Difference between revisions of "Norm"

Revision as of 10:45, 8 January 2016

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 $(V,F)$ (where $F$ is either $\mathbb{R}$ or $\mathbb{C}$ ) is a function

$\|\cdot\|:V\rightarrow\mathbb{R}$ such that^[1]^[2]^[3]^[4]^{See warning notes:^{[Note 1]}^{[Note 2]}}:

$\forall x\in V\ \|x\|\ge 0$
$\|x\|=0\iff x=0$
$\forall \lambda\in F, x\in V\ \|\lambda x\|=|\lambda|\|x\|$ where $|\cdot|$ denotes absolute value
$\forall x,y\in V\ \|x+y\|\le\|x\|+\|y\|$ - a form of the triangle inequality

Often parts 1 and 2 are combined into the statement:

$\|x\|\ge 0\text{ and }\|x\|=0\iff x=0$ so only 3 requirements will be stated.

I don't like this (inline with the Doctrine of monotonic definition)

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 $\langle\cdot,\cdot\rangle:V\times V\rightarrow(\mathbb{R}\text{ or }\mathbb{C})$ induces a norm given by:

$\Vert x\Vert:=\sqrt{\langle x,x\rangle}$

TODO: see inner product (norm induced by) for more details, on that page is a proof that $\langle x,x\rangle\ge 0$ - I cannot think of any complex norms!

Metric induced by a norm

To get a metric space from a norm simply define^[2]^[1]:

$d(x,y):=\|x-y\|$

(See Subtypes of topological spaces for more information, this relationship is very important in Functional analysis)

TODO: Some sort of proof this is never complex

Weaker and stronger norms

Given a norm

$\|\cdot\|_1$ and another

$\|\cdot\|_2$ we say:

$\|\cdot\|_1$ is weaker than $\|\cdot\|_2$ if $\exists C> 0\forall x\in V$ such that $\|x\|_1\le C\|x\|_2$
$\|\cdot\|_2$ is stronger than $\|\cdot\|_1$ in this case

Equivalence of norms

Given two norms

$\|\cdot\|_1$ and

$\|\cdot\|_2$ on a vector space

$V$ we say they are equivalent if:

$\exists c,C\in\mathbb{R}\text{ with }c,C>0\ \forall x\in V:\ c\|x\|_1\le\|x\|_2\le C\|x\|_1$

[Expand]
Theorem: This is an Equivalence relation - so we may write this as
$\|\cdot\|_1\sim\|\cdot\|_2$

Note also that if

$\|\cdot\|_1$ is both weaker and stronger than

$\|\cdot\|_2$ they are equivalent

Examples

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

Common norms

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

Examples

Euclidean norm

Notes

Jump up ↑ A lot of books, including the brilliant Analysis - Part 1: Elements - Krzysztof Maurin referenced here state explicitly that it is possible for $\Vert\cdot,\cdot\Vert:V\rightarrow\mathbb{C}$ they are wrong. I assure you that it is $\Vert\cdot\Vert:V\rightarrow\mathbb{R}_{\ge 0}$ . Other than this the references are valid, note that this is 'obvious' as if the image of $\Vert\cdot\Vert$ could be in $\mathbb{C}$ then the $\Vert x\Vert\ge 0$ would make no sense. What ordering would you use? The canonical ordering used for the product of 2 spaces ( $\mathbb{R}\times\mathbb{R}$ in this case) is the Lexicographic ordering which would put $1+1j\le 1+1000j$ !
Jump up ↑ The other mistake books make is saying explicitly that the field of a vector space needs to be $\mathbb{R}$ , it may commonly be $\mathbb{R}$ but it does not need to be $\mathbb{R}$

References

↑ ^{Jump up to: 1.0} ^1.1 ^1.2 Analysis - Part 1: Elements - Krzysztof Maurin
↑ ^{Jump up to: 2.0} ^2.1 Functional Analysis - George Bachman and Lawrence Narici
Jump up ↑ Functional Analysis - A Gentle Introduction - Volume 1, by Dzung Minh Ha
Jump up ↑ Real and Abstract Analysis - Edwin Hewitt & Karl Stromberg

[5] Jump up ↑ A lot of books, including the brilliant Analysis - Part 1: Elements - Krzysztof Maurin referenced here state explicitly that it is possible for $\Vert\cdot,\cdot\Vert:V\rightarrow\mathbb{C}$ they are wrong. I assure you that it is $\Vert\cdot\Vert:V\rightarrow\mathbb{R}_{\ge 0}$ . Other than this the references are valid, note that this is 'obvious' as if the image of $\Vert\cdot\Vert$ could be in $\mathbb{C}$ then the $\Vert x\Vert\ge 0$ would make no sense. What ordering would you use? The canonical ordering used for the product of 2 spaces ( $\mathbb{R}\times\mathbb{R}$ in this case) is the Lexicographic ordering which would put $1+1j\le 1+1000j$ !

[6] Jump up ↑ The other mistake books make is saying explicitly that the field of a vector space needs to be $\mathbb{R}$ , it may commonly be $\mathbb{R}$ but it does not need to be $\mathbb{R}$

[APIKM-1] {Jump up to: 1.0} ^1.1 ^1.2 Analysis - Part 1: Elements - Krzysztof Maurin

[FA-2] {Jump up to: 2.0} ^2.1 Functional Analysis - George Bachman and Lawrence Narici

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

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

[1]

[2]

[3]

[4]

[Note 1]

[Note 2]

@@ Line 1: / Line 1: @@
-An understanding of a norm is needed to proceed to [[Linear isometry|linear isometries]].
+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]].
-A ''norm'' is a special case of [[Metric space|metrics]]. See [[Subtypes of topological spaces]] for more information
 __TOC__
 ==Definition==
-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<ref name="KMAPI">Analysis - Part 1: Elements - Krzysztof Maurin</ref><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} }}</ref>}}</sup>:
+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 16: / Line 14: @@
 ==Terminology==
-Such a vector space equipped with such a function is called a [[Normed space|normed space]]<ref name="KMAPI"/>
+Such a vector space equipped with such a function is called a [[Normed space|normed space]]<ref name="APIKM"/>
-==Relation to [[inner product]]==
+==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}} - I cannot think of any ''complex'' norms!}}
-==Induced metric==
+===Metric induced by a norm===
-To get a [[Metric space|metric space]] from a norm simply define<ref name="FA"/><ref name="KMAPI"/>:
+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]])
@@ Line 101: / Line 104: @@
 <references/>
-{{Definition|Linear Algebra}}
+{{Definition|Linear Algebra|Topology|Metric Space|Functional Analysis}}

Difference between revisions of "Norm"

Revision as of 10:45, 8 January 2016

Contents

Definition

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

Navigation menu

Views

Personal tools

Navigation

Search

Tools