Category:Definitions/Banach Algebras

This category contains definitions related to Banach Algebras.
Related results can be found in Category:Banach Algebras.

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 $A$.