Norm

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

A norm is a special case of metrics. See Subtypes of topological spaces for more information

Definition

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

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

Terminology

Such a vector space equipped with such a function is called a normed space

Relation to inner product

Every inner product [ilmath]\langle\cdot,\cdot\rangle:V\times V\rightarrow(\mathbb{R}\text{ or }\mathbb{C})[/ilmath] induces a norm given by:

• [ilmath]\Vert x\Vert:=\sqrt{\langle x,x\rangle}[/ilmath]

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TODO: see inner product (norm induced by) for more details, on that page is a proof that [ilmath]\langle x,x\rangle\ge 0[/ilmath] - I cannot think of any complex norms!

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Induced metric

To get a metric space from a norm simply define:

• [math]d(x,y):=\|x-y\|[/math]

(See Subtypes of topological spaces for more information, this relationship is very important in Functional analysis)

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TODO: Some sort of proof this is never complex

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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

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}\text{ with }c,C>0\ \forall x\in V:\ c\|x\|_1\le\|x\|_2\le C\|x\|_1[/math]

Theorem: This is an Equivalence relation - so we may write this as [math]\|\cdot\|_1\sim\|\cdot\|_2[/math]

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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.

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 - 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 sup-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)^\frac{1}{p}[/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

Examples

• Euclidean norm