composition algebra

A real normed algebra, also called a composition algebra, is a multiplication * on R^n that respects the length of vectors, i.e., left bracketing bar x*y right bracketing bar = left bracketing bar x right bracketing bar * left bracketing bar y right bracketing bar for x, y element R^n. The only real normed algebras with a multiplicative identity are the real numbers R, complex numbers C, quaternions H, and octonions O. Hurwitz proved that a real normed algebra must have dimension n = 1, 2, 4, or 8. There are four real normed algebras of dimension 2: the complex numbers and three others.

algebra | complex number | normed space | octonion | quaternion | real number | vector space