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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.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

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.

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 ∥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 ] ) .

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∥∞=maxi{| 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 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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