Spectral radius/Maximum norm/Fact

Let ${\displaystyle {}V}$ be a finite-dimensional ${\displaystyle {}{\mathbb {K} }}$-vector space, and let

${\displaystyle \varphi \colon V\longrightarrow V}$

be an endomorphism. Let a norm on ${\displaystyle {}V}$ be given, and let ${\displaystyle {}\Vert {-}\Vert _{\rm {max}}}$ be the corresponding norm on the endomorphism space.