Normed division algebra

Summary

A normed division algebra is a real algebra with a unit and a norm. The norm of a product must be the product of the norms. The multiplicative norm forbids zero divisors, so in finite dimension such an algebra is a division algebra. By Hurwitz's theorem there are exactly four of finite dimension: $\R$, $\C$, the quaternions and the octonions. They are the positive definite cases of composition algebras over $\R$.