ScalingStacks

Proof. 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. ∎