Definition:Normed Division Algebra
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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 if and only if $A_F$ 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
- Normed Division Algebra is Unitary Division Algebra: $\struct {A_F, \oplus}$ has to be a (unitary) division algebra to be a normed divison algebra
- Results about normed division algebras can be found here.