Difference between revisions of "Norm"

	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	metric space; topological space;
contains all	inner product spaces;
Related objects
Induced metric	[ilmath]d_{\Vert\cdot\Vert}:V\times V\rightarrow\mathbb{R}_{\ge 0}[/ilmath]; [ilmath]d_{\Vert\cdot\Vert}:(x,y)\mapsto\Vert x-y\Vert[/ilmath];
Induced by inner product	[ilmath]\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:V\rightarrow\mathbb{R}_{\ge 0}[/ilmath]; [ilmath]\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:x\mapsto\sqrt{\langle x,x\rangle}[/ilmath];

Latest revision as of 20:33, 9 April 2017

A norm is a an abstraction of the notion of the "length of a vector". Every norm is a metric and every inner product is a norm (see Subtypes of topological spaces for more information), thus every normed vector space is a topological space to, so all the topology theorems apply. Norms are especially useful in functional analysis and also for differentiation.

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

Euclidean norm

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

[5] 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]!

[6] 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]

[APIKM-1] 1.0 ^1.1 ^1.2 Analysis - Part 1: Elements - Krzysztof Maurin

[FA-2] 2.0 ^2.1 Functional Analysis - George Bachman and Lawrence Narici

[FAAGI-3] Functional Analysis - A Gentle Introduction - Volume 1, by Dzung Minh Ha

[RAAAHS-4] Real and Abstract Analysis - Edwin Hewitt & Karl Stromberg

[1]

[2]

[3]

[4]

[Note 1]

[Note 2]

@@ Line 1: / Line 1: @@
+{{:Norm/Heading}}
+__TOC__
 ==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>\|x\|=0\iff x=0</math>
@@ Line 6: / Line 9: @@
 # <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.
-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.}}
-==Common norms==
+===Metric induced by a norm===
-===The 1-norm===
+To get a [[Metric space|metric space]] from a norm simply define<ref name="FA"/><ref name="APIKM"/>:
-<math>\|x\|_1=\sum^n_{i=1}|x_i|</math> - it's just a special case of the p-norm.
+* <math>d(x,y):=\|x-y\|</math>
-===The 2-norm===
+(See [[Subtypes of topological spaces]] for more information, this relationship is very important in [[Functional analysis]])
-<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.
+{{Todo|Move to its own page and do a proof (trivial)}}
-===The 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>)
-===The supremum-norm===
-Also called <math>\infty-</math>norm<br/>
-<math>\|x\|_\infty=\sup(\{x_i\}_{i=1}^n)</math>
-==Examples==
+==Weaker and stronger norms==
-===The Euclidean Norm===
+Given a norm <math>\|\cdot\|_1</math> and another <math>\|\cdot\|_2</math> we say:
-The Euclidean norm is denoted <math>\|\cdot\|_2</math>
+* <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:
-Here for <math>x\in\mathbb{R}^n</math> we have:
+<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>
-<math>\|x\|_2=\sqrt{\sum^n_{i=1}x_i^2}</math>
+{{Begin Theorem}}
-====Proof that it is a norm====
+Theorem: This is an [[Equivalence relation]] - so we may write this as <math>\|\cdot\|_1\sim\|\cdot\|_2</math>
+{{Begin Proof}}
 {{Todo|proof}}
-=====Part 4 - Triangle inequality=====
+{{End Proof}}
-Let <math>x,y\in\mathbb{R}^n</math>
+{{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.
-<math>\|x+y\|_2^2=\sum^n_{i=1}(x_i+y_i)^2</math>
+==Common norms==
-<math>=\sum^n_{i=1}x_i^2+2\sum^n_{i=1}x_iy_i+\sum^n_{i=1}y_i^2</math>
+{| class="wikitable" border="1"
-<math>\le\sum^n_{i=1}x_i^2+2\sqrt{\sum^n_{i=1}x_i^2}\sqrt{\sum^n_{i=1}y_i^2}+\sum^n_{i=1}y_i^2</math> using the [[Cauchy-Schwarz inequality]]
+|-
+! Name
+! Norm
+! Notes
+|-
+!colspan="3"|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<br/>
+|-
+!colspan="3"|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.
+|-
+!colspan="3"|Induced norms
+|-
+| [[Pullback norm]]
+|<math>\|\cdot\|_U</math>
+|For a [[Linear map|linear isomorphism]] <math>L:U\rightarrow V</math> where V is a normed vector space
+|}
-<math>=\left(\sqrt{\sum^n_{i=1}x_i^2}+\sqrt{\sum^n_{i=1}y_i^2}\right)^2</math>
+==Examples==
-<math>=\left(\|x\|_2+\|y\|_2\right)^2</math>
+* [[Euclidean norm]]
-Thus we see: <math>\|x+y\|_2^2\le\left(\|x\|_2+\|y\|_2\right)^2</math>, as norms are always <math>\ge 0</math> we see:
-<math>\|x+y\|_2\le\|x\|_2+\|y\|_2</math> - as required.
-{{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]]

Difference between revisions of "Norm"

Latest revision as of 20:33, 9 April 2017

Contents

Definition

Properties

Terminology

Relation to various subtypes of topological spaces

Relation to inner product

Metric induced by a norm

Weaker and stronger norms

Equivalence of norms

Examples

Common norms

Examples

Notes

References

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relation to other topological spaces
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]
is a	metric space topological space
contains all	inner product spaces
Related objects
Induced metric	[ilmath]d_{\Vert\cdot\Vert}:V\times V\rightarrow\mathbb{R}_{\ge 0}[/ilmath] [ilmath]d_{\Vert\cdot\Vert}:(x,y)\mapsto\Vert x-y\Vert[/ilmath]
Induced by inner product	[ilmath]\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:V\rightarrow\mathbb{R}_{\ge 0}[/ilmath] [ilmath]\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:x\mapsto\sqrt{\langle x,x\rangle}[/ilmath]