# Definition:Norm Axioms (Vector Space)

This page is about the norm axioms on a vector space. For other uses, see Norm Axioms.

## Definition

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

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

Let $\norm {\,\cdot\,}: V \to \R_{\ge 0}$ be a norm on $V$.

The norm axioms are the following conditions on $\norm {\,\cdot\,}$ which define $\norm {\,\cdot\,}$ as being a norm:

 $(\text N 1)$ $:$ 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: $\displaystyle \forall x, y \in V:$ $\displaystyle \norm {x + y}$ $\displaystyle \le$ $\displaystyle \norm x + \norm y$