Normed division algebra

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

This was shown by Mazur in 1938.

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.