Is $f(x) = |\arctan(x)|$ a norm on $\mathbb{R}$?
Im checking if the properties of a norm holds for $f(x) = |\arctan(x)|$.
$1. \ f(x) \ge 0 \Leftrightarrow |\arctan(x)| \ge 0 \\ 2. \ f(x)=0 \Leftrightarrow |\arctan(x)| =0 \Leftrightarrow x=0 \\$
But does $f(\lambda x)=|\arctan(\lambda x)|\Leftrightarrow |\lambda||\arctan(x)|?$ For some $\lambda \in \mathbb{K}$.
Also, how would I check if $|\arctan(x+y)| \le |\arctan(x)|+|\arctan(y)|$?