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Real Normed Algebra

A real normed algebra is a finite-dimensional real algebra on R^n, for n>0, whose bilinear multiplication * satisfies ||x*y||=||x||||y|| for all x,y in R^n, where ||·|| is the Euclidean norm. A real normed algebra with a multiplicative identity is a positive-definite composition algebra. General composition algebras may instead have an indefinite quadratic form and need not satisfy this positive-definite norm condition.

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 1991).

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 1991).

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

See also

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

Portions of this entry contributed by Todd Rowland

Portions of this entry contributed by Skip Garibaldi

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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, 1991.

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Real Normed Algebra

Cite this as:

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

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