Axiom:Quasinorm Axioms
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Definition
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: | \(\ds \exists M \ge 1, \forall x, y \in V:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds M \paren {\norm x + \norm y} \) |