Banach algebra

Summary

A Banach algebra is an associative algebra over $\R$ or $\C$ with a complete norm that satisfies $\norm{xy} \le \norm{x}\norm{y}$. It is a normed algebra that is also a Banach space. Completeness makes power series converge, so in a unital Banach algebra $1 - x$ is invertible when $\norm{x} < 1$. In a nonzero complex unital Banach algebra the spectrum of every element is a nonempty compact set. Algebras of bounded operators, continuous functions and integrable functions on groups are the main examples, and C*-algebras are the most studied special class.