Difference between revisions of "Norm"
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(Created page with "==Definition== A norm on a vector space {{M|(V,F)}} is a function <math>\|\cdot\|:V\rightarrow\mathbb{R}</math> such that: # <math>\forall x\in V\ \|x\|\ge 0<...") |
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# <math>\|x\|=0\iff x=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|absolute value]] | # <math>\forall \lambda\in F, x\in V\ \|\lambda x\|=|\lambda|\|x\|</math> where <math>|\cdot|</math> denotes [[Absolute value|absolute value]] | ||
| + | # <math>\forall x,y\in V\ \|x+y\|\le\|x\|+\|y\|</math> - a form of the [[Triangle inequality|triangle inequality]] | ||
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{{Definition|Linear Algebra}} | {{Definition|Linear Algebra}} | ||
Revision as of 16:13, 7 March 2015
Definition
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
