Definition:Norm on Vector Space/Also presented as

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Norm on Vector Space: 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.


Sources