Difference between revisions of "Norm"

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==Norms may define a metric space==
 
==Norms may define a metric space==
 
To get a [[Metric space|metric space]] from a norm simply define <math>d(x,y)=\|x-y\|</math>
 
To get a [[Metric space|metric space]] from a norm simply define <math>d(x,y)=\|x-y\|</math>
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 +
'''HOWEVER:''' It is only true that a normed vector space is a metric space also, given a metric we may ''not'' be able to get an associated norm.
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==Weaker and stronger norms==
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Given a norm <math>\|\cdot\|_1</math> and another <math>\|\cdot\|_2</math> we say:
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* <math>\|\cdot\|_1</math> is weaker than <math>\|\cdot\|_2</math> if <math>\exists C> 0\forall x\in V</math> such that <math>\|x\|_1\le C\|x\|_2</math>
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* <math>\|\cdot\|_2</math> is stronger than <math>\|\cdot\|_1</math>  in this case
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==Equivalence of norms==
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Given two norms <math>\|\cdot\|_1</math> and <math>\|\cdot\|_2</math> on a [[Vector space|vector space]] {{M|V}} we say they are equivalent if:
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<math>\exists c,C\in\mathbb{R}\ \forall x\in V:\ c\|x\|_1\le\|x\|_2\le C\|x\|_1</math>
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 +
We may write this as <math>\|\cdot\|_1\sim\|\cdot\|_2</math> - this is an [[Equivalence relation]]
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{{Todo|proof}}
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Note also that if <math>\|\cdot\|_1</math> is both weaker and stronger than <math>\|\cdot\|_2</math> they are equivalent
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===Examples===
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*Any two norms on <math>\mathbb{R}^n</math> are equivalent
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*The norms <math>\|\cdot\|_{L^1}</math> and <math>\|\cdot\|_\infty</math> on <math>\mathcal{C}([0,1],\mathbb{R})</math> are not equivalent.
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==Common norms==
 
==Common norms==
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|For a [[Linear map|linear isomorphism]] <math>L:U\rightarrow V</math> where V is a normed vector space
 
|For a [[Linear map|linear isomorphism]] <math>L:U\rightarrow V</math> where V is a normed vector space
 
|}
 
|}
 
==Equivalence of norms==
 
Given two norms <math>\|\cdot\|_1</math> and <math>\|\cdot\|_2</math> on a [[Vector space|vector space]] {{M|V}} we say they are equivalent if:
 
 
<math>\exists c,C\in\mathbb{R}\ \forall x\in V:\ c\|x\|_1\le\|x\|_2\le C\|x\|_1</math>
 
 
We may write this as <math>\|\cdot\|_1\sim\|\cdot\|_2</math> - this is an [[Equivalence relation]]
 
{{Todo|proof}}
 
===Examples===
 
*Any two norms on <math>\mathbb{R}^n</math> are equivalent
 
*The norms <math>\|\cdot\|_{L^1}</math> and <math>\|\cdot\|_\infty</math> on <math>\mathcal{C}([0,1],\mathbb{R})</math> are not equivalent.
 
  
 
==Examples==
 
==Examples==
 
* [[Euclidean norm]]
 
* [[Euclidean norm]]
 
{{Definition|Linear Algebra}}
 
{{Definition|Linear Algebra}}

Revision as of 17:52, 21 April 2015

An understanding of a norm is needed to proceed to linear isometries

Normed vector spaces

A normed vector space is a vector space equipped with a norm [math]\|\cdot\|_V[/math], it may be denoted [math](V,\|\cdot\|_V,F)[/math]

Definition

A norm on a vector space [ilmath](V,F)[/ilmath] is a function [math]\|\cdot\|:V\rightarrow\mathbb{R}[/math] such that:

  1. [math]\forall x\in V\ \|x\|\ge 0[/math]
  2. [math]\|x\|=0\iff x=0[/math]
  3. [math]\forall \lambda\in F, x\in V\ \|\lambda x\|=|\lambda|\|x\|[/math] where [math]|\cdot|[/math] denotes absolute value
  4. [math]\forall x,y\in V\ \|x+y\|\le\|x\|+\|y\|[/math] - a form of the triangle inequality

Often parts 1 and 2 are combined into the statement

  • [math]\|x\|\ge 0\text{ and }\|x\|=0\iff x=0[/math] so only 3 requirements will be stated.

