Difference between revisions of "Norm"
m |
m (Made common norms into table and expanded it) |
||
| Line 11: | Line 11: | ||
==Common norms== | ==Common norms== | ||
| − | === | + | {| class="wikitable" border="1" |
| − | <math>\|x\|_1=\sum^n_{i=1}|x_i|</math> | + | |- |
| − | + | ! Name | |
| − | <math>\|x\|_2=\sqrt{\sum^n_{i=1}x_i^2}</math> | + | ! Norm |
| − | + | ! Notes | |
| − | <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>) | + | |- |
| − | + | !colspan="3"|Norms on <math>\mathbb{R}^n</math> | |
| − | Also called <math>\infty-</math>norm<br/> | + | |- |
| − | <math>\|x\|_\infty=\ | + | | 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<br/> | ||
| + | |- | ||
| + | !colspan="3"|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== | ==Equivalence of norms== | ||
Revision as of 03:17, 8 March 2015
Contents
Definition
A norm on a vector space [ilmath](V,F)[/ilmath] is a function [math]\|\cdot\|:V\rightarrow\mathbb{R}[/math] such that:
- [math]\forall x\in V\ \|x\|\ge 0[/math]
- [math]\|x\|=0\iff x=0[/math]
- [math]\forall \lambda\in F, x\in V\ \|\lambda x\|=|\lambda|\|x\|[/math] where [math]|\cdot|[/math] denotes absolute value
- [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.
