ScalingStacks

Definition 2.34. Let $\mathcal{M}$ be a Banach $\mathcal{A}$-module. It is called a Banach finite $\mathcal{A}$-module if there exists $l\in\mathbb{N}_{+}$ and a surjective homomorphism of Banach $\mathcal{A}$-modules $\mathcal{A}^{\oplus l}\to\mathcal{M}$ where $\mathcal{A}^{\oplus l}$ is the Banach $\mathcal{A}$-module corresponding to the $A$-module $A^{\oplus l}$ equipped with the norm $(a_{1},\dots,a_{l})\mapsto\max\vvvert a_{i}\vvvert$. (Note that such a homomorphism is necessarily admissible.)