Definition:Seminorm

Definition
Let $\left({K, +, \circ}\right)$ be a division ring with norm $\left|{\cdot}\right|_K$.

Let $V$ be a vector space over $K$, with zero $0_V$.

A seminorm on $V$ is a map from $V$ to the positive reals $\left\Vert{\cdot}\right\Vert: V \to \R_{\ge 0}$ satisfying the following properties (for all $x,y \in V$ and $\lambda \in K$):

These may be referred to as the seminorm axioms.

The N2 and N3 markings originate from the fact that these axioms are also used in defining norms.

Also see

 * Definition:Norm on Vector Space