Definition:Normed Division Algebra

Definition
Let $\struct {A_F, \oplus}$ be a unitary algebra where $A_F$ is a vector space over a field $F$.

Then $\struct {A_F, \oplus}$ is a normed divison algebra iff $A_F$ is a normed vector space such that:
 * $\forall a, b \in A_F: \norm {a \oplus b} = \norm a \norm b$

where $\norm a$ denotes the norm of $a$.

Also see
From Normed Division Algebra is Unitary Division Algebra it is proved that $\struct {A_F, \oplus}$ does in fact have to be a (unitary) division algebra to be a normed divison algebra.