I don't like this

Norms may define a metric space

To get a metric space from a norm simply define [math]d(x,y)=\|x-y\|[/math]

HOWEVER: It is only true that a normed vector space is a metric space also, given a metric we may not be able to get an associated norm.

Weaker and stronger norms

Given a norm [math]\|\cdot\|_1[/math] and another [math]\|\cdot\|_2[/math] we say:

  • [math]\|\cdot\|_1[/math] is weaker than [math]\|\cdot\|_2[/math] if [math]\exists C> 0\forall x\in V[/math] such that [math]\|x\|_1\le C\|x\|_2[/math]
  • [math]\|\cdot\|_2[/math] is stronger than [math]\|\cdot\|_1[/math] in this case

Equivalence of norms

Given two norms [math]\|\cdot\|_1[/math] and [math]\|\cdot\|_2[/math] on a vector space [ilmath]V[/ilmath] we say they are equivalent if:

[math]\exists c,C\in\mathbb{R}\ \forall x\in V:\ c\|x\|_1\le\|x\|_2\le C\|x\|_1[/math]

We may write this as [math]\|\cdot\|_1\sim\|\cdot\|_2[/math] - this is an Equivalence relation


TODO: proof



Note also that if [math]\|\cdot\|_1[/math] is both weaker and stronger than [math]\|\cdot\|_2[/math] they are equivalent

Examples

  • Any two norms on [math]\mathbb{R}^n[/math] are equivalent
  • The norms [math]\|\cdot\|_{L^1}[/math] and [math]\|\cdot\|_\infty[/math] on [math]\mathcal{C}([0,1],\mathbb{R})[/math] are not equivalent.


Common norms

Name Norm Notes
Norms on [math]\mathbb{R}^n[/math]
1-norm [math]\|x\|_1=\sum^n_{i=1}|x_i|[/math] it's just a special case of the p-norm.
2-norm [math]\|x\|_2=\sqrt{\sum^n_{i=1}x_i^2}[/math] Also known as the Euclidean norm (see below) - it's just a special case of the p-norm.
p-norm [math]\|x\|_p=\left(\sum^n_{i=1}|x_i|^p\right)^\frac{1}{p}[/math] (I use this notation because it can be easy to forget the [math]p[/math] in [math]\sqrt[p]{}[/math])
[math]\infty-[/math]norm [math]\|x\|_\infty=\sup(\{x_i\}_{i=1}^n)[/math] Also called [math]\infty-[/math]norm
Norms on [math]\mathcal{C}([0,1],\mathbb{R})[/math]
[math]\|\cdot\|_{L^p}[/math] [math]\|f\|_{L^p}=\left(\int^1_0|f(x)|^pdx\right)^\frac{1}{p}[/math] NOTE be careful extending to interval [math][a,b][/math] as proof it is a norm relies on having a unit measure
[math]\infty-[/math]norm [math]\|f\|_\infty=\sup_{x\in[0,1]}(|f(x)|)[/math] Following the same spirit as the [math]\infty-[/math]norm on [math]\mathbb{R}^n[/math]
[math]\|\cdot\|_{C^k}[/math] [math]\|f\|_{C^k}=\sum^k_{i=1}\sup_{x\in[0,1]}(|f^{(i)}|)[/math] here [math]f^{(k)}[/math] denotes the [math]k^\text{th}[/math] derivative.
Induced norms
Pullback norm [math]\|\cdot\|_U[/math] For a linear isomorphism [math]L:U\rightarrow V[/math] where V is a normed vector space

Examples