ScalingStacks

Lemma 2.11. Let $f:V\rightarrow W$ be a surjective $k$-linear map of finite-dimensional vector spaces, with $\mathrm{dim}_{k}W=1$. Let $\lVert\mathord{\cdot}\rVert_{V}$ be a norm on $V$ and let $\lVert\mathord{\cdot}\rVert_{W}$ be its quotient norm for $f$. Then the norm $\lVert\mathord{\cdot}\rVert_{W,k^{\prime}}$ identifies with the quotient norm of $\lVert\mathord{\cdot}\rVert_{V,k^{\prime}}$ induced by the surjective $k^{\prime}$-linear map $f\otimes\mathrm{id}_{k^{\prime}}:V_{k^{\prime}}\rightarrow W_{k^{\prime}}$. ([CMor18, Lemma 2.5])