Definition 6.1.1 (Banach Algebra).label Let $A$ be a Banach space over $\com$. If $A$ is an algebra over $\com$ (a vector space equipped with a bilinear multiplication) in which the multiplication satisfies the inequality
\begin{align*}\norm{xy}{}\leq \norm{x}{}\norm{y}{}\end{align*}
then $A$ is called a Banach algebra. The inequality
\begin{align*}\norm{x_1y_1-x_2y_2}{}\leq \norm{x_1}{}\norm{y_1-y_2}{}+\norm{x_1-x_2}{}\norm{y_2}{}\end{align*}
shows that the product $xy$ is a continuous function of two variables $x$ and $y$. If $E$ is a Banach space over $\com$, then the set $\cali{L}(E)$ of all bounded operators on $E$ is a Banach algebra with the natural algebraic operations and norm.