Difference between revisions of "Norm"
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==Definition== | ==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=" | + | 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>: |
# <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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I don't like this | I don't like this | ||
==Terminology== | ==Terminology== | ||
| − | Such a vector space equipped with such a function is called a [[Normed space|normed space]] | + | Such a vector space equipped with such a function is called a [[Normed space|normed space]]<ref name="KMAPI"/> |
==Relation to [[inner product]]== | ==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: | Every [[inner product]] {{M|\langle\cdot,\cdot\rangle:V\times V\rightarrow(\mathbb{R}\text{ or }\mathbb{C})}} induces a ''norm'' given by: | ||
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{{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!}} | {{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== | ==Induced metric== | ||
| − | To get a [[Metric space|metric space]] from a norm simply define: | + | To get a [[Metric space|metric space]] from a norm simply define<ref name="FA"/><ref name="KMAPI"/>: |
| − | * <math>d(x,y):=\|x-y\|</math | + | * <math>d(x,y):=\|x-y\|</math> |
(See [[Subtypes of topological spaces]] for more information, this relationship is very important in [[Functional analysis]]) | (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}} | {{Todo|Some sort of proof this is ''never'' complex}} | ||
Revision as of 16:25, 24 November 2015
An understanding of a norm is needed to proceed to linear isometries.
A norm is a special case of metrics. See Subtypes of topological spaces for more information
Contents
[hide]Definition
A norm on a vector space (V,F) (where F is either R or C) is a function ∥⋅∥:V→R such that[1][2]:
- ∀x∈V ∥x∥≥0
- ∥x∥=0⟺x=0
- ∀λ∈F,x∈V ∥λx∥=|λ|∥x∥ where |⋅| denotes absolute value
- ∀x,y∈V ∥x+y∥≤∥x∥+∥y∥ - a form of the triangle inequality
Often parts 1 and 2 are combined into the statement
- ∥x∥≥0 and ∥x∥=0⟺x=0 so only 3 requirements will be stated.
I don't like this
Terminology
Such a vector space equipped with such a function is called a normed space[1]
Relation to inner product
Every inner product ⟨⋅,⋅⟩:V×V→(R or C) induces a norm given by:
- ∥x∥:=√⟨x,x⟩
TODO: see inner product (norm induced by) for more details, on that page is a proof that ⟨x,x⟩≥0 - I cannot think of any complex norms!
Induced metric
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 ∥⋅∥1 and another ∥⋅∥2 we say:
- ∥⋅∥1 is weaker than ∥⋅∥2 if ∃C>0∀x∈V such that ∥x∥1≤C∥x∥2
- ∥⋅∥2 is stronger than ∥⋅∥1 in this case
Equivalence of norms
Given two norms ∥⋅∥1 and ∥⋅∥2 on a vector space V we say they are equivalent if:
∃c,C∈R with c,C>0 ∀x∈V: c∥x∥1≤∥x∥2≤C∥x∥1
Theorem: This is an Equivalence relation - so we may write this as ∥⋅∥1∼∥⋅∥2
Note also that if ∥⋅∥1 is both weaker and stronger than ∥⋅∥2 they are equivalent
Examples
- Any two norms on Rn are equivalent
- The norms ∥⋅∥L1 and ∥⋅∥∞ on C([0,1],R) are not equivalent.
Common norms
| Name | Norm | Notes |
|---|---|---|
| Norms on Rn | ||
| 1-norm | ∥x∥1=n∑i=1|xi| | it's just a special case of the p-norm. |
| 2-norm | ∥x∥2=√n∑i=1x2i | Also known as the Euclidean norm - it's just a special case of the p-norm. |
| p-norm | ∥x∥p=(n∑i=1|xi|p)1p | (I use this notation because it can be easy to forget the p in p√) |
| ∞−norm | ∥x∥∞=sup | 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
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
