Let $B$ be the open unit ball in $\mathbb{R}^N$ e consider $$ \mathcal{R}(B) = \{u \in C^{\infty}_c(B) : u \text{ is radially symmetric}\}. $$ I woul like to know if $\mathcal{R}(B)$ is isometric to $C^{\infty}_c(B)$ under the $H^1_0(B)$ norm, that is, $||u||_{H^1_0(B)} = ||\nabla u||_{L^2(B)}$.
The context: In a paper, the author take $E$ as the completion of $\mathcal{R}(B)$ under the $||\cdot||_{H^1_0(B)}$, that is, $E$ is a Banach space which has a dense subspace isometrically to $\mathcal{R}(B)$. My question is justified because $C^\infty_c(B)$ is dense in $H^1_0(B)$. Actually I'm not sure if my question makes sense.