Difference between revisions of "Norm"
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==Definition== | ==Definition== | ||
| − | A norm on a [[Vector space|vector space]] {{M|(V,F)}} is a function <math>\|\cdot\|:V\rightarrow\mathbb{R}</math> such that: | + | A norm on a [[Vector space|vector space]] {{M|(V,F)}} (where {{M|F}} is either {{M|\mathbb{R} }} or {{M|\mathbb{C} }}) is a function <math>\|\cdot\|:V\rightarrow\mathbb{R}</math> such that{{rAPIKM}}<ref name="FA">Functional Analysis - George Bachman and Lawrence Narici</ref><ref name="FAAGI">Functional Analysis - A Gentle Introduction - Volume 1, by Dzung Minh Ha</ref>{{RRAAAHS}}<sup>{{Highlight|See warning notes:<ref group="Note">A lot of books, including the brilliant [[Books:Analysis - Part 1: Elements - Krzysztof Maurin|Analysis - Part 1: Elements - Krzysztof Maurin]] referenced here state ''explicitly'' that it is possible for {{M|\Vert\cdot,\cdot\Vert:V\rightarrow\mathbb{C} }} they are wrong. I assure you that it is {{M|\Vert\cdot\Vert:V\rightarrow\mathbb{R}_{\ge 0} }}. Other than this the references are valid, note that this is 'obvious' as if the image of {{M|\Vert\cdot\Vert}} could be in {{M|\mathbb{C} }} then the {{M|\Vert x\Vert\ge 0}} would make no sense. What ordering would you use? The [[canonical]] ordering used for the product of 2 spaces ({{M|\mathbb{R}\times\mathbb{R} }} in this case) is the [[Lexicographic ordering]] which would put {{M|1+1j\le 1+1000j}}!</ref><ref group="Note">The other mistake books make is saying explicitly that the [[field of a vector space]] needs to be {{M|\mathbb{R} }}, it may commonly be {{M|\mathbb{R} }} but it does not ''need'' to be {{M|\mathbb{R} }}</ref>}}</sup>: |
# <math>\forall x\in V\ \|x\|\ge 0</math> | # <math>\forall x\in V\ \|x\|\ge 0</math> | ||
# <math>\|x\|=0\iff x=0</math> | # <math>\|x\|=0\iff x=0</math> | ||
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# <math>\forall x,y\in V\ \|x+y\|\le\|x\|+\|y\|</math> - a form of the [[Triangle inequality|triangle inequality]] | # <math>\forall x,y\in V\ \|x+y\|\le\|x\|+\|y\|</math> - a form of the [[Triangle inequality|triangle inequality]] | ||
| − | Often parts 1 and 2 are combined into the statement | + | 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. | * <math>\|x\|\ge 0\text{ and }\|x\|=0\iff x=0</math> so only 3 requirements will be stated. | ||
| − | I don't like this | + | I don't like this (inline with the [[Doctrine of monotonic definition]]) |
| + | ==Properties== | ||
| + | * [[The norm of a space is a uniformly continuous map with respect to the topology it induces]] - {{M|\Vert\cdot\Vert:X\rightarrow\mathbb{R} }} is a [[uniformly continuous]] map. | ||
| + | ==Terminology== | ||
| + | Such a vector space equipped with such a function is called a [[Normed space|normed space]]<ref name="APIKM"/> | ||
| + | ==Relation to various [[subtypes of topological spaces]]== | ||
| + | The reader should note that: | ||
| + | * Every [[inner product]] induces a ''norm'' and | ||
| + | * Every ''norm'' induces a [[metric]] | ||
| + | These are outlined below | ||
| + | ===Relation to [[inner product]]=== | ||
| + | Every [[inner product]] {{M|\langle\cdot,\cdot\rangle:V\times V\rightarrow(\mathbb{R}\text{ or }\mathbb{C})}} induces a ''norm'' given by: | ||
| + | * {{M|1=\Vert x\Vert:=\sqrt{\langle x,x\rangle} }} | ||
| + | {{Todo|see [[inner product#Norm induced by|inner product (norm induced by)]] for more details, on that page is a proof that {{M|\langle x,x\rangle\ge 0}}, this needs its own page with a proof.}} | ||
| − | == | + | ===Metric induced by a norm=== |
| − | 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<ref name="FA"/><ref name="APIKM"/>: |
| + | * <math>d(x,y):=\|x-y\|</math> | ||
| + | (See [[Subtypes of topological spaces]] for more information, this relationship is very important in [[Functional analysis]]) | ||
| + | {{Todo|Move to its own page and do a proof (trivial)}} | ||
| + | |||
| + | ==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|vector space]] {{M|V}} we say they are equivalent if: | ||
| + | |||
| + | <math>\exists c,C\in\mathbb{R}\text{ with }c,C>0\ \forall x\in V:\ c\|x\|_1\le\|x\|_2\le C\|x\|_1</math> | ||
| + | |||
| + | {{Begin Theorem}} | ||
| + | Theorem: This is an [[Equivalence relation]] - so we may write this as <math>\|\cdot\|_1\sim\|\cdot\|_2</math> | ||
| + | {{Begin Proof}} | ||
| + | {{Todo|proof}} | ||
| + | {{End Proof}} | ||
| + | {{End Theorem}} | ||
| + | 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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| 2-norm | | 2-norm | ||
|<math>\|x\|_2=\sqrt{\sum^n_{i=1}x_i^2}</math> | |<math>\|x\|_2=\sqrt{\sum^n_{i=1}x_i^2}</math> | ||
| − | | Also known as the Euclidean norm | + | | Also known as the [[Euclidean norm]] - it's just a special case of the p-norm. |
|- | |- | ||
| p-norm | | p-norm | ||
