ScalingStacks

Proof. By the $\mathbb{Q}$-independence assumption and Proposition 5.1, for any two distinct multi-index $J$ and $J^{\prime}$, and any two non-zero coefficients $f_{J}$ and $f_{J^{\prime}}$ in $k$, we have

$$\lVert f_{J}\cdot\boldsymbol{T}^{J}\rVert_{|J|\phi}=\lvert f_{J}\rvert\cdot\prod_{i\in\{0,\dots,d\}}r_{i}^{j_{i}}\neq\lvert f_{J^{\prime}}\rvert\cdot\prod_{i\in\{0,\dots,d\}}r_{i}^{j^{\prime}_{i}}=\lVert f_{J^{\prime}}\cdot\boldsymbol{T}^{J^{\prime}}\rVert_{|J^{\prime}|\phi}.$$

By Lemma 2.13, the elements $\{\boldsymbol{T}^{J}\}_{J\in S}$ form an orthogonal basis for the normed vector space $(\bigoplus_{J\in S}k\cdot\boldsymbol{T}^{J},\vvvert\mathord{\cdot}\vvvert_{\phi})$. So the equality in the conclusion holds. ∎