A normed division algebra A is an associative 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.
While the definition allows normed division algebras to be infinite-dimensional, this does in fact not occur. 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.