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Norms

Module by: Michael Haag, Justin Romberg. E-mail the authors

Summary: 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.

Example 1

In R2 2 , we can define a norm to be a vectors geometric length.

Figure 1
Figure 1 (norm_f1.png)

x= x 0 x 1 T x x 0 x 1 , norm ∥x∥= x 0 2+ x 1 2 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.

To be a norm, ∥·∥ · must satisfy:

  1. the norm of every vector is positive ∀x,x∈S:∥x∥>0 x x S x 0
  2. scaling a vector scales the norm by the same amount ∥α⁢x∥=|α|⁢∥x∥ α x α x for all vectors x x and scalars α α
  3. Triangle Property: ∥x+y∥≤∥x∥+∥y∥ x y x y for all vectors x x, y 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

Example 2

Rn n (or Cn n ), x= x 0 x 1 … x n - 1 x x 0 x 1 … x n - 1 , ∥x∥1=∑i=0n−1| x i | 1 x i 0 n 1 x i , Rn n with this norm is called ℓ 1 ( [ 0 , n - 1 ] ) ℓ 1 ( [ 0 , n - 1 ] ) .

Figure 2: Collection of all x∈R2 x 2 with ∥x∥1=1 1 x 1
Figure 2 (norm_f2.png)

Example 3

Rn n (or Cn n ), with norm ∥x∥2=∑i=0n−1| x i |212 2 x i 0 n 1 x i 2 1 2 , Rn n is called ℓ 2 ( [ 0 , n - 1 ] ) ℓ 2 ( [ 0 , n - 1 ] ) (the usual "Euclidean"norm).

Figure 3: Collection of all x∈R2 x 2 with ∥x∥2=1 2 x 1
Figure 3 (norm_f3.png)

Example 4

Rn n (or Cn n , with norm ∥x∥∞=maxii| x i | x i x i is called ℓ ∞ ( [ 0 , n - 1 ] ) ℓ ∞ ( [ 0 , n - 1 ] )

Figure 4: x∈R2 x 2 with ∥x∥∞=1 x 1
Figure 4 (norm_f4.png)

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 … x n … x -2 x -1 x 0 x 1 x 2 …

  • ∥x⁢n∥1=∑i=−∞∞|x⁢i| 1 x n i x i , x⁢n∈ ℓ 1 ( ℤ ) ⇒(∥x∥1<∞) x n ℓ 1 ( ℤ ) 1 x
  • ∥x⁢n∥2=∑i=−∞∞|x⁢i|212 2 x n i x i 2 1 2 , x⁢n∈ ℓ 2 ( ℤ ) ⇒(∥x∥2<∞) x n ℓ 2 ( ℤ ) 2 x
  • ∥x⁢n∥p=∑i=−∞∞|x⁢i|p1p p x n i x i p 1 p , x⁢n∈ ℓ p ( ℤ ) ⇒(∥x∥p<∞) x n ℓ p ( ℤ ) p x
  • ∥x⁢n∥∞= sup i | x [ i ] | x n sup i | x [ i ] | , x⁢n∈ ℓ ∞ ( ℤ ) ⇒(∥x∥∞<∞) x n ℓ ∞ ( ℤ ) x

For continuous time functions:

  • ∥f⁢t∥p=∫−∞∞|f⁢t|pdt1p p f t t f t p 1 p , f⁢t∈ L p ( ℝ ) ⇒(∥f⁢t∥p<∞) f t L p ( ℝ ) p f t
  • (On the interval) ∥f⁢t∥p=∫0T|f⁢t|pdt1p p f t t 0 T f t p 1 p , f⁢t∈ L p ( [ 0 , T ] ) ⇒(∥f⁢t∥p<∞) f t L p ( [ 0 , T ] ) p f t

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