# Category:Definitions/Examples of Norms

This category contains definitions of 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: $\displaystyle \forall x \in V:$ $\displaystyle \norm x = 0$ $\displaystyle \iff$ $\displaystyle x = \mathbf 0_V$ $(\text N 2)$ $:$ Positive homogeneity: $\displaystyle \forall x \in V, \lambda \in R:$ $\displaystyle \norm {\lambda x}$ $\displaystyle =$ $\displaystyle \norm {\lambda}_R \times \norm x$ $(\text N 3)$ $:$ 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 "Definitions/Examples of Norms"

The following 13 pages are in this category, out of 13 total.