# Norm

relation to other topological spaces Norm [ilmath]\Vert\cdot\Vert:V\rightarrow\mathbb{R}_{\ge 0} [/ilmath]Where [ilmath]V[/ilmath] is a vector space over the field [ilmath]\mathbb{R} [/ilmath] or [ilmath]\mathbb{C} [/ilmath] inner product spaces [ilmath]d_{\Vert\cdot\Vert}:V\times V\rightarrow\mathbb{R}_{\ge 0}[/ilmath] [ilmath]d_{\Vert\cdot\Vert}:(x,y)\mapsto\Vert x-y\Vert[/ilmath] [ilmath]\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:V\rightarrow\mathbb{R}_{\ge 0}[/ilmath] [ilmath]\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:x\mapsto\sqrt{\langle x,x\rangle}[/ilmath]
A norm is a an abstraction of the notion of the "length of a vector". Every norm is a metric and every inner product is a norm (see Subtypes of topological spaces for more information), thus every normed vector space is a topological space to, so all the topology theorems apply. Norms are especially useful in functional analysis and also for differentiation.

## 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 $\|\cdot\|:V\rightarrow\mathbb{R}$ such that[1][2][3][4]See warning notes:[Note 1][Note 2]:

1. $\forall x\in V\ \|x\|\ge 0$
2. $\|x\|=0\iff x=0$
3. $\forall \lambda\in F, x\in V\ \|\lambda x\|=|\lambda|\|x\|$ where $|\cdot|$ denotes absolute value
4. $\forall x,y\in V\ \|x+y\|\le\|x\|+\|y\|$ - a form of the triangle inequality

Often parts 1 and 2 are combined into the statement:

• $\|x\|\ge 0\text{ and }\|x\|=0\iff x=0$ so only 3 requirements will be stated.

I don't like this (inline with the Doctrine of monotonic definition)

## Terminology

Such a vector space equipped with such a function is called a normed space[1]

## Relation to various subtypes of topological spaces

These are outlined below

### 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]

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], this needs its own page with a proof.

### Metric induced by a norm

To get a metric space from a norm simply define[2][1]:

• $d(x,y):=\|x-y\|$

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

TODO: Move to its own page and do a proof (trivial)

## Weaker and stronger norms

Given a norm $\|\cdot\|_1$ and another $\|\cdot\|_2$ we say:

• $\|\cdot\|_1$ is weaker than $\|\cdot\|_2$ if $\exists C> 0\forall x\in V$ such that $\|x\|_1\le C\|x\|_2$
• $\|\cdot\|_2$ is stronger than $\|\cdot\|_1$ in this case

## Equivalence of norms

Given two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on a vector space [ilmath]V[/ilmath] we say they are equivalent if:

$\exists c,C\in\mathbb{R}\text{ with }c,C>0\ \forall x\in V:\ c\|x\|_1\le\|x\|_2\le C\|x\|_1$

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

TODO: proof

Note also that if $\|\cdot\|_1$ is both weaker and stronger than $\|\cdot\|_2$ they are equivalent

### Examples

• Any two norms on $\mathbb{R}^n$ are equivalent
• The norms $\|\cdot\|_{L^1}$ and $\|\cdot\|_\infty$ on $\mathcal{C}([0,1],\mathbb{R})$ are not equivalent.

## Common norms

Name Norm Notes
Norms on $\mathbb{R}^n$
1-norm $\|x\|_1=\sum^n_{i=1}|x_i|$ it's just a special case of the p-norm.
2-norm $\|x\|_2=\sqrt{\sum^n_{i=1}x_i^2}$ Also known as the Euclidean norm - it's just a special case of the p-norm.
p-norm $\|x\|_p=\left(\sum^n_{i=1}|x_i|^p\right)^\frac{1}{p}$ (I use this notation because it can be easy to forget the $p$ in $\sqrt[p]{}$)
$\infty-$norm $\|x\|_\infty=\sup(\{x_i\}_{i=1}^n)$ Also called sup-norm
Norms on $\mathcal{C}([0,1],\mathbb{R})$
$\|\cdot\|_{L^p}$ $\|f\|_{L^p}=\left(\int^1_0|f(x)|^pdx\right)^\frac{1}{p}$ NOTE be careful extending to interval $[a,b]$ as proof it is a norm relies on having a unit measure
$\infty-$norm $\|f\|_\infty=\sup_{x\in[0,1]}(|f(x)|)$ Following the same spirit as the $\infty-$norm on $\mathbb{R}^n$
$\|\cdot\|_{C^k}$ $\|f\|_{C^k}=\sum^k_{i=1}\sup_{x\in[0,1]}(|f^{(i)}|)$ here $f^{(k)}$ denotes the $k^\text{th}$ derivative.
Induced norms
Pullback norm $\|\cdot\|_U$ For a linear isomorphism $L:U\rightarrow V$ where V is a normed vector space

## Notes

1. A lot of books, including the brilliant Analysis - Part 1: Elements - Krzysztof Maurin referenced here state explicitly that it is possible for [ilmath]\Vert\cdot,\cdot\Vert:V\rightarrow\mathbb{C} [/ilmath] they are wrong. I assure you that it is [ilmath]\Vert\cdot\Vert:V\rightarrow\mathbb{R}_{\ge 0} [/ilmath]. Other than this the references are valid, note that this is 'obvious' as if the image of [ilmath]\Vert\cdot\Vert[/ilmath] could be in [ilmath]\mathbb{C} [/ilmath] then the [ilmath]\Vert x\Vert\ge 0[/ilmath] would make no sense. What ordering would you use? The canonical ordering used for the product of 2 spaces ([ilmath]\mathbb{R}\times\mathbb{R} [/ilmath] in this case) is the Lexicographic ordering which would put [ilmath]1+1j\le 1+1000j[/ilmath]!
2. The other mistake books make is saying explicitly that the field of a vector space needs to be [ilmath]\mathbb{R} [/ilmath], it may commonly be [ilmath]\mathbb{R} [/ilmath] but it does not need to be [ilmath]\mathbb{R} [/ilmath]

## References

1. Analysis - Part 1: Elements - Krzysztof Maurin
2. Functional Analysis - George Bachman and Lawrence Narici
3. Functional Analysis - A Gentle Introduction - Volume 1, by Dzung Minh Ha
4. Real and Abstract Analysis - Edwin Hewitt & Karl Stromberg