Difference between revisions of "Norm"

Revision as of 17:52, 21 April 2015 (view source) Alec (Talk | contribs) m ← Older edit		Latest revision as of 20:33, 9 April 2017 (view source) Alec (Talk | contribs) m (Adding uniform continuity property)
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−	An understanding of a norm is needed to proceed to [[Linear isometry|linear isometries]]	+	{{:Norm/Heading}}
−		+
−	==Normed vector spaces==	+
−	A normed vector space is a [[Vector space|vector space]] equipped with a norm <math>\|\cdot\|_V</math>, it may be denoted <math>(V,\|\cdot\|_V,F)</math>	+
		+	__TOC__
	==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.}}
−	==Norms may define a metric space==	+	===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>
−	'''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.	+	(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==		==Weaker and stronger 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:		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>	+	<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>
−	We may write this as <math>\|\cdot\|_1\sim\|\cdot\|_2</math> - this is an [[Equivalence relation]]	+	{{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}}		{{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		Note also that if <math>\|\cdot\|_1</math> is both weaker and stronger than <math>\|\cdot\|_2</math> they are equivalent
	===Examples===		===Examples===
	*Any two norms on <math>\mathbb{R}^n</math> are equivalent		*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.		*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 (see below) - it's just a special case of the p-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 <math>\infty-</math>norm<br/>	+	|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>
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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
$\Vert\cdot\Vert:V\rightarrow\mathbb{R}_{\ge 0}$ Where $V$ is a vector space over the field $\mathbb{R}$ or $\mathbb{C}$
relation to other topological spaces
is a	metric space topological space
contains all	inner product spaces
Related objects
Induced metric	$d_{\Vert\cdot\Vert}:V\times V\rightarrow\mathbb{R}_{\ge 0}$ $d_{\Vert\cdot\Vert}:(x,y)\mapsto\Vert x-y\Vert$
Induced by inner product	$\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:V\rightarrow\mathbb{R}_{\ge 0}$ $\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:x\mapsto\sqrt{\langle x,x\rangle}$

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]}:

1. $$\forall x\in V\ \|x\|\ge 0$$
2. $$\|x\|=0\iff x=0$$
3. $$\forall \lambda\in F, x\in V\ \|\lambda x\|=|\lambda|\|x\|$$ where $$|\cdot|$$ denotes absolute value
4. $$\forall x,y\in V\ \|x+y\|\le\|x\|+\|y\|$$ - 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)

Properties

• The norm of a space is a uniformly continuous map with respect to the topology it induces - $\Vert\cdot\Vert:X\rightarrow\mathbb{R}$ is a uniformly continuous map.

Terminology

Such a vector space equipped with such a function is called a normed space^[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 $\langle x,x\rangle\ge 0$, 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]:

• $$d(x,y):=\|x-y\|$$

(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 $$\|\cdot\|_1$$ 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:

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

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

1. 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 $1+1j\le 1+1000j$!
2. 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

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

[Expand] v • d • e Normed and Banach spaces Overview of the basic and important concepts of normed vector spaces, which if complete are called Banach spaces Primitives Norm, Vector space, Field, Completeness Spaces Normed space, Banach space (if the metric induced by the norm is complete) Supertypes Metric space, Complete metric space, Topological space Subtypes Inner product space (all IPSes induce a norm), Hilbert space (if the metric induced by inner product is complete) Easy examples Euclidean norm Examples Analysis: Functional Analysis: Measure Theory: $\Vert\cdot\Vert_\infty$ examples (see: Category:Supremum norms) Supremum norm on $\mathbb{R}^n$ or $\mathbb{C}^n$

[Expand] v • d • e Metric spaces Overview of the basic and important concepts of metric spaces

[Expand] v • d • e Topology Overview of the structures studied in topology Primitives topological space, open set, neighbourhood Global properties of topological spaces Compactness, Connectedness (Path-Connectedness), Hausdorff-ness Local properties of topological spaces Constructs product (see also box, difference between), subspace (AKA: induced), quotient, disjoint union, adjunction, cone over a topological space Subtypes of topological spaces inner product spaces $\subset$ normed spaces $\subset$ metric spaces $\subset$ topological spaces Maps between topological spaces Continuous map (AKA: topological homomorphism), topological homeomorphism