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

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.

The only commutative Banach division algebras over the reals (up to isomorphism) are the reals themselves, and the complex numbers. This is known as the Gelfand-Mazur theorem. It was proved by Gelfand in 1941, based on the 1938 work of Mazur mentioned above.