Encyclopedia > Normed division algebra

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Normed division algebra

A normed division algebra A is a division algebra over the real or complex numbers that is also a normed vector space and satisfies ||xy|| = ||x||.||y|| for all x and y in A.

The only normed division algebras over the reals (up to isomorphism) are

  • the reals themselves
  • the complex numbers
  • the quaternions
The only normed division algebra over the complex numbers are the complex numbers themselves. In all of the above cases, the norm is given by the absolute value.

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