Informally, a (vector) norm is a function which assigns a length to vectors.
Any sensible measure of length should satisfy the following basic properties: it should be a convex function of its argument (that is, the length of an average of two vectors should be always less than the average of their lengths); it should be positive-definite (always non-negative, and zero only when the argument is the zero vector), and preserve positive scaling (so that multiplying a vector by a positive number scales its norm accordingly).
Formally, a vector norm is a function [latex]f:\mathbb{R}^n \rightarrow \mathbb{R}[/latex] which satisfies the following properties.
Definition of a vector norm
|
1. Positive homogeneity : for every [latex]x \in \mathbb{R}^n[/latex], [latex]\alpha \ge 0[/latex], we have [latex]f(\alpha x) = \alpha f(x)[/latex]. 2. Triangle inequality : for every [latex]x, y \in \mathbb{R}^n[/latex], we have [latex]\begin{align*} f(x+y) &\leq f(x)+f(y) \end{align*}[/latex] 3. Definiteness : for every [latex]x \in \mathbb{R}^n[/latex], [latex]f(x)=0[/latex] implies [latex]x=0[/latex]. |
A consequence of the first two conditions is that a norm only assumes non-negative values, and that it is convex.
Popular norms include the so-called [latex]l_p[/latex]-norms, where [latex]p=1,2[/latex] or [latex]p=\infty[/latex]:
[latex]\begin{align*} \|x\|_p &:= \left(\sum\limits_{i=1}^{n} |x_i|^p\right)^{1/p}, \end{align*}[/latex]
with the convention that when
[latex]\begin{align*} p = \infty, \quad \|x\|_\infty &= \max_{1\leq i \leq n} |x_i|. \end{align*}[/latex]
