Axiom:Seminorm Axioms

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 on $V$ $\norm {\, \cdot \,}$ satisfies the following axioms:

These criteria are called 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 see

 * Definition:Seminorm


 * Definition:Norm on Vector Space