Difference between revisions of "Norm"

Revision as of 03:51, 8 March 2015

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

A normed vector space is a vector space equipped with a norm $\|\cdot\|_V$ , it may be denoted $(V,\|\cdot\|_V,F)$

A norm on a vector space $(V,F)$ is a function $\|\cdot\|:V\rightarrow\mathbb{R}$ such that:

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

Often parts 1 and 2 are combined into the statement

I don't like this

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 (see below) - 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 $\infty-$ norm
Norms on $\mathcal{C}([0,1],\mathbb{R})$
$\\|\cdot\\|_{L^p}$	$\\|f\\|_{L^p}=\left(\int^1_0\|f(x)\|^pdx\right)$	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

Given two norms and on a vector space $V$ we say they are equivalent if:

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

We may write this as $\|\cdot\|_1\sim\|\cdot\|_2$ - this is an Equivalence relation

TODO: proof

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.

@@ Line 53: / Line 53: @@
 | <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.
+|-
+!colspan="3"|Induced norms
+|-
+| [[Pullback norm]]
+|<math>\|\cdot\|_U</math>
+|For a [[Linear map|linear isomorphism]] <math>L:U\rightarrow V</math> where V is a normed vector space
 |}