Definition:Seminorm

Definition
Let $\struct {K, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_K$.

Let $V$ be a vector space over $\struct {K, \norm {\,\cdot\,}_K}$, with zero vector $0_V$.

A seminorm on $V$ is a map from $V$ to the positive reals $\norm {\, \cdot \,}: V \to \R_{\ge 0}$ satisfying the following properties:

These may be referred to as the seminorm axioms.

The $(\text N 2)$ and $(\text N 3)$ labels originate from the fact that these axioms are also used in defining norms.

Also defined as
It is usual to define a seminorm when $K$ is $\R$ or $\C$.

In this context, $\norm {\,\cdot\,}_\R$ is the absolute value and $\norm {\,\cdot\,}_\C$ is the modulus.

Also see

 * Definition:Norm on Vector Space