Norm

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An understanding of a norm is needed to proceed to linear isometries

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