Take a complete non-Archimedean valued field extension $(K,\lvert\mathord{\cdot}\rvert_{K})$ of $(k,\lvert\mathord{\cdot}\rvert_{k})$ such that

$$\forall i\in\{0,\dots,d\},\quad\{r_{i}\}_{i\in\{0,\dots,d\}}\subseteq\lvert k^{\times}\rvert_{K},$$

hence for any $i\in\{0,\dots,d\}$, there exist elements $\kappa_{i}\in K$ such that $\lvert\kappa_{i}\rvert_{K}=r_{i}$. One denotes by $x(\boldsymbol{r})\in(\mathbb{P}^{d}_{k})^{\mathrm{an}}$ the point given by coordinates $[\kappa_{0}:\dots:\kappa_{d}]$.

By this Claim, for any multi-index $J$, the function $\lvert\boldsymbol{T}^{J}\rvert_{\phi}(x)$ can attain its maximum value $\prod_{i\in\{0,\dots,d\}}r_{i}^{j_{i}}$ at the point $x(\boldsymbol{r})\in(\mathbb{P}_{k}^{d})^{an}$ as the product of maximum of factors of the monomial. By definition,

$$\lVert\boldsymbol{T}^{J}\rVert_{|J|\phi}=\sup_{x\in(\mathbb{P}^{d})^{an}}\lvert\boldsymbol{T}^{J}\rvert_{\phi}(x)=\prod_{i\in\{0,\dots,d\}}r_{i}^{j_{i}}=\prod_{i\in\{0,\dots,d\}}\lVert T_{i}\rVert_{\phi}^{j_{i}}.$$

∎

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. ∎

Choose an arbitrary $\boldsymbol{\delta}$ with $|\boldsymbol{\delta}|\leq\epsilon$. Then

$$\dist(\lVert\mathord{\cdot}\rVert_{\phi},\lVert\mathord{\cdot}\rVert_{\phi(\boldsymbol{\delta})})\leq\epsilon.$$

By Proposition 3.12, we have

$$\dist(\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{\phi}),\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{\phi(\boldsymbol{\delta})}))=\dist(\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{\phi}),\phi(\boldsymbol{\delta}))\leq\epsilon,$$

by the assumption and Proposition 3.11,

$$\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{\phi})=\phi,$$

so the conlusion holds. ∎

Since $\lvert\mathord{\cdot}\rvert_{k}$ is discrete, for any $\epsilon>0$, there exists $\boldsymbol{\delta}$ with $|\boldsymbol{\delta}|\leq\epsilon$ such that the elements $\{\alpha(\lVert T_{i}\rVert_{\phi(\boldsymbol{\delta})})\}_{i\in\{0,\dots,d\}}$ are $\mathbb{Q}$-independent in $\mathbb{R}/H(k,\lvert\mathord{\cdot}\rvert)$. By Proposition 5.5, for any $n\in\mathbb{N}$ and any $s_{n}=\sum_{\lvert J\rvert=n}f_{J}\cdot\boldsymbol{T}^{J}\in V_{n}(\mathscr{O}(1))$,

$$\mathrm{e}^{-n\epsilon}\Big\lVert\sum_{\lvert J\rvert=n}f_{J}\cdot\boldsymbol{T}^{J}\Big\rVert_{n\phi(\boldsymbol{\delta})}\leq\Big\lVert\sum_{\lvert J\rvert=n}f_{J}\cdot\boldsymbol{T}^{J}\Big\rVert_{n\phi}\leq\mathrm{e}^{n\epsilon}\Big\lVert\sum_{\lvert J\rvert=n}f_{J}\cdot\boldsymbol{T}^{J}\Big\rVert_{n\phi(\boldsymbol{\delta})}.$$

By Proposition 5.3, one has

$$\max_{\lvert J\rvert=n}\Big\{\mathrm{e}^{-n\epsilon}\lvert f_{J}\rvert\prod_{i\in\{0,\dots,d\}}(\mathrm{e}^{\delta_{i}}r_{i})^{j_{i}}\Big\}\leq\Big\lVert\sum_{\lvert J\rvert=n}f_{J}\cdot\boldsymbol{T}^{J}\Big\rVert_{n\phi}\leq\max_{\lvert J\rvert=n}\Big\{\mathrm{e}^{n\epsilon}\lvert f_{J}\rvert\prod_{i\in\{0,\dots,d\}}(\mathrm{e}^{\delta_{i}}r_{i})^{j_{i}}\Big\}.$$

Fix $n$ and let $\epsilon\to 0$, one gets

$$\Big\lVert\sum_{\lvert J\rvert=n}f_{J}\cdot\boldsymbol{T}^{J}\Big\rVert_{n\phi}=\max_{\lvert J\rvert=n}\Big\{\lvert f_{J}\rvert\prod_{i\in\{0,\dots,d\}}r_{i}^{j_{i}}\Big\}.$$

