00LZ Proposition 5.3. Assume that {α(∥Ti∥ϕ)}i∈{0,…,d}\{\alpha(\lVert T_{i}\rVert_{\phi})\}_{i\in\{0,\dots,d\}} are ℚ\mathbb{Q}-independent in ℝ/H(k,|⋅|)\mathbb{R}/H(k,\lvert\mathord{\cdot}\rvert). Let S⊆ℕd+1S\subseteq\mathbb{N}^{d+1} be a finite set of multi-indices, then for any J∈SJ\in S any fJ∈kf_{J}\in k, one has ⦀∑J∈SfJ⋅𝑻J⦀=supJ∈S∥fJ⋅𝑻J∥|J|ϕ.\Big\vvvert\sum_{\begin{subarray}{c}J\in S\end{subarray}}f_{J}\cdot\boldsymbol{T}^{J}\Big\vvvert=\sup_{\begin{subarray}{c}J\in S\end{subarray}}\ \lVert f_{J}\cdot\boldsymbol{T}^{J}\rVert_{|J|\phi}. In other words, the algebra norm ⦀⋅⦀ϕ\vvvert\mathord{\cdot}\vvvert_{\phi} on V∙(𝒪(1))V_{{\scriptscriptstyle\bullet}}(\mathscr{O}(1)) is a Gauss norm on k[T0,…,Td]k[T_{0},\dots,T_{d}] of multi-radius 𝒓\boldsymbol{r}. The Banach kk-algebra V^∙(𝒪(1),ϕ)\widehat{V}_{{\scriptscriptstyle\bullet}}(\mathscr{O}(1),\phi) is an affinoid algebra.
00M0 Proof. By the ℚ\mathbb{Q}-independence assumption and Proposition 5.1, for any two distinct multi-index JJ and J′J^{\prime}, and any two non-zero coefficients fJf_{J} and fJ′f_{J^{\prime}} in kk, we have ∥fJ⋅𝑻J∥|J|ϕ=|fJ|⋅∏i∈{0,…,d}riji≠|fJ′|⋅∏i∈{0,…,d}riji′=∥fJ′⋅𝑻J′∥|J′|ϕ.\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 {𝑻J}J∈S\{\boldsymbol{T}^{J}\}_{J\in S} form an orthogonal basis for the normed vector space (⨁J∈Sk⋅𝑻J,⦀⋅⦀ϕ)(\bigoplus_{J\in S}k\cdot\boldsymbol{T}^{J},\vvvert\mathord{\cdot}\vvvert_{\phi}). So the equality in the conclusion holds. ∎