Let $K$ be a field equipped with a valuation $\lvert {}\cdot{} \rvert : K \to \R$ and $V$ be a vector space over $K$. A function $\lVert {}\cdot{} \rVert : V \to \R$ is a seminorm on $V$ if:
(Absolute homogeneity) For all $\lambda \in K$ and $v \in V$,
\[\lVert \lambda v\rVert = \lvert \lambda \rvert \lVert v \rVert.\](Subadditivity) For all $v$, $w \in V$,
\[\lVert v + w \rVert \leq \lVert v \rVert + \lVert w \rVert.\]$(V, \lVert {}\cdot{} \rVert)$ is then called a seminormed space.
If $(V, \lVert {}\cdot{} \rVert)$ is a seminormed space, then
\[\lVert 0 \rVert = 0.\]
Proof. If $V$ is over $K$,
\[\lVert 0 \rVert =% \lVert 0_K \cdot 0 \rVert =% \lvert 0_K \rvert \lVert 0 \rVert =% 0 \cdot \lVert 0 \rVert = 0.\]​
Proposition 2. Non-Negativity.
Let $(V, \lVert {}\cdot{} \rVert)$ be a seminormed space. For all $v \in V$,
\[\lVert v \rVert \geq 0.\]
Proof.
\[2 \lVert v \rVert =% \lVert v \rVert + \lVert v \rVert =% \lVert v \rVert + \lVert -v \rVert \geq% \lVert v - v \rVert =% \lVert 0 \rVert = 0.\]​