Definition:Trivial Norm

Division Ring
Let $\left({R, +, \circ}\right)$ be a division ring, and denote its zero by $0_R$.

Then the map $\left \Vert{\cdot}\right \Vert: R \to \R_{\ge 0}$ given by:


 * $\left\Vert{x}\right\Vert = \begin{cases}

0 & \text{ if } x = 0_R\\ 1 & \text{ otherwise} \end{cases}$

defines a norm on $R$, called the trivial norm.

Some authors refer to this norm as the trivial absolute value.

That $\left\Vert{\cdot}\right\Vert$ is in fact a norm is proved in Trivial Norm on Division Ring is Norm.

Vector Space
Let $\left({K, +, \circ}\right)$ be a division ring endowed with the trivial norm as defined above.

Let $V$ be a vector space over $K$, with zero $0_V$.

Then the map $\left\Vert{\cdot}\right\Vert: V \to \R_+ \cup \{0\}$ given by:


 * $\left\Vert{x}\right\Vert = \begin{cases}

0 & \text{ if } x = 0_V\\ 1 & \text{ otherwise} \end{cases}$

defines a norm on $V$, called the trivial norm.

That $\left\Vert{\cdot}\right\Vert$ is in fact a norm is proved in Trivial Norm on Vector Space is Norm.

Also see

 * Definition:Discrete Metric