Assume that $\{\alpha(\lVert T_{i}\rVert_{\phi})\}_{i\in\{0,\dots,d\}}$ are $\mathbb{Q}$-independent in $\mathbb{R}/H(k,\lvert\mathord{\cdot}\rvert)$. Let $S\subseteq\mathbb{N}^{d+1}$ be a finite set of multi-indices, then for any $J\in S$ any $f_{J}\in k$, one has

$$\Big\vvvert\sum_{\begin{subarray}{c}J\in S\end{subarray}}f_{J}\cdot\boldsymbol{T}^{J}\Big\vvvert=\sup_{\begin{subarray}{c}J\in S\end{subarray}}\ \lVert f_{J}\cdot\boldsymbol{T}^{J}\rVert_{|J|\phi}.$$

In other words, the algebra norm $\vvvert\mathord{\cdot}\vvvert_{\phi}$ on $V_{{\scriptscriptstyle\bullet}}(\mathscr{O}(1))$ is a Gauss norm on $k[T_{0},\dots,T_{d}]$ of multi-radius $\boldsymbol{r}$. The Banach $k$-algebra $\widehat{V}_{{\scriptscriptstyle\bullet}}(\mathscr{O}(1),\phi)$ is an affinoid algebra.