Definition:Trivial Norm

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

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


 * $\left|{x}\right| = \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 $\left|{\cdot}\right|$ is in fact a norm is proved in Trivial Norm 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 is Norm.

Caution
The trivial norm as referred to on ProofWiki is not the zero norm.

When dealing with the trivial norm, make sure you understand which of these two is meant.