Difference between revisions of "Norm"

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m (Made common norms into table and expanded it)
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==Common norms==
 
==Common norms==
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===The 1-norm===
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{| class="wikitable" border="1"
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<math>\|x\|_1=\sum^n_{i=1}|x_i|</math> - it's just a special case of the p-norm.
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|-
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===The 2-norm===
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! Name
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<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.
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! Norm
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===The p-norm===
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! Notes
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<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>)
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|-
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===The supremum-norm===
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!colspan="3"|Norms on <math>\mathbb{R}^n</math>
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Also called <math>\infty-</math>norm<br/>
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|-
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<math>\|x\|_\infty=\sup(\{x_i\}_{i=1}^n)</math>
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| 1-norm
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|<math>\|x\|_1=\sum^n_{i=1}|x_i|</math>
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|it's just a special case of the p-norm.
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|-
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| 2-norm
 +
|<math>\|x\|_2=\sqrt{\sum^n_{i=1}x_i^2}</math>
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| Also known as the Euclidean norm (see below) - it's just a special case of the p-norm.
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|-
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| p-norm
 +
|<math>\|x\|_p=\left(\sum^n_{i=1}|x_i|^p\right)^\frac{1}{p}</math>
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|(I use this notation because it can be easy to forget the <math>p</math> in <math>\sqrt[p]{}</math>)
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|-
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| <math>\infty-</math>norm
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|<math>\|x\|_\infty=\sup(\{x_i\}_{i=1}^n)</math>
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|Also called <math>\infty-</math>norm<br/>
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|-
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!colspan="3"|Norms on <math>\mathcal{C}([0,1],\mathbb{R})</math>
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|-
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| <math>\|\cdot\|_{L^p}</math>
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| <math>\|f\|_{L^p}=\left(\int^1_0|f(x)|^pdx\right)</math>
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| '''NOTE''' be careful extending to interval <math>[a,b]</math> as proof it is a norm relies on having a unit measure
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|-
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| <math>\infty-</math>norm
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| <math>\|f\|_\infty=\sup_{x\in[0,1]}(|f(x)|)</math>
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| Following the same spirit as the <math>\infty-</math>norm on <math>\mathbb{R}^n</math>
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|-
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| <math>\|\cdot\|_{C^k}</math>
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| <math>\|f\|_{C^k}=\sum^k_{i=1}\sup_{x\in[0,1]}(|f^{(i)}|)</math>
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| here <math>f^{(k)}</math> denotes the <math>k^\text{th}</math> derivative.
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|}
  
 
==Equivalence of norms==
 
==Equivalence of norms==

Revision as of 03:17, 8 March 2015

Definition

A norm on a vector space [ilmath](V,F)[/ilmath] is a function [math]\|\cdot\|:V\rightarrow\mathbb{R}[/math] such that:

  1. [math]\forall x\in V\ \|x\|\ge 0[/math]
  2. [math]\|x\|=0\iff x=0[/math]
  3. [math]\forall \lambda\in F, x\in V\ \|\lambda x\|=|\lambda|\|x\|[/math] where [math]|\cdot|[/math] denotes absolute value
  4. [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

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 (see below) - 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 [math]\infty-[/math]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)[/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.

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}\ \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

TODO: proof

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.

Examples

The Euclidean Norm

TODO: Migrate this norm to its own page

The Euclidean norm is denoted [math]\|\cdot\|_2[/math]


Here for [math]x\in\mathbb{R}^n[/math] we have:

[math]\|x\|_2=\sqrt{\sum^n_{i=1}x_i^2}[/math]

Proof that it is a norm

TODO: proof

Part 4 - Triangle inequality

Let [math]x,y\in\mathbb{R}^n[/math]

[math]\|x+y\|_2^2=\sum^n_{i=1}(x_i+y_i)^2[/math] [math]=\sum^n_{i=1}x_i^2+2\sum^n_{i=1}x_iy_i+\sum^n_{i=1}y_i^2[/math] [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

[math]=\left(\sqrt{\sum^n_{i=1}x_i^2}+\sqrt{\sum^n_{i=1}y_i^2}\right)^2[/math] [math]=\left(\|x\|_2+\|y\|_2\right)^2[/math]

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.

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