Let $\phi:\mathcal{A}_{1}\to\mathcal{A}_{2}$ be a homomorphism between Banach $k$-algebras. It is called Banach finite if $\mathcal{A}_{2}$ is a Banach finite $\mathcal{A}_{1}$-module. In this case $\mathcal{A}_{2}$ is called a Banach finite $\mathcal{A}_{1}$-algebra.