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| <math>\infty-</math>norm | | <math>\infty-</math>norm | ||
|<math>\|x\|_\infty=\sup(\{x_i\}_{i=1}^n)</math> | |<math>\|x\|_\infty=\sup(\{x_i\}_{i=1}^n)</math> | ||
| − | |Also called | + | |Also called sup-norm<br/> |
|- | |- | ||
!colspan="3"|Norms on <math>\mathcal{C}([0,1],\mathbb{R})</math> | !colspan="3"|Norms on <math>\mathcal{C}([0,1],\mathbb{R})</math> | ||
|- | |- | ||
| <math>\|\cdot\|_{L^p}</math> | | <math>\|\cdot\|_{L^p}</math> | ||
| − | | <math>\|f\|_{L^p}=\left(\int^1_0|f(x)|^pdx\right)</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 | | '''NOTE''' be careful extending to interval <math>[a,b]</math> as proof it is a norm relies on having a unit measure | ||
|- | |- | ||
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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}} | + | |
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
| + | ==References== | ||
| + | <references/> | ||
| + | {{Normed and Banach spaces navbox}} | ||
| + | {{Metric spaces navbox}} | ||
| + | {{Topology navbox}} | ||
| + | {{Definition|Linear Algebra|Topology|Metric Space|Functional Analysis}} | ||
| + | [[Category:Exemplary pages]] | ||
| + | [[Category:First-year friendly]] | ||
Latest revision as of 20:33, 9 April 2017
| Norm | |
| [ilmath]\Vert\cdot\Vert:V\rightarrow\mathbb{R}_{\ge 0} [/ilmath] Where [ilmath]V[/ilmath] is a vector space over the field [ilmath]\mathbb{R} [/ilmath] or [ilmath]\mathbb{C} [/ilmath] | |
| relation to other topological spaces | |
|---|---|
| is a | |
| contains all | |
| Related objects | |
| Induced metric |
|
| Induced by inner product |
|
Contents
Definition
A norm on a vector space [ilmath](V,F)[/ilmath] (where [ilmath]F[/ilmath] is either [ilmath]\mathbb{R} [/ilmath] or [ilmath]\mathbb{C} [/ilmath]) is a function [math]\|\cdot\|:V\rightarrow\mathbb{R}[/math] such that[1][2][3][4]See warning notes:[Note 1][Note 2]:
- [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 (inline with the Doctrine of monotonic definition)
Properties
- The norm of a space is a uniformly continuous map with respect to the topology it induces - [ilmath]\Vert\cdot\Vert:X\rightarrow\mathbb{R} [/ilmath] is a uniformly continuous map.
Terminology
Such a vector space equipped with such a function is called a normed space[1]
Relation to various subtypes of topological spaces
The reader should note that:
- Every inner product induces a norm and
- Every norm induces a metric
These are outlined below
Relation to inner product
Every inner product [ilmath]\langle\cdot,\cdot\rangle:V\times V\rightarrow(\mathbb{R}\text{ or }\mathbb{C})[/ilmath] induces a norm given by:
- [ilmath]\Vert x\Vert:=\sqrt{\langle x,x\rangle}[/ilmath]
TODO: see inner product (norm induced by) for more details, on that page is a proof that [ilmath]\langle x,x\rangle\ge 0[/ilmath], this needs its own page with a proof.
Metric induced by a norm
To get a metric space from a norm simply define[2][1]:
- [math]d(x,y):=\|x-y\|[/math]
(See Subtypes of topological spaces for more information, this relationship is very important in Functional analysis)
TODO: Move to its own page and do a proof (trivial)
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}\text{ with }c,C>0\ \forall x\in V:\ c\|x\|_1\le\|x\|_2\le C\|x\|_1[/math]
Theorem: This is an Equivalence relation - so we may write this as [math]\|\cdot\|_1\sim\|\cdot\|_2[/math]
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 - 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 sup-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
Notes
- ↑ A lot of books, including the brilliant Analysis - Part 1: Elements - Krzysztof Maurin referenced here state explicitly that it is possible for [ilmath]\Vert\cdot,\cdot\Vert:V\rightarrow\mathbb{C} [/ilmath] they are wrong. I assure you that it is [ilmath]\Vert\cdot\Vert:V\rightarrow\mathbb{R}_{\ge 0} [/ilmath]. Other than this the references are valid, note that this is 'obvious' as if the image of [ilmath]\Vert\cdot\Vert[/ilmath] could be in [ilmath]\mathbb{C} [/ilmath] then the [ilmath]\Vert x\Vert\ge 0[/ilmath] would make no sense. What ordering would you use? The canonical ordering used for the product of 2 spaces ([ilmath]\mathbb{R}\times\mathbb{R} [/ilmath] in this case) is the Lexicographic ordering which would put [ilmath]1+1j\le 1+1000j[/ilmath]!
- ↑ The other mistake books make is saying explicitly that the field of a vector space needs to be [ilmath]\mathbb{R} [/ilmath], it may commonly be [ilmath]\mathbb{R} [/ilmath] but it does not need to be [ilmath]\mathbb{R} [/ilmath]
References
- ↑ 1.0 1.1 1.2 Analysis - Part 1: Elements - Krzysztof Maurin
- ↑ 2.0 2.1 Functional Analysis - George Bachman and Lawrence Narici
- ↑ Functional Analysis - A Gentle Introduction - Volume 1, by Dzung Minh Ha
- ↑ Real and Abstract Analysis - Edwin Hewitt & Karl Stromberg
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