Axiom:Quasinorm Axioms

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Let $\struct {R, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_R$.

Let $V$ be a vector space over $R$.

Let $\norm {\,\cdot\,}: V \to \R_{\ge 0}$ be a mapping from $V$ to the nonnegative reals $\R_{\ge 0}$.

$\norm{\,\cdot\,}$ satisfies the quasinorm axioms on $V$ if and only if $\norm{\,\cdot\,}$ satisifes the axioms:

\((\text Q 1)\)   $:$   Positive Homogeneity:      \(\ds \forall x \in V, \lambda \in R:\)    \(\ds \norm {\lambda x} \)   \(\ds = \)   \(\ds \norm {\lambda}_R \times \norm x \)      
\((\text Q 2)\)   $:$   Weak Triangle Inequality:      \(\ds \exists M \ge 1, \forall x, y \in V:\)    \(\ds \norm {x + y} \)   \(\ds \le \)   \(\ds M \paren {\norm x + \norm y} \)      

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