Difference between revisions of "Norm"
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==Norms may define a metric space== | ==Norms may define a metric space== | ||
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 <math>d(x,y)=\|x-y\|</math> | ||
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| + | '''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. | ||
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| + | ==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 | ||
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| + | ==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: | ||
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| + | <math>\exists c,C\in\mathbb{R}\ \forall x\in V:\ c\|x\|_1\le\|x\|_2\le C\|x\|_1</math> | ||
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| + | We may write this as <math>\|\cdot\|_1\sim\|\cdot\|_2</math> - this is an [[Equivalence relation]] | ||
| + | {{Todo|proof}} | ||
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| + | 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. | ||
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==Common norms== | ==Common norms== | ||
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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}} | {{Definition|Linear Algebra}} | ||
Revision as of 17:52, 21 April 2015
An understanding of a norm is needed to proceed to linear isometries
Contents
[hide]Normed vector spaces
A normed vector space is a vector space equipped with a norm ∥⋅∥V, it may be denoted (V,∥⋅∥V,F)
Definition
A norm on a vector space (V,F) is a function ∥⋅∥:V→R such that:
- ∀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
Norms may define a metric space
To get a metric space from a norm simply define d(x,y)=∥x−y∥
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.
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 ∀x∈V: c∥x∥1≤∥x∥2≤C∥x∥1
We may write this as ∥⋅∥1∼∥⋅∥2 - this is an Equivalence relation
TODO: proof
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 (see below) - 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 \infty-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 |
