Home > Normed division algebra

||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.

See also: Cayley-Dickson constructionIn mathematics, the Cayley-Dickson construction produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by this process are known as Cayley-Dickson algebras since they extend