ScalingStacks

Definition 2.3. Let $(V,\lVert\mathord{\cdot}\rVert_{V})$ and $(W,\lVert\mathord{\cdot}\rVert_{W})$ be seminormed vector spaces over $k$, and $f:V\rightarrow W$ be a $k$-linear map. We say that $f$ is bounded if there exists a constant $C>0$ such that $\lVert f(x)\rVert_{W}\leq C\lVert x\rVert_{V}$ for any $x\in V$. Note that this condition holds if and only if $f$ is continuous with respect to the topologies on $V$ and $W$ induced by the seminorms $\lVert\mathord{\cdot}\rVert_{V}$ and $\lVert\mathord{\cdot}\rVert_{W}$ respectively. We say that $f$ is admissible if it is bounded and if on the image of $f$, the quotient seminorm of $\lVert\mathord{\cdot}\rVert_{V}$ and the restriction of $\lVert\mathord{\cdot}\rVert_{W}$ are equivalent.