ScalingStacks

Proposition 5.9. Assume that $\phi$ is a Fubini-Study metric. Then there exist $C(\phi,Y)>0$ such that for any $\underline{t}\in V_{{\scriptscriptstyle\bullet}}(L_{X|Y})$, there exists $\underline{s}\in V_{{\scriptscriptstyle\bullet}}(L)$ with

$$\vvvert\underline{s}\vvvert_{\phi}\leq C(\phi,Y,X)\cdot\vvvert\underline{t}\vvvert_{\phi|_{Y}}.$$

In particular, for any $n\in\mathbb{N}$ and $t_{n}\in V_{{\scriptscriptstyle\bullet}}(L_{X|Y})$, there exists $s_{n}\in V_{{\scriptscriptstyle\bullet}}(L)$ with

$$\lVert s_{n}\rVert_{n\phi}\leq C(\phi,Y,X)\cdot\lVert t_{n}\rVert_{n\phi|_{Y}}.$$