Definition:Trivial Norm
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Definition
Division Ring
Let $\struct {R, +, \circ}$ be a division ring, and denote its zero by $0_R$.
Then the map $\norm {\cdot}: R \to \R_{\ge 0}$ given by:
- $\norm x = \begin{cases}
0 & : \text{if $x = 0_R$}\\ 1 & : \text{otherwise}
\end{cases}$
defines a norm on $R$, called the trivial norm.
Vector Space
Let $\struct {K, +, \circ}$ be a division ring endowed with the trivial norm.
Let $V$ be a vector space over $K$, with zero $0_V$.
Then the map $\norm {\cdot}: V \to \R_+ \cup \set 0$ given by:
- $\norm x = \begin{cases}
0 & : \text {if $x = 0_V$} \\ 1 & : \text {otherwise}
\end{cases}$
defines a norm on $V$, called the trivial norm.