Axiom:Norm Axioms (Vector Space)

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This page is about the norm axioms on a vector space. For other uses, see Norm Axioms.


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

Let $V$ be a vector space over $R$, with zero $\mathbf 0_V$.

Let $\norm {\,\cdot\,}: V \to \R_{\ge 0}$ be a norm on $V$.

The norm axioms are the following conditions on $\norm {\,\cdot\,}$ which define $\norm {\,\cdot\,}$ as being a norm:

\((\text N 1)\)   $:$   Positive Definiteness:      \(\ds \forall x \in V:\)    \(\ds \norm x = 0 \)   \(\ds \iff \)   \(\ds x = \mathbf 0_V \)      
\((\text N 2)\)   $:$   Positive Homogeneity:      \(\ds \forall x \in V, \lambda \in R:\)    \(\ds \norm {\lambda x} \)   \(\ds = \)   \(\ds \norm {\lambda}_R \times \norm x \)      
\((\text N 3)\)   $:$   Triangle Inequality:      \(\ds \forall x, y \in V:\)    \(\ds \norm {x + y} \)   \(\ds \le \)   \(\ds \norm x + \norm y \)      

Also see