Normed division algebra

In mathematics, a normed division algebra A is a division algebra over the real or complex numbers which is also a normed vector space, with norm || · || satisfying the following property:

$\|xy\| = \|x\| \|y\|$ for all x and y in A.

While the definition allows normed division algebras to be infinite-dimensional, this, in fact, does not occur. The only normed division algebras over the reals (up to isomorphism) are:

• the real numbers, denoted by R
• the complex numbers, denoted by C
• the quaternions, denoted by H
• the octonions, denoted by O,

a result known as Hurwitz's theorem. In all of the above cases, the norm is given by the absolute value. Note that the first three of these are actually associative algebras, while the octonions form an alternative algebra (a weaker form of associativity).

The only associative normed division algebra over the complex numbers are the complex numbers themselves.

Normed division algebras are a special case of composition algebras. Composition algebras are unital algebras with a multiplicative quadratic form. General composition algebras need not be division algebras, however—they may contain zero divisors. Over the real numbers this gives rise to three additional algebras: the split-complex numbers, the split-quaternions, and the split-octonions.

See also

• Cayley–Dickson construction
• Composition algebra