Definition: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.

Definition

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:      \(\displaystyle \forall x \in V:\)    \(\displaystyle \norm x = 0 \)   \(\displaystyle \iff \)   \(\displaystyle x = \mathbf 0_V \)             
\((\text N 2)\)   $:$   Positive homogeneity:      \(\displaystyle \forall x \in V, \lambda \in R:\)    \(\displaystyle \norm {\lambda x} \)   \(\displaystyle = \)   \(\displaystyle \norm {\lambda}_R \times \norm x \)             
\((\text N 3)\)   $:$   Triangle inequality:      \(\displaystyle \forall x, y \in V:\)    \(\displaystyle \norm {x + y} \)   \(\displaystyle \le \)   \(\displaystyle \norm x + \norm y \)             


Also see