# 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:

${\displaystyle \|xy\|=\|x\|\|y\|}$ for all x and y in A.[1]

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:[2][3]

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.

## Classification

The classification of real division algebras began with Georg Frobenius,[4] continued with Hurwitz[5] and was set in general form by Max Zorn.[6] A brief historical summary may be found in Badger.[7]

A full proof can be found in Kantor and Solodovnikov,[8] and in Shapiro.[9] As a basic idea, if an algebra A is proportional to 1 then it is isomorphic to the real numbers. Otherwise we extend the subalgebra isomorphic to 1 using the Cayley–Dickson construction and introducing a vector e which is orthogonal to 1. This subalgebra is isomorphic to the complex numbers. If this is not all of A then we once again use the Cayley–Dickson construction and another vector orthogonal to the complex numbers and get a subalgebra isomorphic to the quaternions. If this is not all of A then we double up once again and get a subalgebra isomorphic to the Cayley numbers (or Octonions). We now have a theorem which says that every subalgebra of A that contains 1 and is not A is associative. The Cayley numbers are not associative and therefore must be A.

## Hurwitz's theorem

Hurwitz's theorem (also called the "1,2,4 8 Theorem"), named after Adolf Hurwitz, who proved it in 1898, shows that the product of the sum of n squares by the sum of n squares is the sum of n squares in a bilinear way only when n is equal to 1, 2, 4 or 8.[10] The original proof is for quadratic forms with coefficients taken in C[11] but extends to any field of characteristic not 2.[12]

## Composition algebras

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.

## References

1. Porteous (1969) p.277
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10. Lam (2005) p.130
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