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

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

The basic characteristics of a norm are −

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

• $left | x right |geq 0$

• $left | x right |= 0 Leftrightarrow x= 0forall x in X$

• $left |alpha x right |=left | alpha right |left | x right |forall 😡 in X and :alpha :is :a :scalar$

• $left | x+y right |leq left | x right |+left | y right | forall x,y in X$

• $left | x-y right |geq left | left | x right |-left | y right | right |$

By definition, norm is calculated as follows −

• $left | x right |_1=displaystylesumlimits_{i=1}^nleft | x_i right |$

• $left | x right |_2=left ( displaystylesumlimits_{i=1}^nleft | x_i right |^2 right )^{frac{1}{2}}$

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

Norm is a continuous function.

Proof

By definition, if $x_nrightarrow x$ in $XRightarrow fleft ( x_n right )rightarrow fleft ( x right ) $ then $fleft ( x right )$ is a constant function.

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

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

Since $x_n rightarrow x$ thus, $left | x_n-x right |rightarrow 0$

Therefore $left | fleft ( x_n right )-fleft ( x right ) right |leq 0Rightarrow left | fleft ( x_n right )-fleft ( x right ) right |=0Rightarrow fleft ( x_n right )rightarrow fleft ( x right )$

Hence, norm is a continuous function.