ScalingStacks

Corollary 2.58. Let $\mathcal{A}$ be a reduced general affinoid algebra. Then there exists $C>0$ such that $\vvvert f\vvvert\leq C\vvvert f\vvvert_{\mathrm{sp}}$ for all $f\in\mathcal{A}$. In particular, $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ is complete on $\mathcal{A}$ , and is equivalent to $\vvvert\mathord{\cdot}\vvvert$. ([Ber, Proposition 2.1.4.ii])