I understand that one way to define the radial Lebesgue space $L_{rad}^{p}(\mathbb{R}^n)$$L_\text{rad}^{p}(\mathbb{R}^n)$ is by the completation of the space of radial smooth function with compact support, i.e, $L_{rad}^p(\mathbb{R}^n)=\overline{C_{rad,0}^{\infty}(\mathbb{R}^n)}^{\left\|\cdot\right\|_{p}}$$L_\text{rad}^p(\mathbb{R}^n)=\overline{C_\text{rad,0}^{\infty}(\mathbb{R}^n)}^{\left\|\cdot\right\|_{p}}$ but, it can also be defined as $L_{rad}^p(\mathbb{R}^n)=\left\{u\in L^p(\mathbb{R}^n)\colon u\text{ radial function }\right\}$$L_\text{rad}^p(\mathbb{R}^n)=\left\{u\in L^p(\mathbb{R}^n)\mid \text{$u$ is radial function }\right\}$.
Question. Do these two definitions coincide? Thanks.
