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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