Difference between revisions of "Norm"

Revision as of 10:31, 8 January 2016

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 $(V,F)$ (where $F$ is either $\mathbb{R}$ or $\mathbb{C}$) 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 inner product

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

• $\Vert x\Vert:=\sqrt{\langle x,x\rangle}$

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

Induced metric

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: Some sort of proof this is never complex

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 $V$ we say they are equivalent if:

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

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

Examples

• Euclidean norm

Notes

1. Jump up ↑ A lot of books, including the brilliant Analysis - Part 1: Elements - Krzysztof Maurin referenced here state explicitly that it is possible for $\Vert\cdot,\cdot\Vert:V\rightarrow\mathbb{C}$ they are wrong. I assure you that it is $\Vert\cdot\Vert:V\rightarrow\mathbb{R}_{\ge 0}$. Other than this the references are valid, note that this is 'obvious' as if the image of $\Vert\cdot\Vert$ could be in $\mathbb{C}$ then the $\Vert x\Vert\ge 0$ would make no sense. What ordering would you use? The canonical ordering used for the product of 2 spaces ($\mathbb{R}\times\mathbb{R}$ in this case) is the Lexicographic ordering which would put $1+1j\le 1+1000j$!
2. Jump up ↑ The other mistake books make is saying explicitly that the field of a vector space needs to be $\mathbb{R}$

References

1. ↑ ^{Jump up to: 1.0 1.1 1.2} Analysis - Part 1: Elements - Krzysztof Maurin
2. ↑ ^{Jump up to: 2.0 2.1} Functional Analysis - George Bachman and Lawrence Narici
3. Jump up ↑ Functional Analysis - A Gentle Introduction - Volume 1, by Dzung Minh Ha
4. Jump up ↑ Real and Abstract Analysis - Edwin Hewitt & Karl Stromberg