In the script for my lecture, there is an exercise asking for a proof that$$\|\varphi\|_{\alpha, \beta} = \sup_{x \in \mathbb{R}^d} |x^{\alpha} \partial^{\beta}\varphi(x)|$$is not a norm on $\mathcal{S}(\mathbb{R}^d)$. The solution says that $\varphi \equiv 1$ is a counterexample for positive-definiteness because $|x^{\alpha} \partial^{\beta}\varphi(x)| = 0$ for $\beta \geq 1$, so $\|\varphi\|_{\alpha, \beta}=0$. However, I don't understand why $\varphi$ should even be an element of $\mathcal{S}(\mathbb{R}^d)$, given that $x \cdot \varphi(x)$ is not bounded. I also found this post where the user apparently asks for a proof that $\| \cdot\|_{\alpha, \beta}$is a norm, at least on $\mathcal{S}(\mathbb{R})$. Other posts however talk about how the Schwartz space isn't even normable and call $\| \cdot\|_{\alpha, \beta}$ merely a seminorm.
Now I am very confused. Is $\| \cdot\|_{\alpha, \beta}$ a norm on $\mathcal{S}(\mathbb{R}^d)$ or not?