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 $0_V$.

Let $\norm {\, \cdot \,}: V \to \R_{\ge 0}$ be a mapping from $V$ to the positive reals $\R_{\ge 0}$.

The mapping $\norm {\, \cdot \,}$ is a seminorm $\norm {\, \cdot \,}$ satisfies the seminorm axioms:

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