ScalingStacks

Proof. By assumption, there exists $j\in\mathbb{N}$ and a homomorphism of $k$-algebras and elements $\{e_{i}\}_{i\in\{1,\dots,j\}}\subseteq B$ such that

$$F:\bigoplus_{i\in\{1,\dots,j\}}A\rightarrow B,1_{i}\mapsto e_{i}$$

Moreover, $F$ is bounded

$$\vvvert\sum_{i\in\{1,\dots,j\}}a_{i}\cdot e_{i}\vvvert\leq\max_{i\in\{1,\dots,j\}}\vvvert a_{i}\cdot e_{i}\vvvert\leq\max_{i\in\{1,\dots,j\}}\vvvert e_{i}\vvvert\cdot\max_{i\in\{1,\dots,j\}}\vvvert a_{i}\vvvert$$

So $F$ extends to a homomorphism of Banach $\mathcal{A}$-modules

$$\mathcal{F}:\bigoplus_{i\in\{1,\dots,j\}}\mathcal{A}\rightarrow\mathcal{B},1_{i}\mapsto e_{i}$$

Let $\mathcal{B}^{-}$ be the image of $\mathcal{F}$, it is a Banach finite $\mathcal{A}$-module with the quotient norm $\lVert\mathord{\cdot}\rVert_{\mathcal{F}}$ induced by $\mathcal{F}$. As $\mathcal{B}^{-}$ is Banach finite over $\mathcal{A}$, it is an affinoid algebra with an affinoid algebra spectral norm $\vvvert\mathord{\cdot}\vvvert^{-}$, which is equivalent to $\lVert\mathord{\cdot}\rVert_{\mathcal{F}}$. Now on $\mathcal{B}^{-}$, $\vvvert\mathord{\cdot}\vvvert$ is bounded with respect to $\vvvert\mathord{\cdot}\vvvert^{-}$ by the continuity of $\mathcal{F}$. To show the reverse, note that $\mathcal{B}^{-}$ is dense in $\mathcal{B}$, so by Theorem 2.30 one has for any $b\in\mathcal{B}^{-}$

$$\vvvert b\vvvert^{-}=\max_{z\in\mathfrak{M}(\mathcal{B}^{-})}\lvert b(z)\rvert=\max_{z\in\mathfrak{M}(\mathcal{B})}\lvert b(z)\rvert=\vvvert b\vvvert_{\mathrm{sp}}\leq\vvvert b\vvvert$$

Therefore $\vvvert\mathord{\cdot}\vvvert$ and $\vvvert\mathord{\cdot}\vvvert^{-}$ are equivalent norms on $\mathcal{B}^{-}$, so $\mathcal{B}^{-}$ is closed in $\mathcal{B}$, hence coincides with it. ∎