So one has

$$\Big\vvvert\sum_{\lvert J\rvert<\infty}f_{J}\cdot\boldsymbol{T}^{J}\Big\vvvert_{n\phi}=\sup_{n\in\mathbb{N}}\max_{\lvert J\rvert=n}\Big\{\lvert f_{J}\rvert\prod_{i\in\{0,\dots,d\}}r_{i}^{j_{i}}\Big\}=\max_{\lvert J\rvert<\infty}\Big\{\lvert f_{J}\rvert\prod_{i\in\{0,\dots,d\}}r_{i}^{j_{i}}\Big\}.$$

Hence $\vvvert\mathord{\cdot}\vvvert_{\phi}$ is a Gauss norm of multi-radius $\boldsymbol{r}$ on $V_{{\scriptscriptstyle\bullet}}(\mathscr{O}(1))$. ∎

By the assumption, the elements of $V_{1}(L)$ induces an embedding

$$\iota_{1}:X\rightarrow\mathbb{P}^{d_{1}}_{k}$$

such that $\iota_{1}^{*}\mathscr{O}(1)=L$ with $\mathrm{dim}_{k}V_{1}=d_{1}+1$. Moreover there exists a norm $\lVert\mathord{\cdot}\rVert_{1}$ on $V_{1}(L)$ such that $\phi=\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{1})$. View $\lVert\mathord{\cdot}\rVert_{1}$ as a norm on $V_{1}(\mathscr{O}(1))$, we get a metric $\psi=\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{1})$ on $\mathscr{O}(1)$. By construction $\psi|_{X}=\phi$.

By Proposition 5.6, the Banach algebra $\widehat{V}_{{\scriptscriptstyle\bullet}}(\mathscr{O}(1),\vvvert\mathord{\cdot}\vvvert_{\psi})$ is an affinoid algebra. Hence the quotient Banach algebra $\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\vvvert\mathord{\cdot}\vvvert_{\psi,\mathbb{P}^{d_{1}}|X})$ is an affinoid algebra.

By Proposition 3.27, the algebra norm $\vvvert\mathord{\cdot}\vvvert_{\phi}$ is the spectral norm of $\vvvert\mathord{\cdot}\vvvert_{\psi,\mathbb{P}^{d_{1}}|X}$. The later is an affinoid norm, hence is equivalent to its spectral norm by Proposition 2.58. So $\vvvert\mathord{\cdot}\vvvert_{\phi}$ is also an affinoid algebra norm. Thus $\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi)$ is an affinoid algebra.

Similarly, by Proposition 3.27, on $V_{{\scriptscriptstyle\bullet}}(L_{X|Y})$, the algebra norm $\vvvert\mathord{\cdot}\vvvert_{\phi|_{Y}}$ is the spectral norm of $\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y}$, hence is itself an affinoid algebra norm. ∎

By assumption, $L^{\otimes M}$ is very ample. So $\widehat{V}_{{\scriptscriptstyle\bullet}}^{(M)}(L,\phi)$, $\widehat{V}_{{\scriptscriptstyle\bullet}}^{(M)}(L_{X|Y},\phi_{X|Y})$ and $\widehat{V}_{{\scriptscriptstyle\bullet}}^{(M)}(L|_{Y},\phi|_{Y})$ are affinoid algebras. Since $V_{{\scriptscriptstyle\bullet}}(L)$ is integral and is finite over $V_{{\scriptscriptstyle\bullet}}^{(M)}(L)$, by Proposition 2.45, the Banach algebras $\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi)$, $\widehat{V}(L_{X|Y},\phi_{X|Y})$ and $\widehat{V}_{{\scriptscriptstyle\bullet}}(L|_{Y},\phi|_{Y})$ are Banach finite over $\widehat{V}_{{\scriptscriptstyle\bullet}}^{(M)}(L,\phi)$, $\widehat{V}_{{\scriptscriptstyle\bullet}}^{(M)}(L_{X|Y},\phi_{X|Y})$ and $\widehat{V}_{{\scriptscriptstyle\bullet}}^{(M)}(L|_{Y},\phi|_{Y})$ respectively. Hence they are affinoid algebras. ∎

By Corollary 5.8, the Banach algebra norm $\vvvert\mathord{\cdot}\vvvert_{\phi_{X|Y}}$ is an affinoid algebra norm. By Proposition 2.58, there exists $C(\phi,Y)>0$ such that

$$\vvvert\mathord{\cdot}\vvvert_{\phi_{X|Y}}\leq C(\phi,Y,X)\cdot\vvvert\mathord{\cdot}\vvvert_{\phi_{X|Y};\mathrm{sp}}.$$

Since $\vvvert\mathord{\cdot}\vvvert_{\phi_{X|Y};\mathrm{sp}}=\vvvert\mathord{\cdot}\vvvert_{\phi|_{Y}}$ by Corollary 3.27, one gets the bounds. ∎