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# 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.

$x= x 0 x 1 T x x 0 x 1$, 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
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
Example 4

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

# 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) $∥f⁢t∥p=∫0T|f⁢t|pdt1p p f t t 0 T f t p 1 p$, $f⁢t∈ L p ( [ 0 , T ] ) ⇒(∥f⁢t∥p<∞) f t L p ( [ 0 , T ] ) p f t$