ScalingStacks

Proposition 2.35. Let $\mathcal{A}$ be a Banach $k$-algebra and $\mathcal{M}$ be a Banach $\mathcal{A}$-module. If $A$ is Noetherian as a $k$-algebra and $M$ is finitely generated as $A$-module, then any $\mathcal{A}$-sub-module of $\mathcal{M}$ is closed, and $\mathcal{M}$ is a Banach finite $\mathcal{A}$-module. ([FvdP, Lemma 1.2.3]