Axiom:Non-Archimedean Norm Axioms

From ProofWiki
Jump to navigation Jump to search

Definition

Non-Archimedean Norm Axioms (Division Ring)

Let $\struct {R, +, \circ}$ be a division ring whose zero is $0_R$.

Let $\norm {\, \cdot \,}: R \to \R_{\ge 0}$ be a non-Archimedean norm on $R$.


The non-Archimedean norm axioms are the conditions on $\norm {\, \cdot \,}$ which are satisfied for all elements of $R$ in order for $\norm {\, \cdot \,}$ to be a non-Archimedean norm:

\((\text N 1)\)   $:$   Positive Definiteness:      \(\ds \forall x \in R:\)    \(\ds \norm x = 0 \)   \(\ds \iff \)   \(\ds x = 0_R \)      
\((\text N 2)\)   $:$   Multiplicativity:      \(\ds \forall x, y \in R:\)    \(\ds \norm {x \circ y} \)   \(\ds = \)   \(\ds \norm x \times \norm y \)      
\((\text N 4)\)   $:$   Ultrametric Inequality:      \(\ds \forall x, y \in R:\)    \(\ds \norm {x + y} \)   \(\ds \le \)   \(\ds \max \set {\norm x, \norm y} \)      

Non-Archimedean Norm Axioms (Vector Space)

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

Let $X$ be a vector space over $R$, with zero $0_X$.

Let $\norm {\, \cdot \,}: X \to \R_{\ge 0}$ be a mapping from $X$ to the non-negative reals.


$\norm {\, \cdot \,}$ is satisfies the non-Archimedean norm axioms if and only if $\norm {\, \cdot \,}$ satisfies the following contitions:

\((\text N 1)\)   $:$   Positive Definiteness:      \(\ds \forall x \in X:\)    \(\ds \norm x = 0 \)   \(\ds \iff \)   \(\ds x = 0_R \)      
\((\text N 2)\)   $:$   Positive Homogeneity:      \(\ds \forall x \in X, \lambda \in R:\)    \(\ds \norm {\lambda x} \)   \(\ds = \)   \(\ds \norm {\lambda}_R \times \norm x \)      
\((\text N 4)\)   $:$   Ultrametric Inequality:      \(\ds \forall x, y \in X:\)    \(\ds \norm {x + y} \)   \(\ds \le \)   \(\ds \max \set {\norm x, \norm y} \)