# Linear norm

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This is the wiki page for understanding seminorms of linear growth on a group $G$ (such as the free group on two generators). These are functions $\| \|: G \to [0,+\infty)$ that obey the triangle inequality

$\|xy\| \leq \|x\| + \|y\|$

and the linear growth condition

$\|x^n \| = |n| \|x\|$

for all $x,y \in G$ and $n \in {\bb Z}$.