Normed division algebra

Remarks

If $xy = 0$, then $\norm{x}\norm{y} = 0$, so $x = 0$ or $y = 0$. Hence a finite-dimensional normed division algebra is a division algebra. Example: $\C$ with $\norm{z} = \abs{z}$, since $\abs{zw} = \abs{z}\abs{w}$. Non-example: the $2 \times 2$ real matrices with the operator norm satisfy only $\norm{xy} \le \norm{x}\norm{y}$; equality fails for $x = y = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, where $xy = 0$ but $\norm{x}\norm{y} = 1$. In finite dimension the norm comes from an inner product, and the algebra is $\R$, $\C$, the quaternions or the octonions, by Hurwitz's theorem.