Seminorm

Definition 1. 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:

1. (Absolute homogeneity) For all $\lambda \in K$ and $v \in V$,

   \[\lVert \lambda v\rVert = \lvert \lambda \rvert \lVert v \rVert.\]

2. (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.

Proposition 1. If $(V, \lVert {}\cdot{} \rVert)$ is a seminormed space, then

\[\lVert 0 \rVert = 0.\]

Proposition 2 (Non-Negativity). Let $(V, \lVert {}\cdot{} \rVert)$ be a seminormed space. For all $v \in V$,

\[\lVert v \rVert \geq 0.\]