Banach algebra

Remarks

Example: for a compact Hausdorff space $X$, the algebra $C(X)$ of continuous functions $X \to \C$ with $\norm{f} = \sup_{x \in X} \abs{f(x)}$. Submultiplicativity holds since $\abs{f(x) g(x)} \le \norm{f}\norm{g}$ for every $x$, and completeness holds because a uniform limit of continuous functions is continuous. Example: the bounded linear operators on a Banach space with the operator norm. Non-example: the polynomial functions on $[0, 1]$ with the same sup norm form a normed algebra but not a Banach algebra. They are dense in $C([0, 1])$ by the Weierstrass approximation theorem, and $e^x$ is a uniform limit of polynomials that is not a polynomial, so completeness fails. In a unital Banach algebra, $1 - x$ is invertible with inverse $\sum_{n \ge 0} x^n$ when $\norm{x} < 1$.