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 $K$,
\[\lVert 0\rVert = \lVert 0_K\cdot 0\rVert = \lvert 0_K\rvert\lVert 0\rVert = 0\cdot\lVert 0\rVert = 0.\]
Let $(V,\lVert{}\cdot{}\rVert)$ be a seminormed space. For all $v\in V$,
\[\lVert v\rVert \geq 0.\]
Proof.
\[2\lVert v\rVert = \lVert v\rVert + \lVert v\rVert = \lVert v\rVert + \lVert -v\rVert \geq \lVert v-v\rVert = \lVert 0\rVert = 0.\]