Definition:Banach Algebra

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