Normed division algebra

A normed division algebra A is a division algebra over the real or complex numbers that is also a normed vector space and satisfies ||xy|| = ||x||.||y|| for all x and y in A.

The only normed division algebras over the reals (up to isomorphism) are

• the reals themselves
• the complex numbers
• the quaternions

The only normed division algebra over the complex numbers are the complex numbers themselves. In all of the above cases, the norm is given by the absolute value.