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Norms

Module by: Michael Haag, Justin Romberg

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

Introduction

Much of the language in this section will be familiar to you - you should have previously been exposed to the concepts of in the context of ℝn n . We're going to take what we know about vectors and apply it to functions (continuous time signals).

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.
norm_f1.png
Figure 1
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 ∥αx∥=|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 ] ) .
norm_f2.png
Figure 2: Collection of all x∈ℝ2 x 2 with ∥x∥1=1 1 x 1
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).
norm_f3.png
Figure 3: Collection of all x∈ℝ2 x 2 with ∥x∥2=1 2 x 1
Example 4 
ℝn n (or ℂn n , with norm ∥x∥∞=maxi{| x i |} x i x i is called ℓ ∞ ( [ 0 , n - 1 ] ) ℓ ∞ ( [ 0 , n - 1 ] )
norm_f4.png
Figure 4: x∈ℝ2 x 2 with ∥x∥∞=1 x 1

Spaces of Sequences and Functions

We can define similar norms for spaces of sequences and functions.
Discrete time signals = sequences of numbers xn=… x -2 x -1 x 0 x 1 x 2 … x n … x -2 x -1 x 0 x 1 x 2 …
  • ∥xn∥1=∑i=-∞∞|xi| 1 x n i x i , xn∈ ℓ 1 ( ℤ ) ⇒∥x∥1<∞ x n ℓ 1 ( ℤ ) 1 x
  • ∥xn∥2=∑i=-∞∞|xi|212 2 x n i x i 2 1 2 , xn∈ ℓ 2 ( ℤ ) ⇒∥x∥2<∞ x n ℓ 2 ( ℤ ) 2 x
  • ∥xn∥p=∑i=-∞∞|xi|p1p p x n i x i p 1 p , xn∈ ℓ p ( ℤ ) ⇒∥x∥p<∞ x n ℓ p ( ℤ ) p x
  • ∥xn∥∞= sup i | x [ i ] | x n sup i | x [ i ] | , xn∈ ℓ ∞ ( ℤ ) ⇒∥x∥∞<∞ x n ℓ ∞ ( ℤ ) x
For continuous time functions:
  • ∥ft∥p=∫-∞∞|ft|pdt1p p f t t f t p 1 p , ft∈ L p ( ℝ ) ⇒∥ft∥p<∞ f t L p ( ℝ ) p f t
  • (On the interval) ∥ft∥p=∫0T|ft|pdt1p p f t t 0 T f t p 1 p , ft∈ L p ( [ 0 , T ] ) ⇒∥ft∥p<∞ f t L p ( [ 0 , T ] ) p f t

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