# Definition:Banach Algebra

## Definition

Let $R$ be either the real numbers $\R$ or the complex numbers $\C$..

Let $A$ be an algebra over $R$ which is also a Banach space.

Then $A$ is a Banach algebra if and only if:

$\forall a, b \in R: \norm {a b} \le \norm a \norm b$

where $\norm {\, \cdot \,}$ denotes the norm on $A$.

## Also known as

Some sources refer to a Banach algebra as a normed ring, but on $\mathsf{Pr} \infty \mathsf{fWiki}$ this term is used to mean a ring with a norm on it.

## Also see

• Results about Banach algebras can be found here.

## Source of Name

This entry was named for Stefan Banach.