Let $\mathbb{S}(\mathbb{R^d})$ be the Schwartz class on $\mathbb{R^d}$ and define the following two families of seminorms.$$\rho_{\alpha\beta}(f):= \|x^\alpha\partial^\beta f\|_\infty \\\sigma_{\alpha\beta}(f):=\|x^\alpha\partial^\beta(f)\|_{L^1}$$, where $\alpha,\beta\in\mathbb{Z^d}$ are multi-indices.
I know that $\mathbb{S}(\mathbb{R^d})$ is a Frechet space when equipped with the first norm.Let's call the topologies induced by them $T_1$ and $T_2$ respectively. Now I am trying to show the $\sigma_{\alpha\beta}$ induces the same norm on $\mathbb{S}(\mathbb{R^d})$. Would it suffice to show the existence of two constants $c_{\alpha\beta}$ an $C_{\alpha\beta}$ such that $$c_{\alpha\beta}\rho_{\alpha\beta}(f)\leq\sigma_{\alpha\beta}(f)\leq C_{\alpha\beta}(f)\rho_{\alpha\beta}(f)$$, for every $f\in\mathbb{S}(\mathbb{R^d})$, ?
Also, I am wondering is there a more concise way of proving it without going back the open sets?