Axiom:Vector Space Norm Axioms

From ProofWiki
Jump to navigation Jump to search

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:      \(\ds \forall x \in V:\)    \(\ds \norm x = 0 \)   \(\ds \iff \)   \(\ds x = \mathbf 0_V \)      
\((\text N 2)\)   $:$   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 presented as

The norm axioms for a vector space are usually defined over the real numbers $\R$ or complex numbers $\C$, and so presented in the form:

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

where $\size \lambda$ denotes either:

the absolute value function on $\R$
the complex modulus on $\C$.

The equivalence of the two definitions is confirmed by:

Absolute Value is Norm
Complex Modulus is Norm.


Also see


Sources