# Definition:Banach Algebra

Jump to navigation
Jump to search

## 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.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Banach algebra**