Difference between revisions of "Norm"

Revision as of 16:18, 7 March 2015

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

[math]\forall x\in V\ \|x\|\ge 0[/math]
[math]\|x\|=0\iff x=0[/math]
[math]\forall \lambda\in F, x\in V\ \|\lambda x\|=|\lambda|\|x\|[/math] where [math]|\cdot|[/math] denotes absolute value
[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

The Euclidean norm is denoted [math]\|\cdot\|_2[/math]

Here for [math]x\in\mathbb{R}^n[/math] we have:

[math]\|x\|_2=\sqrt{\sum^n_{i=1}x_i^2}[/math]

TODO: proof

@@ Line 6: / Line 6: @@
 # <math>\forall x,y\in V\ \|x+y\|\le\|x\|+\|y\|</math> - a form of the [[Triangle inequality|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
+==Examples==
+===The Euclidean Norm===
+The Euclidean norm is denoted <math>\|\cdot\|_2</math>
+Here for <math>x\in\mathbb{R}^n</math> we have:
+<math>\|x\|_2=\sqrt{\sum^n_{i=1}x_i^2}</math>
+{{Todo|proof}}
 {{Definition|Linear Algebra}}