Field Norm of Quaternion is not Norm
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Theorem
Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion.
Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$.
The field norm of $\mathbf x$:
- $\map n {\mathbf x} := \size {\mathbf x \overline {\mathbf x} }$
is not a norm in the abstract algebraic context of a division ring.
Proof
Each of the norm axioms is examined in turn:
Norm Axiom $\text N 1$: Positive Definiteness
This is proved in Field Norm of Quaternion is Positive Definiteā.
$\Box$
Norm Axiom $\text N 2$: Multiplicativity
This is proved in Field Norm of Quaternion is Multiplicative.
$\Box$
Norm Axiom $\text N 3$: Triangle Inequality
For example:
- $\map n {1 + 1} = 4 > 2 = \map n 1 + \map n 1$
and so Norm Axiom $\text N 3$: Triangle Inequality is not satisfied.
$\Box$
Not all the norm axioms are fulfilled.
Hence the result.
$\blacksquare$