Axiom:Seminorm Axioms

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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 $V$ if and only if $\norm {\, \cdot \,}$ satisfies the following axioms:

\((\text N 2)\)   $:$   Positive Homogeneity:      \(\ds \forall x \in V, \lambda \in K:\)    \(\ds \norm {\lambda x} \)   \(\ds = \)   \(\ds \norm \lambda_K \times \norm x \)      
\((\text N 3)\)   $:$   Triangle Inequality:      \(\ds \forall x, y \in V:\)    \(\ds \norm {x + y} \)   \(\ds \le \)   \(\ds \norm x + \norm y \)      

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

  • Results about seminorms can be found here.