ScalingStacks

Proposition 2.56. For any $f\in\mathcal{T}_{n}$, there exists $z\in\mathrm{Max}(\mathcal{T}_{n})$ such that $\lvert f(z)\rvert_{z}=\vvvert f\vvvert_{\mathcal{T}_{n}}$ ([BGR, Proposition 5.1.4.3]). On $\mathcal{T}_{n}$, the three norms are equal: $\vvvert\mathord{\cdot}\vvvert_{\mathcal{T}_{n}}=\vvvert\mathord{\cdot}\vvvert_{\mathcal{T}_{n},\text{sp}}=\vvvert\mathord{\cdot}\vvvert_{\mathcal{T}_{n},\mathrm{spM}}$.