# 15.2 Norms

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This module will define a norm and give examples and properties of it.

## Introduction

This module will explain norms, a mathematical concept that provides a notion of the size of a vector. Specifically, the general definition of a norm will be discussed and discrete time signal norms will be presented.

## Norms

The norm of a vector is a real number that represents the "size" of the vector.

In $\mathbb{R}^{2}$ , we can define a norm to be a vectors geometric length.

$x=\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\end{array}\right)$ , norm $(x)=\sqrt{{x}_{0}^{2}+{x}_{1}^{2}}$

Mathematically, a norm $(·)$ is just a function (taking a vector and returning a real number) that satisfies three rules.

To be a norm, $(·)$ must satisfy:

1. the norm of every vector is positive $\forall x, x\in S\colon (x)> 0$
2. scaling a vector scales the norm by the same amount $(\alpha x)=\left|\alpha \right|(x)$ for all vectors $x$ and scalars $\alpha$
3. Triangle Property: $(x+y)\le (x)+(y)$ for all vectors $x$ , $y$ . "The "size" of the sum of two vectors is less than or equal to the sum of their sizes"

A vector space with a well defined norm is called a normed vector space or normed linear space .

## Examples

$\mathbb{R}^{n}$ (or $\mathbb{C}^{n}$ ), $x=\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\\ \dots \\ {x}_{n-1}\end{array}\right)$ , $(, x)=\sum_{i=0}^{n-1} \left|{x}_{i}\right|$ , $\mathbb{R}^{n}$ with this norm is called ${\ell }^{1}\left(\left[0,n-1\right]\right)$ .

$\mathbb{R}^{n}$ (or $\mathbb{C}^{n}$ ), with norm $(, x)=\sum_{i=0}^{n-1} \left|{x}_{i}\right|^{2}^{\left(\frac{1}{2}\right)}$ , $\mathbb{R}^{n}$ is called ${\ell }^{2}\left(\left[0,n-1\right]\right)$ (the usual "Euclidean"norm).

$\mathbb{R}^{n}$ (or $\mathbb{C}^{n}$ , with norm $()$ x i x i is called ${\ell }^{\infty }\left(\left[0,n-1\right]\right)$

## Spaces of sequences and functions

We can define similar norms for spaces of sequences and functions.

Discrete time signals = sequences of numbers $x(n)=\{\dots , {x}_{-2}, {x}_{-1}, {x}_{0}, {x}_{1}, {x}_{2}, \dots \}$

• $(, x(n))=\sum_{i=()}$ x i , $x(n)\in {\ell }^{1}\left(ℤ\right)\implies ((, x))$
• $(, x(n))=\sum_{i=()} ^{}$ x i 2 1 2 , $x(n)\in {\ell }^{2}\left(ℤ\right)\implies ((, x))$
• $(, x(n))=\sum_{i=()} ^{}$ x i p 1 p , $x(n)\in {\ell }^{p}\left(ℤ\right)\implies ((, x))$
• $()$ x n sup i | x [ i ] | , $x(n)\in {\ell }^{\infty }\left(ℤ\right)\implies (())$ x

For continuous time functions:

• $(, f(t))=\int_{()} \,d t^{}$ f t p 1 p , $f(t)\in {L}^{p}\left(ℝ\right)\implies ((, f(t)))$
• (On the interval) $(, f(t))=\int_{0}^{T} \left|f(t)\right|^{p}\,d t^{\left(\frac{1}{p}\right)}$ , $f(t)\in {L}^{p}\left(\left[0,T\right]\right)\implies ((, f(t)))$

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