{\infty}, supp(G) \subset \Omega ~compact\}$. My doubt is:
Is it true or false that Sobolev space $H_{1}^{0}$ and the Hilbert space $L^{2}(\Omega)$ are isometrically isomorphic?
Thanks!