28.1 Banach Algebras

Definition 28.1.1 (Banach Algebra).label Let $A$ be an associative algebra over $\complex$ and $\norm{\cdot}_{A}: A \to [0, \infty)$ be a norm, then $A$ is a Banach algebra if:

1. (1)

   $A$ is complete with respect to $\norm{\cdot}_{A}$.

2. (2)

   For any $x, y \in A$, $\norm{xy}_{A} \le \norm{x}_{A}\norm{y}_{A}$.

Definition 28.1.2 (Unital Banach Algebra).label Let $A$ be a Banach algebra, then $A$ is unital if there exists $1 \in A$ such that for any $x \in A$, $x1 = 1x = x$. In which case, $1$ is the unique multiplicative identity of $A$.