# Why $\displaystyle\rho_{\alpha, \beta}(f)=\sup_{x\in\mathbb R^n}|x^\beta \partial^\alpha f(x)|$ is not a norm on $\mathcal{S}(\mathbb R^n)$?

The Schwartz space $\mathcal{S}(\mathbb R^n)$ is the set of all function $f:\mathbb R^n\longrightarrow \mathbb C$ such that $\displaystyle \sup_{x\in \mathbb R^n}|x^\beta \partial^\alpha f(x)|<\infty$ for all $\alpha, \beta\in\mathbb N_0^n$ ($\mathbb N_0=\mathbb N\cup\{0\}$). Can anyone give me an example why $$\rho_{\alpha, \beta}(f)=\sup_{x\in \mathbb R^n} |x^\beta \partial^\alpha f(x)|,$$ is not a norm on $\mathcal{S}(\mathbb R^n)$ for each $\alpha, \beta\in\mathbb N_0^n$?

The $\rho_{\alpha,\beta}$ are norms on $\mathcal{S}(\mathbb{R}^n)$. That they are seminorms is straightforward to verify, and $\rho_{\alpha,\beta}(f) = 0$ only for $f \equiv 0$ follows since $x^\beta$ is only zero on a nowhere-dense subset (the union of finitely many coordinate hyperplanes $\{x_i = 0\}$), so $\rho_{\alpha,\beta}(f) = 0 \Rightarrow \partial^\alpha f \equiv 0$, and that implies that $f$ is a polynomial, but the only polynomial in $\mathcal{S}(\mathbb{R}^n)$ is $0$.