Definition:Seminorm
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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 if and only if $\norm {\, \cdot \,}$ satisfies the seminorm 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 | \(\ds \forall x, y \in V:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds \norm x + \norm y \) |
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
- Results about seminorms can be found here.
Sources
- 2011: Graham R. Allan and H. Garth Dales: Introduction to Banach Spaces and Algebras ... (previous) ... (next): $2.1$: Normed Spaces