# Definition:Banach Algebra

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## Definition

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

Let $\struct {A, \circ}$ be an algebra over $R$ which is also a Banach space.

Then $\struct {A, \circ}$ is a **Banach algebra** if and only if:

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

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):**Banach algebra** - 2017: Amol Sasane:
*A Friendly Approach to Functional Analysis*... (previous) ... (next): Chapter $\S 2.4$: Composition of continuous linear transformations