# Category:Seminorms

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This category contains results about **Seminorms**.

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: | \(\ds \forall x, y \in V:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds \norm x + \norm y \) |

## Pages in category "Seminorms"

The following 15 pages are in this category, out of 15 total.