ScalingStacks

Definition 2.1. Let $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ be seminorms on $V$. We say that $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ are equivalent if there exist two constants $C_{1}>0$ and $C_{2}>0$ such that $C_{1}\lVert\mathord{\cdot}\rVert_{1}\leq\lVert\mathord{\cdot}\rVert_{2}\leq C_{2}\lVert\mathord{\cdot}\rVert_{1}$. Note that this condition holds if and only if the seminorms $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ induce the same topology on the vector space $V$ ([Bou, Corollaire I.3.3.1])(note that the absolute value $\lvert\mathord{\cdot}\rvert$ is supposed to be non-trivial).