Convex Optimization - Norm

A norm is a function that gives a strictly positive value to a vector or a variable.

Norm is a function $f:mathbb{R}^n ightarrow mathbb{R}$

The basic characteristics of a norm are −

Let $X$ be a vector such that $Xin mathbb{R}^n$

$left | x ight |geq 0$

$left | x ight |= 0 Leftrightarrow x= 0forall x in X$

$left |alpha x ight |=left | alpha ight |left | x ight |forall :x in X and :alpha :is :a :scalar$

$left | x+y ight |leq left | x ight |+left | y ight | forall x,y in X$

$left | x-y ight |geq left | left | x ight |-left | y ight | ight |$

By definition, norm is calculated as follows −

$left | x ight |_1=displaystylesumpmits_{i=1}^nleft | x_i ight |$

$left | x ight |_2=left ( displaystylesumpmits_{i=1}^nleft | x_i ight |^2 ight )^{frac{1}{2}}$

$left | x ight |_p=left ( displaystylesumpmits_{i=1}^nleft | x_i ight |^p ight )^{frac{1}{p}},1 leq p leq infty$

Norm is a continuous function.

Proof

By definition, if $x_n ightarrow x$ in $XRightarrow fleft ( x_n ight ) ightarrow fleft ( x ight ) $ then $fleft ( x ight )$ is a constant function.

Let $fleft ( x ight )=left | x ight |$

Therefore, $left | fleft ( x_n ight )-fleft ( x ight ) ight |=left | left | x_n ight | -left | x ight | ight |leq left | left | x_n-x ight | : ight |$

Since $x_n ightarrow x$ thus, $left | x_n-x ight | ightarrow 0$

Therefore $left | fleft ( x_n ight )-fleft ( x ight ) ight |leq 0Rightarrow left | fleft ( x_n ight )-fleft ( x ight ) ight |=0Rightarrow fleft ( x_n ight ) ightarrow fleft ( x ight )$

Hence, norm is a continuous function.