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== | ||
| + | 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 | ||
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| + | ==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}} | ||
| + | |||
| + | 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== | ==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 | ||
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==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
Contents
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:
- [math]\forall x\in V\ \|x\|\ge 0[/math]
- [math]\|x\|=0\iff x=0[/math]
- [math]\forall \lambda\in F, x\in V\ \|\lambda x\|=|\lambda|\|x\|[/math] where [math]|\cdot|[/math] denotes absolute value
- [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 |
