TOPICS
Search

Real Normed Algebra

A real normed algebra, also called a composition algebra, is a multiplication * on R^n that respects the length of vectors, i.e., |x*y|=|x|*|y| for x,y in R^n.

The only real normed algebras with a multiplicative identity are the real numbers R, complex numbers C, quaternions H, and octonions O (Koecher and Remmert 1988).

Hurwitz (1898) 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 (Koecher and Remmert 1988).

Real normed algebras have no zero divisors since the equation |x|=0 implies that x=0.

See also

Algebra, Complex Number, Octonion, Normed Space, Quaternion, Real Number, Vector Space

Portions of this entry contributed by Todd Rowland

Portions of this entry contributed by Skip Garibaldi

Explore with Wolfram|Alpha

References

Hurwitz, A. "Ueber die Composition der quadratischen Formen von beliebig vielen Variabeln." Nachr. Königl. Gesell. Wiss. Göttingen. Math.-Phys. Klasse, 309-316, 1898.Koecher, M. and Remmert, R. "Composition Algebras. HURWITZ's Theorem--Vector-Product Algebras." Ch. 10 in Ebbinghaus, H.-D.; Hermes, H.; Hirzebruch, F.; Koecher, M.; Mainzer, K.; Neukirch, J.; Prestel, A.; and Remmert, R. Numbers. New York: Springer-Verlag, pp. 265-280, 1988.

Referenced on Wolfram|Alpha

Real Normed Algebra

Cite this as:

Garibaldi, Skip; Rowland, Todd; and Weisstein, Eric W. "Real Normed Algebra." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RealNormedAlgebra.html

Subject classifications