Definition:Normed Division Algebra

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Let $\left({A_F, \oplus}\right)$ be a unitary 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:

$\forall a, b \in A_F: \left \Vert{a \oplus b}\right \Vert = \left \Vert{a}\right \Vert \left \Vert{b}\right \Vert$

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

Also see

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