# Category:Examples of Norms

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This category contains examples of norms.

Let $\struct {R, +, \circ}$ be a division ring with norm $\norm{\,\cdot\,}_R$.

Let $V$ be a vector space over $R$, with zero $0_V$.

A **norm** on $V$ is a map from $V$ to the nonnegative reals:

- $\norm{\,\cdot\,}: V \to \R_{\ge 0}$

satisfying the (vector space) norm axioms:

\((N1)\) | $:$ | Positive definiteness: | \(\displaystyle \forall x \in V:\) | \(\displaystyle \norm x = 0 \) | \(\displaystyle \iff \) | \(\displaystyle x = \mathbf 0_V \) | ||

\((N2)\) | $:$ | Positive homogeneity: | \(\displaystyle \forall x \in V, \lambda \in R:\) | \(\displaystyle \norm {\lambda x} \) | \(\displaystyle = \) | \(\displaystyle \norm {\lambda}_R \times \norm x \) | ||

\((N3)\) | $:$ | Triangle inequality: | \(\displaystyle \forall x, y \in V:\) | \(\displaystyle \norm {x + y} \) | \(\displaystyle \le \) | \(\displaystyle \norm x + \norm y \) |

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Examples of Norms"

The following 6 pages are in this category, out of 6 total.