Denote the Schwartz space by $\mathcal S(\mathbb{R})$. I want to show that $\forall n,k \in \mathbb{N} \cup \{0\}$, $\|\cdot\|^{(n,k)} : \mathcal S(\mathbb{R}) \rightarrow [0, \infty)$ defined by $$\|f\|^{(n,k)} = \sup_{x \in \mathbb{R}} |x^n f^{(k)}(x)|$$ is a norm on $\mathcal S(\mathbb{R})$ and therefore is a countably normed space.
I have never worked with this space before so I am a little unsure. The help would be appreciated!
The most difficult part of the exercise is to show that if $\|f\|^{(n,k)} =0$ for some $f\in\mathcal S(\mathbb R)$, then $f \equiv 0$.
For each $x$, we have $x^nf^{(k)}(x)=0$, hence for each $x\neq 0$, we get $f^{(k)}(x)=0$. Since $f^{(k)}$ is a continuous function, the equality $f^{(k)}(x)=0$ actually hold for each $x\in\mathbb R$. If $k=0$ we are done, and if $k\neq 0$, then defining $g:=f^{(k-1)}$, we have $g\in\mathcal S(\mathbb R^n)$ and $g'=0$, hence $g(x)=c$ for some constant $c$. Since $g$ vanishes at infinity, we thus have $g(x)=f^{(k-1)}(x)=0$ for each $x$.
The proof can thus be completed by a clean induction.