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:
- 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.
Read more about Normed Division Algebra: Classification, Hurwitz's Theorem, Composition Algebras
Famous quotes containing the words division and/or algebra:
“Major [William] McKinley visited me. He is on a stumping tour.... I criticized the bloody-shirt course of the canvass. It seems to me to be bad “politics,” and of no use.... It is a stale issue. An increasing number of people are interested in good relations with the South.... Two ways are open to succeed in the South: 1. A division of the white voters. 2. Education of the ignorant. Bloody-shirt utterances prevent division.”
—Rutherford Birchard Hayes (1822–1893)
“Poetry has become the higher algebra of metaphors.”
—José Ortega Y Gasset (1883–1955)