I'm supposed to prove that the topology of the Schwartz-Space $\mathcal{S}:=\mathcal{S}(\mathbb{R}^n)$ is not induced by a norm by assuming that there exists such a norm $||\cdot||$.
As a first step, I'm supposed to prove that for every $N \in \mathbb{N}$ there is a constant $C_N$ such that $|f|_N \leq C_N \cdot ||f||$ for every $f \in \mathcal{S}$, where
$|f|_N = \sup \{ (1+|x|)^N \cdot |\partial^\alpha f(x)| ; x \in \mathbb{R}^n, \alpha \in \mathbb{N}_0^n, |\alpha| \leq N \}$.
My Question is:
How do I prove the inequality involving $||f||$ without knowing what $||f||$ looks like? I only now that it meets the following condition (per assumption):
A sequence of Schwartz-functions $(f_k)_{k \in \mathbb{N} } \subset \mathcal{S}$ converges to f in the Schwarz-space (i.e. in $| \cdot |_N) \Longleftrightarrow ||f_k - f|| \longrightarrow 0$, as $k \to \infty$.