Norm
An understanding of a norm is needed to proceed to linear isometries
Contents
Normed vector spaces
A normed vector space is a vector space equipped with a norm [math]\|\cdot\|_V[/math], it may be denoted [math](V,\|\cdot\|_V,F)[/math]
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
Norms may define a metric space
To get a metric space from a norm simply define [math]d(x,y)=\|x-y\|[/math]
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. |
| Induced norms | ||
| Pullback norm | [math]\|\cdot\|_U[/math] | For a linear isomorphism [math]L:U\rightarrow V[/math] where V is a normed vector space |
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.
