# 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**:

\((\text N 1)\) | $:$ | Positive Definiteness: | \(\ds \forall x \in V:\) | \(\ds \norm x = 0 \) | \(\ds \iff \) | \(\ds x = \mathbf 0_V \) | |||

\((\text N 2)\) | $:$ | Positive Homogeneity: | \(\ds \forall x \in V, \lambda \in R:\) | \(\ds \norm {\lambda x} \) | \(\ds = \) | \(\ds \norm {\lambda}_R \times \norm x \) | |||

\((\text N 3)\) | $:$ | Triangle Inequality: | \(\ds \forall x, y \in V:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds \norm x + \norm y \) |

## Subcategories

This category has the following 9 subcategories, out of 9 total.

### C

### P

- P-adic Norm is Norm (3 P)
- P-Norms (8 P)

### Q

- Quotient Norms (1 P)

### S

- Supremum Norm is Norm (3 P)

### T

- Taxicab Norm (1 P)
- Trivial Norms (2 P)

## Pages in category "Examples of Norms"

The following 17 pages are in this category, out of 17 total.