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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 2 , we can define a norm to be a vectors geometric length.

x x 0 x 1 , norm x 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.

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To be a norm, · must satisfy:

  1. the norm of every vector is positive x x S x 0
  2. scaling a vector scales the norm by the same amount α x α x for all vectors x and scalars α
  3. Triangle Property: x y 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

n (or n ), x x 0 x 1 x n - 1 , 1 x i 0 n 1 x i , n with this norm is called 1 ( [ 0 , n - 1 ] ) .

Collection of all x 2 with 1 x 1
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n (or n ), with norm 2 x i 0 n 1 x i 2 1 2 , n is called 2 ( [ 0 , n - 1 ] ) (the usual "Euclidean"norm).

Collection of all x 2 with 2 x 1
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n (or n , with norm x i x i is called ( [ 0 , n - 1 ] )

x 2 with x 1
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Spaces of sequences and functions

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

Discrete time signals = sequences of numbers x n x -2 x -1 x 0 x 1 x 2

  • 1 x n i x i , x n 1 ( ) 1 x
  • 2 x n i x i 2 1 2 , x n 2 ( ) 2 x
  • p x n i x i p 1 p , x n p ( ) p x
  • x n sup i | x [ i ] | , x n ( ) x

For continuous time functions:

  • p f t t f t p 1 p , f t L p ( ) p f t
  • (On the interval) p f t t 0 T f t p 1 p , f t L p ( [ 0 , T ] ) p f t

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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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