Definition:Normed Division Algebra

Definition
Let $\left({A_F, \oplus}\right)$ be a division algebra where $A_F$ is a vector space over a field $F$.

Then $\left({A_F, \oplus}\right)$ is a normed divison algebra iff $A_F$ is a normed vector space such that:
 * $\forall a, b \in A_F: \left \Vert{a b}\right \Vert = \left \Vert{a}\right \Vert \left \Vert{b}\right \Vert$

where $\left \Vert{a}\right \Vert$ denotes the norm of $a$.