Can a norm "grow exponentially"?
Let $||\cdot||_*: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0} $ be a norm such that:
$ \lim_{|x| \rightarrow \infty } \frac{ ||x||_* }{ e^{|x|} } > 0 $
where $|\cdot| \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0} $ is a Euclidean norm.
Is that possible?
What about so called "Nagumo norms"?