Let $f$ be in the Shwartz space $\mathcal S(\Bbb R)$.
Why does the $\mathcal S$-norm $$ \|f\|_{a,b}=\sup_{x \in \mathbb R} |x^af^{(b)}(x)|, \text{ for } a,b \in \Bbb Z_+, $$ implies that $f$ vanish at infinity?
The norm gives a bound on $$ \lim_{|x| \to \infty} f(x) $$ but that doesn't show the function vanish.
This post raised this question.