# Definition:Trivial Norm/Vector Space

## Definition

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.