ScalingStacks

Theorem 2.5. Let $V$ be a vector space over $k$ and $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ be complete norms on $V$. If there exists $C>0$ such that $\lVert\mathord{\cdot}\rVert_{2}\leq C\lVert\mathord{\cdot}\rVert_{1}$, then the norms $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ are equivalent.