Let $A$ be a $k$-algebra (the unit of which is denoted by $\mathbf{1}$) and $\lVert\mathord{\cdot}\rVert$ be a seminorm on $A$ (viewed as a vector space over $k$).

1. (1)

   The seminorm $\lVert\mathord{\cdot}\rVert$ is said to be sub-multiplicative if for any $(a,b)\in A\times A$ one has $\lVert ab\rVert\leq\lVert a\rVert\cdot\lVert b\rVert$.

2. (2)

   The seminorm $\lVert\mathord{\cdot}\rVert$ is called power-multiplicative if $\lVert a^{n}\rVert=\lVert a\rVert^{n}$ for any $a\in A$ and any $n\in\mathbb{N}\setminus\{0\}$.

3. (3)

   The seminorm $\lVert\mathord{\cdot}\rVert$ is called multiplicative if $\lVert ab\rVert=\lVert a\rVert\cdot\lVert b\rVert$ for any $(a,b)\in A^{2}$.

A $k$-algebra seminorm (resp. $k$-algebra norm) on $A$ is defined to be a sub-multiplicative seminorm (resp. sub-multiplicative norm) $\lVert\mathord{\cdot}\rVert$ on $A$ such that $\lVert\mathbf{1}\rVert=1$. We denote by $\vvvert\mathord{\cdot}\vvvert$ an algebra seminorm. Any $k$-algebra equipped with a complete $k$-algebra norm is called a Banach $k$-algebra.

We use calligraphic letters to denote Banach algebras and Banach modules (defined below) and use the corresponding capital letters to denote the underlying $k$-algebra or the underlying module of a $k$-algebra. For example, a Banach $k$-algebra $(A,\vvvert\mathord{\cdot}\vvvert)$ is denoted by $\mathcal{A}$. If $A^{\prime}$ is a sub-$k$-algebra of $A$, then the restriction of $\vvvert\mathord{\cdot}\vvvert$ on $A^{\prime}$ is a $k$-algebra norm. If this norm is complete, we say that $\mathcal{A}^{\prime}$ ($A^{\prime}$ equipped with the restricted norm) is a Banach $k$-sub-algebra of $\mathcal{A}$. Similarly, if $Q$ is a quotient $k$-algebra of $A$, then the quotient of the norm $\vvvert\mathord{\cdot}\vvvert$ on $Q$ is a sub-multiplicative seminorm. If it is a complete norm, we say that $\mathcal{Q}$ ($Q$ equipped with the quotient norm) is a Banach quotient $k$-algebra of $\mathcal{A}$.